Arithmetical Properties of Commutative Rings and Monoids

Arithmetical Properties of Commutative Rings and Monoids

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, New Jersey

Zuhair Nashed University of Central Florida Orlando, Florida

EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology

Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi University of California, Berkeley

David L. Russell Virginia Polytechnic Institute and State University

Marvin Marcus University of California, Santa Barbara

Walter Schempp Universität Siegen

W. S. Massey Yale University

Mark Teply University of Wisconsin, Milwaukee

LECTURE NOTES IN PURE AND APPLIED MATHEMATICS

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.

N. Jacobson, Exceptional Lie Algebras L.-Å. Lindahl and F. Poulsen, Thin Sets in Harmonic Analysis I. Satake, Classification Theory of Semi-Simple Algebraic Groups F. Hirzebruch et al., Differentiable Manifolds and Quadratic Forms I. Chavel, Riemannian Symmetric Spaces of Rank One R. B. Burckel, Characterization of C(X) Among Its Subalgebras B. R. McDonald et al., Ring Theory Y.-T. Siu, Techniques of Extension on Analytic Objects S. R. Caradus et al., Calkin Algebras and Algebras of Operators on Banach Spaces E. O. Roxin et al., Differential Games and Control Theory M. Orzech and C. Small, The Brauer Group of Commutative Rings S. Thomier, Topology and Its Applications J. M. Lopez and K. A. Ross, Sidon Sets W. W. Comfort and S. Negrepontis, Continuous Pseudometrics K. McKennon and J. M. Robertson, Locally Convex Spaces M. Carmeli and S. Malin, Representations of the Rotation and Lorentz Groups G. B. Seligman, Rational Methods in Lie Algebras D. G. de Figueiredo, Functional Analysis L. Cesari et al., Nonlinear Functional Analysis and Differential Equations J. J. Schäffer, Geometry of Spheres in Normed Spaces K. Yano and M. Kon, Anti-Invariant Submanifolds W. V. Vasconcelos, The Rings of Dimension Two R. E. Chandler, Hausdorff Compactifications S. P. Franklin and B. V. S. Thomas, Topology S. K. Jain, Ring Theory B. R. McDonald and R. A. Morris, Ring Theory II R. B. Mura and A. Rhemtulla, Orderable Groups J. R. Graef, Stability of Dynamical Systems H.-C. Wang, Homogeneous Branch Algebras E. O. Roxin et al., Differential Games and Control Theory II R. D. Porter, Introduction to Fibre Bundles M. Altman, Contractors and Contractor Directions Theory and Applications J. S. Golan, Decomposition and Dimension in Module Categories G. Fairweather, Finite Element Galerkin Methods for Differential Equations J. D. Sally, Numbers of Generators of Ideals in Local Rings S. S. Miller, Complex Analysis R. Gordon, Representation Theory of Algebras M. Goto and F. D. Grosshans, Semisimple Lie Algebras A. I. Arruda et al., Mathematical Logic F. Van Oystaeyen, Ring Theory F. Van Oystaeyen and A. Verschoren, Reflectors and Localization M. Satyanarayana, Positively Ordered Semigroups D. L Russell, Mathematics of Finite-Dimensional Control Systems P.-T. Liu and E. Roxin, Differential Games and Control Theory III A. Geramita and J. Seberry, Orthogonal Designs

46. J. Cigler, V. Losert, and P. Michor, Banach Modules and Functors on Categories of Banach Spaces 47. P.-T. Liu and J. G. Sutinen, Control Theory in Mathematical Economics 48. C. Byrnes, Partial Differential Equations and Geometry 49. G. Klambauer, Problems and Propositions in Analysis 50. J. Knopfmacher, Analytic Arithmetic of Algebraic Function Fields 51. F. Van Oystaeyen, Ring Theory 52. B. Kadem, Binary Time Series 53. J. Barros-Neto and R. A. Artino, Hypoelliptic Boundary-Value Problems 54. R. L. Sternberg et al., Nonlinear Partial Differential Equations in Engineering and Applied Science 55. B. R. McDonald, Ring Theory and Algebra III 56. J. S. Golan, Structure Sheaves Over a Noncommutative Ring 57. T. V. Narayana et al., Combinatorics, Representation Theory and Statistical Methods in Groups 58. T. A. Burton, Modeling and Differential Equations in Biology 59. K. H. Kim and F. W. Roush, Introduction to Mathematical Consensus Theory 60. J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces 61. O. A. Nielson, Direct Integral Theory 62. J. E. Smith et al., Ordered Groups 63. J. Cronin, Mathematics of Cell Electrophysiology 64. J. W. Brewer, Power Series Over Commutative Rings 65. P. K. Kamthan and M. Gupta, Sequence Spaces and Series 66. T. G. McLaughlin, Regressive Sets and the Theory of Isols 67. T. L. Herdman et al., Integral and Functional Differential Equations 68. R. Draper, Commutative Algebra 69. W. G. McKay and J. Patera, Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras 70. R. L. Devaney and Z. H. Nitecki, Classical Mechanics and Dynamical Systems 71. J. Van Geel, Places and Valuations in Noncommutative Ring Theory 72. C. Faith, Injective Modules and Injective Quotient Rings 73. A. Fiacco, Mathematical Programming with Data Perturbations I 74. P. Schultz et al., Algebraic Structures and Applications 75. L Bican et al., Rings, Modules, and Preradicals 76. D. C. Kay and M. Breen, Convexity and Related Combinatorial Geometry 77. P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces 78. C.-C. Yang, Factorization Theory of Meromorphic Functions 79. O. Taussky, Ternary Quadratic Forms and Norms 80. S. P. Singh and J. H. Burry, Nonlinear Analysis and Applications 81. K. B. Hannsgen et al., Volterra and Functional Differential Equations 82. N. L. Johnson et al., Finite Geometries 83. G. I. Zapata, Functional Analysis, Holomorphy, and Approximation Theory 84. S. Greco and G. Valla, Commutative Algebra 85. A. V. Fiacco, Mathematical Programming with Data Perturbations II 86. J.-B. Hiriart-Urruty et al., Optimization 87. A. Figa Talamanca and M. A. Picardello, Harmonic Analysis on Free Groups 88. M. Harada, Factor Categories with Applications to Direct Decomposition of Modules 89. V. I. Istra’tescu, Strict Convexity and Complex Strict Convexity 90. V. Lakshmikantham, Trends in Theory and Practice of Nonlinear Differential Equations

91. H. L. Manocha and J. B. Srivastava, Algebra and Its Applications 92. D. V. Chudnovsky and G. V. Chudnovsky, Classical and Quantum Models and Arithmetic Problems 93. J. W. Longley, Least Squares Computations Using Orthogonalization Methods 94. L. P. de Alcantara, Mathematical Logic and Formal Systems 95. C. E. Aull, Rings of Continuous Functions 96. R. Chuaqui, Analysis, Geometry, and Probability 97. L. Fuchs and L. Salce, Modules Over Valuation Domains 98. P. Fischer and W. R. Smith, Chaos, Fractals, and Dynamics 99. W. B. Powell and C. Tsinakis, Ordered Algebraic Structures 100. G. M. Rassias and T. M. Rassias, Differential Geometry, Calculus of Variations, and Their Applications 101. R.-E. Hoffmann and K. H. Hofmann, Continuous Lattices and Their Applications 102. J. H. Lightbourne III and S. M. Rankin III, Physical Mathematics and Nonlinear Partial Differential Equations 103. C. A. Baker and L. M. Batten, Finite Geometrics 104. J. W. Brewer et al., Linear Systems Over Commutative Rings 105. C. McCrory and T. Shifrin, Geometry and Topology 106. D. W. Kueke et al., Mathematical Logic and Theoretical Computer Science 107. B.-L. Lin and S. Simons, Nonlinear and Convex Analysis 108. S. J. Lee, Operator Methods for Optimal Control Problems 109. V. Lakshmikantham, Nonlinear Analysis and Applications 110. S. F. McCormick, Multigrid Methods 111. M. C. Tangora, Computers in Algebra 112. D. V. Chudnovsky and G. V. Chudnovsky, Search Theory 113. D. V. Chudnovsky and R. D. Jenks, Computer Algebra 114. M. C. Tangora, Computers in Geometry and Topology 115. P. Nelson et al., Transport Theory, Invariant Imbedding, and Integral Equations 116. P. Clément et al., Semigroup Theory and Applications 117. J. Vinuesa, Orthogonal Polynomials and Their Applications 118. C. M. Dafermos et al., Differential Equations 119. E. O. Roxin, Modern Optimal Control 120. J. C. Díaz, Mathematics for Large Scale Computing 121. P. S. Milojevic, Nonlinear Functional Analysis 122. C. Sadosky, Analysis and Partial Differential Equations 123. R. M. Shortt, General Topology and Applications 124. R. Wong, Asymptotic and Computational Analysis 125. D. V. Chudnovsky and R. D. Jenks, Computers in Mathematics 126. W. D. Wallis et al., Combinatorial Designs and Applications 127. S. Elaydi, Differential Equations 128. G. Chen et al., Distributed Parameter Control Systems 129. W. N. Everitt, Inequalities 130. H. G. Kaper and M. Garbey, Asymptotic Analysis and the Numerical Solution of Partial Differential Equations 131. O. Arino et al., Mathematical Population Dynamics 132. S. Coen, Geometry and Complex Variables 133. J. A. Goldstein et al., Differential Equations with Applications in Biology, Physics, and Engineering 134. S. J. Andima et al., General Topology and Applications 135. P Clément et al., Semigroup Theory and Evolution Equations

136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178.

K. Jarosz, Function Spaces J. M. Bayod et al., p-adic Functional Analysis G. A. Anastassiou, Approximation Theory R. S. Rees, Graphs, Matrices, and Designs G. Abrams et al., Methods in Module Theory G. L. Mullen and P. J.-S. Shiue, Finite Fields, Coding Theory, and Advances in Communications and Computing M. C. Joshi and A. V. Balakrishnan, Mathematical Theory of Control G. Komatsu and Y. Sakane, Complex Geometry I. J. Bakelman, Geometric Analysis and Nonlinear Partial Differential Equations T. Mabuchi and S. Mukai, Einstein Metrics and Yang–Mills Connections L. Fuchs and R. Göbel, Abelian Groups A. D. Pollington and W. Moran, Number Theory with an Emphasis on the Markoff Spectrum G. Dore et al., Differential Equations in Banach Spaces T. West, Continuum Theory and Dynamical Systems K. D. Bierstedt et al., Functional Analysis K. G. Fischer et al., Computational Algebra K. D. Elworthy et al., Differential Equations, Dynamical Systems, and Control Science P.-J. Cahen, et al., Commutative Ring Theory S. C. Cooper and W. J. Thron, Continued Fractions and Orthogonal Functions P. Clément and G. Lumer, Evolution Equations, Control Theory, and Biomathematics M. Gyllenberg and L. Persson, Analysis, Algebra, and Computers in Mathematical Research W. O. Bray et al., Fourier Analysis J. Bergen and S. Montgomery, Advances in Hopf Algebras A. R. Magid, Rings, Extensions, and Cohomology N. H. Pavel, Optimal Control of Differential Equations M. Ikawa, Spectral and Scattering Theory X. Liu and D. Siegel, Comparison Methods and Stability Theory J.-P. Zolésio, Boundary Control and Variation M. Kr’íz’’ek et al., Finite Element Methods G. Da Prato and L. Tubaro, Control of Partial Differential Equations E. Ballico, Projective Geometry with Applications M. Costabel et al., Boundary Value Problems and Integral Equations in Nonsmooth Domains G. Ferreyra, G. R. Goldstein, and F. Neubrander, Evolution Equations S. Huggett, Twistor Theory H. Cook et al., Continua D. F. Anderson and D. E. Dobbs, Zero-Dimensional Commutative Rings K. Jarosz, Function Spaces V. Ancona et al., Complex Analysis and Geometry E. Casas, Control of Partial Differential Equations and Applications N. Kalton et al., Interaction Between Functional Analysis, Harmonic Analysis, and Probability Z. Deng et al., Differential Equations and Control Theory P. Marcellini et al. Partial Differential Equations and Applications A. Kartsatos, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type

179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222.

M. Maruyama, Moduli of Vector Bundles A. Ursini and P. Aglianò, Logic and Algebra X. H. Cao et al., Rings, Groups, and Algebras D. Arnold and R. M. Rangaswamy, Abelian Groups and Modules S. R. Chakravarthy and A. S. Alfa, Matrix-Analytic Methods in Stochastic Models J. E. Andersen et al., Geometry and Physics P.-J. Cahen et al., Commutative Ring Theory J. A. Goldstein et al., Stochastic Processes and Functional Analysis A. Sorbi, Complexity, Logic, and Recursion Theory G. Da Prato and J.-P. Zolésio, Partial Differential Equation Methods in Control and Shape Analysis D. D. Anderson, Factorization in Integral Domains N. L. Johnson, Mostly Finite Geometries D. Hinton and P. W. Schaefer, Spectral Theory and Computational Methods of Sturm–Liouville Problems W. H. Schikhof et al., p-adic Functional Analysis S. Sertöz, Algebraic Geometry G. Caristi and E. Mitidieri, Reaction Diffusion Systems A. V. Fiacco, Mathematical Programming with Data Perturbations M. Kr’íz’cek et al., Finite Element Methods: Superconvergence, Post-Processing, and A Posteriori Estimates S. Caenepeel and A. Verschoren, Rings, Hopf Algebras, and Brauer Groups V. Drensky et al., Methods in Ring Theory W. B. Jones and A. Sri Ranga, Orthogonal Functions, Moment Theory, and Continued Fractions P. E. Newstead, Algebraic Geometry D. Dikranjan and L. Salce, Abelian Groups, Module Theory, and Topology Z. Chen et al., Advances in Computational Mathematics X. Caicedo and C. H. Montenegro, Models, Algebras, and Proofs C. Y. Yédérém and S. A. Stepanov, Number Theory and Its Applications D. E. Dobbs et al., Advances in Commutative Ring Theory F. Van Oystaeyen, Commutative Algebra and Algebraic Geometry J. Kakol et al., p-adic Functional Analysis M. Boulagouaz and J.-P. Tignol, Algebra and Number Theory S. Caenepeel and F. Van Oystaeyen, Hopf Algebras and Quantum Groups F. Van Oystaeyen and M. Saorin, Interactions Between Ring Theory and Representations of Algebras R. Costa et al., Nonassociative Algebra and Its Applications T.-X. He, Wavelet Analysis and Multiresolution Methods H. Hudzik and L. Skrzypczak, Function Spaces: The Fifth Conference J. Kajiwara et al., Finite or Infinite Dimensional Complex Analysis G. Lumer and L. Weis, Evolution Equations and Their Applications in Physical and Life Sciences J. Cagnol et al., Shape Optimization and Optimal Design J. Herzog and G. Restuccia, Geometric and Combinatorial Aspects of Commutative Algebra G. Chen et al., Control of Nonlinear Distributed Parameter Systems F. Ali Mehmeti et al., Partial Differential Equations on Multistructures D. D. Anderson and I. J. Papick, Ideal Theoretic Methods in Commutative Algebra Á. Granja et al., Ring Theory and Algebraic Geometry A. K. Katsaras et al., p-adic Functional Analysis

223. 224. 225. 226. 227. 228. 229. 230. 231. 232. 233. 234. 235. 236. 237. 238. 239. 240. 241.

R. Salvi, The Navier-Stokes Equations F. U. Coelho and H. A. Merklen, Representations of Algebras S. Aizicovici and N. H. Pavel, Differential Equations and Control Theory G. Lyubeznik, Local Cohomology and Its Applications G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications W. A. Carnielli et al., Paraconsistency A. Benkirane and A. Touzani, Partial Differential Equations A. Illanes et al., Continuum Theory M. Fontana et al., Commutative Ring Theory and Applications D. Mond and M. J. Saia, Real and Complex Singularities V. Ancona and J. Vaillant, Hyperbolic Differential Operators and Related Problems G. R. Goldstein et al., Evolution Equations A. Giambruno et al., Polynomial Identities and Combinatorial Methods A. Facchini et al., Rings, Modules, Algebras, and Abelian Groups J. Bergen et al., Hopf Algebras A. C. Krinik and R. J. Swift, Stochastic Processes and Functional Analysis: A Volume of Recent Advances in Honor of M. M. Rao S. Caenepeel and F. van Oystaeyen, Hopf Algebras in Noncommutative Geometry and Physics J. Cagnol and J.-P. Zolésio, Control and Boundary Analysis S. T. Chapman, Arithmetical Properties of Commutative Rings and Monoids

Arithmetical Properties of Commutative Rings and Monoids

Scott T. Chapman Trinity University San Antonio, Texas, U.S.A.

Boca Raton London New York Singapore

Published in 2005 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW Boca Raton, FL 33487-2742 © 2005 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8247-2327-9 (Hardcover) International Standard Book Number-13: 978-0-8247-2327-9 (Hardcover) Library of Congress Card Number 2004061830 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Arithmetic properties of commutative rings and monoids / edited by Scott T. Chapman. p. cm. — (Lecture notes in pure and applied mathematics v. 240) Includes bibliographical references and index. ISBN 0-8247-2327-9 (alk. paper) 1. Commutative rings—Congresses. 2. Monoids—Congresses. I. Chapman, Scott T. II. Series. QA251.3.A72 2005 512'.44--dc22

2004061830

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Preface This volume contains the Proceedings of two related meetings which took place in October 2003. The first event was a one day Mini-Conference on Factorization Properties of Commutative Rings and Monoids hosted by the Department of Mathematics at the University of North Carolina at Chapel Hill. The Mini-Conference was followed by a regular Special Session on Commutative Rings and Monoids at the Fall 2003 Southeast Sectional Meeting of the American Mathematical Society. A major focus of these two meetings was the study of properties related to nonunique factorizations of elements into irreducible elements in commutative rings and monoids. Such problems have emerged as an independent area of research only over the past 30 years. Almost every result obtained in this area since 1970 can be traced to the following Theorem from a short two page paper by Leonard Carlitz (Proc. Amer. Math. Soc. 11(1960), 391–392). Theorem: The algebraic number field Z has class number ≤ 2 if and only if for every nonzero integer α ∈ Z the number of primes πj in every factorization α = π1 π2 · · · πk only depends on α. While Carlitz’s result has been referenced well over 50 times, it did not initially gain much attention. None the less, according to Math Science Net, since the early 1970’s well over 200 papers dedicated to the study of factorization properties of integral domains and monoids, and related topics, have appeared in refereed mathematical journals or conference proceedings. During this same period, over 14 Doctoral Dissertations which touch on factorization properties have been completed. The institutions where such degrees were awarded include The University of Iowa, The University of Tennessee at Knoxville, The University of Nebraska at Lincoln, The University of North Carolina at Chapel Hill, North Dakota State University, Karl-Franzens-Universit¨at-Graz and Universite D’Aix-Marseille III. That this has become a highly active area of mathematical research is demonstrated by Marcel Dekker’s publication in 1997 of the monograph Factorization in Integral Domains, edited by D.D. Anderson. This monograph contains the Proceedings of a 1996 Mini-Conference held at the University of Iowa on factorization problems followed by a Special Session on Commutative Algebra at the Midwestern Regional Meeting of the American Mathematical Society. The meetings in Iowa were followed by a flurry of research activity and the 2003 Meetings in Chapel Hill xi

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were organized with the intent of allowing some of the leading researchers in this field to review the major results of this period. I invited D. D. Anderson (University of Iowa), D. F. Anderson (University of Tennessee at Knoxville), Jim Coykendall (North Dakota State University), Alfred Geroldinger (Karl-Franzens-Universit¨at, Graz), Franz Halter-Koch (Karl-FranzensUniversit¨ at, Graz) and Ulrich Krause (Universit¨at Bremen) to deliver 45 minute addresses at the Mini-Conference. These authors have produced a series of 7 papers based on these lectures which make up the introductory chapters of this volume. These chapters demonstrate the diverse approach that many authors have taken in studying nonunique factorizations. Several are written in a purely monoidal setting, while several others demonstrate that ring theoretic properties play a key role in how elements factor. Moreover, these chapters exhibit the broad range of mathematical techniques necessary to obtain results in this area. Such techniques can range from purely algebraic and combinatorial, to number theoretic. It is hoped that these chapters not only offer mathematicians new to this area the opportunity to survey its current trends and major results, but also offer an up to date introduction to factorization theory for beginning graduate students. The remaining chapters in this volume reflect research which is motivated by arithmetical properties of commutative rings and monoids. This is not restricted to factorization problems, as aspects of the following topics are all considered: multiplicative ideal theory, the factorization of ideals and ideal generation problems, integer-valued polynomials, Pr¨ ufer rings and Domains, block monoids and their combinatorial invariants, and numerical monoids. The two sessions in Chapel Hill and subsequent Proceedings Volume have generated a significant amount of interest. The sessions attracted 30 speakers from 6 different countries. This volume features 26 papers from 33 different contributors from 7 different countries. I wish to take this opportunity to thank all those who participated in the sessions, and to all those who submitted papers for consideration to this volume. I am indebted to a large number of referees, who as always will remain nameless. A special thanks goes to the Department of Mathematics at the University of North Carolina at Chapel Hill, and in particular to Professor William W. Smith who was instrumental in planning and organizing this event. I also wish to thank the Department of Mathematics at Trinity University for their continued support and help in preparing this manuscript. The organization and preparation of the finished work would not have been possible without the help and patience of Denise Wilson, our Departmental Secretary and technical typist. I am of course indebted to Maria Allegra and her staff at Marcel Dekker for giving me the support and opportunity to complete this manuscript. I hope the readers of this volume enjoy its contents as much as I enjoyed putting it together. Scott Chapman San Antonio, Texas January 14, 2005

Contents Preface

xi

Contributors

xv

1 Non-Atomic Unique Factorization in Integral Domains Daniel D. Anderson

1

2 Divisibility Properties in Graded Integral Domains David F. Anderson

22

3 Extensions of Half-Factorial Domains: A Survey Jim Coykendall

46

4 C-Monoids and Congruence Monoids in Krull Domains Franz Halter-Koch

71

5 Monotone Chains of Factorizations in C-Monoids Andreas Foroutan and Alfred Geroldinger

99

6 Transfer Principles in the Theory of Non-unique Factorizations Alfred Geroldinger and Franz Halter-Koch

114

7 Cale Monoids, Cale Domains, and Cale Varieties Scott T. Chapman and Ulrich Krause

142

8 Weakly Krull Inside Factorial Domains Daniel D. Anderson, Muhammed Zafrullah, and Gyu Whan Chang

172

9 The m-Complement of a Multiplicative Set David F. Anderson and Gyu Whan Chang

180

10 Some Remarks on Infinite Products Jim Coykendall

188

11 Rings with Prime Nilradical Ayman Badawi and Thomas G. Lucas

198

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12 On the Ideal Generated by the Values of a Polynomial Jean-Luc Chabert and Sabine Evrard

213

13 Using Factorizations to Prove a Partition Identity David E. Dobbs and Timothy P. Kilbourn

226

14 On Inside Factorial Integral Domains David E. Dobbs, Gabriel Picavet, and Martine Picavet-L’Hermitte

233

15 Polynomial Separation of Points in Algebras Sophie Frisch

253

16 k-Factorized Elements in Telescopic Numerical Semigroups Jose C. Rosales, Pedro A. Garc´ıa-S´ anchez, and Juan I. Garc´ıa-Garc´ıa

260

17 Pr¨ ufer Conditions in Rings with Zero-Divisors Sarah Glaz

272

18 Unmixedness and the Generalized Principal Ideal Theorem Tracy Dawn Hamilton

282

19 A Note on Sets of Lengths of Powers of Elements of Finitely Generated Monoids Wolfgang Hassler

293

20 UMV-Domains Evan Houston and Muhammad Zafrullah

304

21 On Local Half-Factorial Orders Florian Kainrath

316

22 On Factorization in Krull Domains with Divisor Class Group Z2k Karl M. Kattchee

325

23 Integral Morphisms Jack Maney

337

24 A Special Type of Invertible Ideal Stephen McAdam and Richard G. Swan

356

25 Factorization into Radical Ideals Bruce Olberding

363

26 Strongly Primary Ideals Gyu Whan Chang, Hoyoung Nam, and Jeanam Park

378

Index

389

Contributors Daniel D. Anderson, Department of Mathematics, University of Iowa, Iowa City, Iowa, 52242, [email protected] David F. Anderson, Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300, [email protected] Ayman Badawi, Department of Mathematics and Statistics, American University of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates, [email protected] Jean-Luc Chabert, Laboratoire Ami´enois de Math´ematiques Fondamentale et Appliqu´ee, CNRS-UMR 6140, Universit´e de Picardie, 33 rue Saint Leu, 80039 Amiens, France, [email protected] Gyu Whan Chang, Department of Mathematics, University of Incheon, Incheon, 402-748 Korea, [email protected] Scott T. Chapman, Department of Mathematics, Trinity University, One Trinity Place, San Antonio, Texas 78212-7200, [email protected] Jim Coykendall, Department of Mathematics, North Dakota State University,Fargo, North Dakota 58105-5075, [email protected] David E. Dobbs, Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300, [email protected] Sabine Evrard, Laboratoire Ami´enois de Math´ematiques Fondamentale et Appliqu´ee, CNRS-UMR 6140, Universit´e de Picardie, 33 rue Saint Leu, 80039 Amiens, France, [email protected] Andreas Foroutan, Institut f¨ ur Mathematik, Karl-Franzens-Universit¨at, Heinrichstrasse 36, 8010 Graz, Austria, a− [email protected] Sophie Frisch, Institut f¨ ur Mathematik, Technische Universit¨at Graz, A-8010 Graz, Austria, [email protected] ´ Pedro A. Garc´ıa-S´anchez, Departamento de Algebra, Universidad de Granada, Facultad de Ciencias, Campus Fuentenueva s/n, 18071 Granada, Spain, [email protected] xv

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Contributors

´ Juan I. Garc´ıa-Garc´ıa, Departamento de Algebra, Facultad de Ciencias, Universidad de Granada, Campus Fuentenueva s/n, 18071 Granada, Spain, [email protected] Alfred Geroldinger, Institut f¨ ur Mathematik, Karl-Franzens-Universit¨at, Heinrichstrasse 36, 8010 Graz, Austria, [email protected] Sarah Glaz, Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269, [email protected] Franz Halter-Koch, Institut f¨ ur Mathematik, Karl-Franzens-Universit¨at, Heinrichstrasse 36, 8010 Graz, Austria, [email protected] Tracy Dawn Hamilton, Department of Mathematics and Statistics, California State University Sacramento, 6000 J Street, Sacramento, California 95819, [email protected] Wolfgang Hassler, Institut f¨ ur Mathematik, Karl-Franzens Universit¨ at Graz, Heinrichstraße 36/IV, A-8010 Graz, Austria, [email protected] Evan Houston, University of North Carolina at Charlotte, Department of Mathematics, Charlotte, North Carolina 28223-0001, [email protected] Florian Kainrath, Institut f¨ ur Mathematik, Karl-Franzens-Universit¨at Graz, Heinrichstraße 36, A-8010 Graz, Austria, fl[email protected] Karl M. Kattchee, Mathematics Department, University of Wisconsin-La Crosse, 1725 State Street, La Crosse, Wisconsin 54601, [email protected] Ulrich Krause, Fachbereich Mathematik/Informatik, Universit¨at Bremen, 28334 Bremen, Germany, [email protected] Timothy P. Kilbourn, Department of Mathematics, University of Illinois at UrbanaChampaign, Urbana, Illinois 61801-2975, [email protected] Thomas G. Lucas, Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, North Carolina 28223, [email protected] Jack Maney, Department of Mathematics, North Dakota State University, Fargo, North Dakota 58105-5075, [email protected] Stephen McAdam, Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, Texas 78712-0257, [email protected] Hoyoung Nam, Department of Mathematics, Inha University, Incheon, 402-751, Korea Bruce Olberding, Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003-8001, [email protected]

Contributors

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Jeanam Park, Department of Mathematics, Inha University, Incheon, 402-751, Korea Gabriel Picavet, Laboratoire de Math´ematiques Pures, Universit´e Blaise Pascal, 63177 Aubi`ere Cedex, France, [email protected] Martine Picavet-L’Hermitte, Laboratoire de Math´ematiques Pures, Universit´e Blaise Pascal, 63177 Aubi`ere Cedex, France, [email protected] ´ Jose C. Rosales, Departamento de Algebra, Universidad de Granada, Facultad de Ciencias, Campus Fuentenueva s/n, 18071 Granada, Spain, [email protected] Richard G. Swan, Department of Mathematics, The University of Chicago, Chicago, Illinois 60637, [email protected] Muhammed Zafrullah, Department of Mathematics, Idaho State University, Pocatello, Idaho 83209-8085, [email protected]

Chapter 1

Non-Atomic Unique Factorization in Integral Domains by

Daniel D. Anderson Abstract UFDs can be characterized by the property that every nonzero nonunit is a product of principal prime elements or equivalently that every nonzero nonunit x can be written in the form x = upa1 1 · · · pann where u is a unit, p1 , . . . , pn are nonassociate principal primes, and each ai ≥ 1. Each pai i , in addition to being a power of a prime, has a number of other properties, each of which is subject to generalization. We survey various generalizations of (unique) factorization into prime powers in integral domains.

1

Introduction

Unique factorization domains are of course the integral domains in which every nonzero nonunit element has a unique factorization (up to order and associates) into irreducible elements or atoms. Now UFDs can also be characterized by the property that every nonzero nonunit is a product of principal primes or equivalently that every nonzero nonunit has the form upa1 1 · · · pann where u is a unit, p1 , . . . , pn are nonassociate principal primes, and each ai ≥ 1. Each of the pai i , in addition to being a power of a prime, has other properties, each of which is subject to generalization. For example, each pai i is primary, each is contained in a unique maximal t-ideal, and the pai i are pairwise coprime. The goal of this chapter is to survey various generalizations of (unique) factorization into prime powers in integral domains. This follows the thesis of M. Zafrullah that the pai i are the building blocks in a UFD. The author would like to thank M. Zafrullah for a number of discussions of these topics over the past several years. 1

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This chapter consists of four sections besides the introduction. Section 2 covers integral domains whose elements are products of primary elements, Section 3 relates locally finite intersections of localizations to factorizations, Section 4 surveys factorizations into pairwise comaximal elements, and Section 5 summarizes the various generalizations of prime powers and the integral domains whose elements are products of these various generalizations. A fairly complete list of references is also given. The next four paragraphs outline in more detail these four sections. Then a brief review of some of the terms and notation used throughout the chapter is given. In particular, star-operations, the t-operation, the group of divisibility, and splitting sets are discussed. In Section 2 we consider weakly factorial domains, integral domains with the property that every nonzero nonunit is a product of primary elements. While factorization into primary elements need not be unique, it is unique (up to order and associates) once we combine primaries with the same radical, that is, it is unique when the primary elements involved are pairwise v-coprime. Now UFDs are also characterized as being the Krull domains D with trivial divisor class group Cl(D) or the integral domains D whose group of divisibility G(D) is a cardinal sum of copies of Z (with the usual order). In a similar manner, an integral domain D is weakly factorial if and only if D is weakly Krull, that is, D = ∩P ∈X (1) (D) DP (here X (1) (D) is the set of height-one primes of D) where the intersection is locally finite and D has t-class group Clt (D) = 0 or if and only if the natural map G(D) → ⊕P ∈X (1) (D) G(DP ) is an order isomorphism (or equivalently, is just surjective). Also, D is weakly factorial if and only if every saturated multiplicatively closed subset of D is a splitting set. In the case of a weakly factorial domain D, the (unique) factorization of an element into v-coprime primary elements is given by the intersection D = ∩P ∈X (1) (D) DP in the following sense. Let x be a nonzero nonunit of D that is contained in the height-one primes P1 , . . . , Pn . Then each xDPi ∩ D = (xi ) is a principal Pi primary ideal and (x) = ∩P ∈X (1) (D) xDP = (x1 ) ∩ · · · ∩ (xn ) = (x1 ) · · · (xn ) where (x1 ) ∩ · · · ∩ (xn ) is also the reduced primary decomposition for (x). In Section 3 we consider factorizations induced by other locally finite intersections of localizations D = ∩P ∈S DP . For example, call a nonzero nonunit element t-pure if it is contained in a unique maximal t-ideal (as is the case for a nonzero primary element). Then every nonzero nonunit of D is a product of t-pure elements if and only if the intersection D = ∩P ∈t-Max(D) DP (where t-M ax(D) is the set of maximal t-ideals of D) is locally finite, is independent (distinct maximal t-ideals contain no common nonzero prime ideal) and Clt (D) = 0. Again, this factorization into t-pure elements is unique once elements contained in the same maximal t-ideal are combined, or equivalently, when the elements in the product are pairwise v-coprime. We also consider factorization into homogeneous elements and into rigid elements. In Section 4 we consider the comaximal factorizations recently introduced by McAdam and Swan [27]. They defined a nonzero nonunit element d of an integral domain D to be pseudo-irreducible (pseudo-prime) if d = ab (d|ab) with a and b comaximal implies that a or b is a unit (d|a or d|b). A factorization d = d1 · · · dn is a (complete) comaximal factorization if each di is a nonzero nonunit (pseudoirreducible) and the di ’s are pairwise comaximal. The integral domain D is a comaximal factorization domain (CFD) if each nonzero nonunit has a complete comaximal factorization and a unique comaximal factorization domain (UCFD) is

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a CFD in which complete comaximal factorizations of an element are unique up to order and associates. The main result is that for a CFD D the following are equivalent: (1) every two-generated invertible ideal of D is principal, (2) every nonzero nonunit of D has a comaximal factorization into pseudo-prime elements, and (3) D is a UCFD. We give a star-operation generalization, and relate these comaximal factorizations to the factorizations given in Sections 2 and 3. It is interesting to note that a one-dimensional domain is a UCFD if and only if it is weakly factorial. In Section 5 we summarize the various prime power generalizations given in Sections 2–4 and give several additional ones. Then the integral domains whose nonzero nonunits are products of these prime power generalizations are considered. Diagrams are given to show the various implications among the prime power generalizations and the integral domains defined. We next review some of the terms and notation used in this chapter. Throughout D is an integral domain with quotient field K. Let F (D) denote the set of nonzero fractional ideals of D and let f (D) be the set of finitely generated members of F (D). A star-operation ∗ on D is a closure operation on F (D) that further satisfies D∗ = D and (xA)∗ = xA∗ for all 0 = x ∈ K and A ∈ F (D). Here A ∈ F (D) is called a ∗-ideal if A∗ = A and A has finite type if A∗ = B ∗ for some B ∈ f (D). The star-operation ∗ has finite character if for each A ∈ F (D), A∗ = ∪{B ∗ |B ⊆ A with B ∈ f (D)}. Suppose that ∗ has finite character. Then every proper ∗-ideal is contained in a maximal ∗-ideal and a maximal ∗-ideal is prime. A prime ideal minimal over a ∗-ideal is a ∗-ideal. Hence a prime ideal minimal over a principal ideal is a ∗-ideal. We use ∗-Max(D) to denote the set of maximal ∗-ideals. Here D = ∩P ∈∗-Max(D) DP . With any star-operation ∗ is associated the finite character star-operation ∗s where A∗s = ∪{B ∗ |B ⊆ A where B ∈ f (D)}. Evidently ∗ has finite character precisely when ∗ = ∗s . Examples of star-operations include (1) the d-operation Ad = A, (2) the voperation Av = (A−1 )−1 = ∩{Dx|Dx ⊇ A where 0 = x ∈ K}, (3) the t-operation At = Avs = ∪{Bv |B ⊆ A with B ∈ f (D)}, and (4) for a set {Dα } of overrings of D with D = ∩Dα , the star-operation induced by ∩Dα A∗ = ∩ADα . Here (1) and (3) have finite character while (2) generally does not. An intersection D = ∩Dα has finite character or is locally finite if each 0 = x ∈ D is a unit in almost all Dα . If the intersection D = ∩Dα is locally finite, then the induced star-operation has finite character, but not conversely [1, Theorem 1]. An ideal A ∈ F (D) is ∗-invertible if there is a B ∈ F (D) with (AB)∗ = D. In this case we can take B = A−1 . The set Inv∗ (D) of ∗-invertible ∗-ideals forms a group under the ∗-product A ∗ B = (AB)∗ . The ∗-class group Cl∗ (D) of D is Inv∗ (D)/ Princ(D) where Princ(D) is the subgroup of nonzero principal fractional ideals. Of particular importance is the t-class group Clt (D) (or just the class group) of D. For D a Krull domain Clt (D) is the usual divisor class group while for a Pr¨ ufer domain D or one-dimensional domain D Clt (D) is the Picard group Pic(D) = Inv(D)/ Princ(D) = Cld (D). If ∗ is a finite character star-operation, then a ∗-invertible ideal has finite type, in fact A ∈ F (D) is ∗-invertible if and only if A has finite type and AP is principal for each P ∈ ∗-Max(D). Two ideals A and B of D are ∗-comaximal if (A, B)∗ = D. If A and B are ∗-comaximal, then it is easily proved that A∗ ∩ B ∗ = (A ∩ B)∗ = (AB)∗ . In the case where ∗ = d, we just say comaximal. Two elements a, b ∈ D are ∗-coprime if

4

Anderson

(a, b)∗ = ((a), (b))∗ = D and a and b are coprime if [a, b] = 1 (here [a, b] denotes the GCD of a and b). Thus a, b ∈ D are v-coprime if (a, b)v = D, or equivalently, a and b are not contained in any maximal t-ideal. Of particular interest are star-operations induced by intersections of localizations D = ∩P ∈S DP : A → A∗S = ∩P ∈S ADP . For any star-operation ∗ we have the associated finite character star-operation ∗w defined by A∗w = {x ∈ K|xI ⊆ A for some I ∈ f (D) with I ∗ = D} = ∩P ∈∗s -Max(D) ADP . Here ∗s -Max(D) = ∗w -Max(D), A is ∗s -invertible if and only if A is ∗w -invertible and Cl∗s (D) = Cl∗w (D). See [7] for details. General references for star-operations include [21], [23], and [24]. For results on ∗-invertibility, class groups, and t-ideals see [18] and [35]. For star-operations induced by overrings, see [1] and [7]. For an integral domain D with quotient field K let K ∗ = K − {0} and U (D) be the group of units of D. Then the multiplicative group G(D) = K ∗ /U (D) is called the group of divisibility of D. Now G(D) is partially ordered by aU (D) ≤ bU (D) ⇐⇒ a|b in D ⇐⇒ Da ⊇ Db. Note that the map G(D) → Princ(D) given by aU (D) → Da is an order isomorphism where Princ(D) is ordered by reverse inclusion. Given a family {(Gλ , ≤λ˙ )} of partially ordered abelian groups, the cardinal product is ΠGλ with the order (aλ ) ≤ (bλ ) ⇐⇒ each aλ ≤ bλ . The cardinal sum is defined in a similar manner. Suppose that D = ∩P ∈S DP . Then we have an order preserving monomorphism ϕ: G(D) → ΠP ∈S G(DP ) (cardinal product) given by ϕ(xU (D)) = (xU (DP )). Note that if D = ∩P ∈S DP has finite character, then im ϕ ⊆ ⊕P ∈S G(DP ). A subgroup H of a partially ordered abelian group (G, ≤) is convex if whenever 0 ≤ a ≤ h with h ∈ H, then a ∈ H and H is directed (or filtered ) if given a, b ∈ H, there exists c ∈ H with a ≤ c and b ≤ c or equivalently H+ = {h ∈ H|h ≥ 0} generates H as a group. Let S be a saturated multiplicatively closed subset of D. Then < S >= {s1 s−1 2 U (D)|s1 , s2 ∈ S} is a convex directed subgroup of G(D). The converse is also true, see [28]. A convex directed subgroup H of G(D) is a cardinal summand of G(D) if there is a convex directed subgroup K of G(D) with G(D) = H ⊕ K where H ⊕ K is the cardinal sum of H and K. A saturated multiplicatively closed subset S of D is a splitting set if every nonzero element x of D can be written in the form x = st where s ∈ S and t is v-coprime to each element of S. It is not hard to show that a saturated multiplicatively closed set S is a splitting set if and only if < S > is a cardinal summand of G(D) ([29, Proposition 4.1] and [4, Theorem 22]). Perhaps the main example of a splitting set is a multiplicatively
closed subset generated by a set {pα } of height-one principal primes satisfying pαn D = 0 for each countable subcollection {pαn } of {pα }. Splitting sets were introduced in [29] and studied in [4] and [17]. There are of course many other generalizations of UFDs. A UFD is characterized by the property that every nonzero prime ideal contains a nonzero principal prime (ideal). Thus for each of the generalizations of a prime power, we can ask what domains have the property that each nonzero prime ideal contains such an element. While we will not pursue that theme here, the reader is referred to [14] for various “Kaplansky-like” theorems. Recall that an integral domain D is atomic if every nonzero nonunit of D is a product of irreducible elements or atoms. The theme of studying integral domains with various good atomic factorization properties is put

Non-Atomic Unique Factorization in Integral Domains

5

forth in [3]. For a survey of extensions of unique factorization to integral domains, rings with zerodivisors, rings without identity, or to modules, the reader is referred to [2]. In fact, this chapter is an elaboration of Section 2 of that chapter. Portions of this chapter were presented at a mini-conference on factorization organized by Professor Scott Chapman held October 23, 2003 preceding the regional American Mathematical Society meeting at the University of North Carolina.

2

Weakly Factorial Domains

Throughout D will denote an integral domain with quotient field K. Recall that D is weakly factorial if every (nonzero) nonunit of D is a product of primary elements or equivalently every proper principal ideal of D is a product of principal primary ideals. Weakly factorial domains were introduced by D. D. Anderson and L. A. Mahaney [9] and further studied by Anderson and Zafrullah [13]. We begin by looking at factorizations into primary elements in an integral domain. Suppose that (q) is a nonzero P -primary ideal of D. Then P is the unique prime t-ideal containing (q) [5, Lemma 1]. Thus nonzero primary elements with distinct radicals are v-coprime. If (q1 ) and (q2 ) are P -primary, then so is (q1 ) (q2 ) = (q1 q2 ) [9, Corollary 2]. Thus if a nonzero nonunit x ∈ D is a product of primary elements, then by combining primary elements with the same radical, we can write x = q1 · · · qn where (qi ) is Pi -primary and the Pi ’s are distinct, or equivalently, the qi ’s are pairwise v-coprime. In this case, (x) = (q1 ) · · · (qn ) = (q1 ) ∩ · · · ∩ (qn ) where (q1 ) ∩ · · · ∩ (qn ) is the unique normal decomposition for (x) [9, Corollary 5]. It follows that the factorization of an element into a product of pairwise v-coprime primary elements is unique up to order and associates. We next give a number of conditions equivalent to D being weakly factorial and then sketch the proof of several of the implications. Theorem 2.1. For an integral domain D the following conditions are equivalent. (1 ) D is weakly factorial. (2 ) Every nonzero nonunit of D has a unique factorization (up to order and associates) into a product of pairwise v-coprime primary elements. (3 ) D = ∩P ∈X (1) (D) DP is locally finite and Clt (D) = 0. (4 ) For every nonzero nonunit x ∈ D, if P is a prime ideal minimal over (x), then ht P = 1 and xDP ∩ D is principal. (5 ) D = ∩P ∈X (1) (D) DP is locally finite and the natural map G(D) → ⊕P ∈X (1) (D) G(DP ) is surjective (and hence an order isomorphism). (6 ) Every convex directed subgroup of G(D) is a cardinal summand of G(D). (7 ) For (nonzero) prime ideal P of D, D − P  =  every U(D)|s is a cardinal summand of G(D). s1 s−1 1 , s2 ∈ D − P 2 (8 ) Every saturated multiplicatively closed subset of D is a splitting set. (9 ) For every (nonzero) prime ideal P of D, D − P is a splitting set. Proof. We have already remarked the equivalence of (1) and (2). The equivalence of (1) and (3)-(7) is [13,  convex directed subgroup of G(D)  Theorem]. Since every U (D) |s , s ∈ S for some saturated multiplicatively is of the form S = s1 s−1 1 2 2

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closed subset S of D and conversely and since S is a cardinal summand of G(D) if and only if S is a splitting set, we get the equivalence of (6) and (8) and of (7) and (9).

We next sketch the proof of several of the implications of the previous theorem. Suppose that D is weakly factorial. Since each nonzero prime ideal contains a nonzero primary element, each nonzero prime ideal contains the radical of a nonzero principal primary ideal which is a maximal t-ideal. Hence X (1) (D) = t-Max (D). As we always have D = ∩P ∈t-Max(D) DP , we have D = ∩P ∈X (1) (D) DP . Moreover, each proper principal ideal having a primary decomposition involving only heightone primes gives that each nonzero principal ideal is contained in only finitely many height-one primes and hence the intersection is locally finite. (In fact, D = ∩P ∈X (1) (D) DP with the intersection being locally finite if and only if every nonzero principal ideal has a primary decomposition involving only height-one primes [9, Theorem 13].) So D is weakly Krull. Let A be an integral t-invertible ideal of D. Let P1 , · · · , Pn be the maximal t-ideals (=height-one primes) containing A. Now APi is principal, so APi = qi DPi where qi D is Pi -primary. Then A ⊆ (AP1 ∩ D) ∩ · · · ∩ (APn ∩ D) = q1 D ∩ · · · ∩ qn D = q1 · · · qn D. So A = q1 · · · qn A where A is contained in no height-one prime. Hence At = D, so At = q1 · · · qn D is principal. Thus Clt (D) = 0. Conversely, suppose that D = ∩P ∈X (1) (D) DP where the intersection is locally finite and that Clt (D) = 0. Let x be a nonzero nonunit of D. Then xD = (xDP1 ∩ D) ∩ · · · ∩ (xDPn ∩ D) where P1 , · · · , Pn are the height-one primes containing xD. But each xDPi ∩D is Pi -primary and xDP1 ∩D, · · · , xDPn ∩D are v-coprime. Hence xD = (xDP1 ∩ D) ∩ · · · ∩ (xDPn ∩ D) = ((xDP1 ∩ D) · · · (xDPn ∩ D))t ; so each xDPi ∩ D is t-invertible. Hence Clt (D) = 0 gives (xDPi ∩ D)t = qi D where qi D is Pi -primary. So xD = q1 D · · · qn D. Thus D is weakly factorial. We next show that (1) and (4) are equivalent. Suppose that D is weakly factorial. If P is minimal over (x), then ht P = 1 and (x)P ∩D is principal being the P -primary component in the reduced primary decomposition of (x). Conversely, suppose that (4) holds. Let S be the multiplicatively closed subset of D consisting of all products of primary elements and units. Then S is saturated. Let y be a nonunit factor of an element x ∈ S. Then (x) being a product of principal primary ideals has only finitely many minimal primes each of which by hypothesis has height-one. Since y is a factor of x, it too is contained in only finitely many height-one primes P1 , · · · , Pn . Now y ∈ (y)P1 ∩ D = (q1 ) where (q1 ) is P1 -primary and y = q1 t1 where t1 ∈ P2 ∩ · · · ∩ Pn − P1 . Continuing in this manner we get y = q1 · · · qn tn where (qi ) is Pi -primary and tn is contained in no height-one prime, hence a unit. So y ∈ S. Suppose that some nonzero nonunit of D is not in S. Then (a) ∩ S = ∅ since S is saturated. Hence (a) can be enlarged to a prime ideal P ⊇ (a) maximal with respect to P ∩ S = ∅. Shrinking P to a prime ideal Q minimal over (a), we have a height-one prime ideal Q containing no primary elements. But this is absurd, for (a)Q ∩ D is itself a principal primary ideal contained in Q. We next give some examples of weakly factorial domains.

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Example 2.2. (1) A one-dimensional quasilocal domain has every nonunit element primary and hence is weakly factorial. Conversely, a domain in which every nonunit is primary is either a field or one-dimensional quasilocal. (2) [9, page 149] A one-dimensional domain D is weakly factorial if and only if each nonunit of D is contained in only finitely many maximal ideals and P ic(D) = 0. (3) [9, Theorem 21] Given any collection {Hλ } of rank-one totally ordered abelian groups, there is a weakly factorial Bezout domain D with G(D) order-isomorphic to the cardinal direct sum ⊕Hλ . (4) [13, Example] Let K ⊆ L be fields and D = K +XL[X] = {f (X) ∈ L[X]|f (0) ∈ K}. Then D is an atomic weakly factorial domain. It is easily seen that every nonzero nonunit of D may be written in one of the following two forms: p1 · · · pn or p1 · · · pn (aX r ) where p1 , · · · , pn are principal primes and aX r (r ≥ 1, a ∈ L) is primary. (5) [5, Lemma 3] An atomic domain in which almost all atoms are prime is weakly factorial. (6) [5, Corollary 2] An atomic domain is weakly factorial ⇐⇒ every atom is primary ⇐⇒ each atom is contained in a unique maximal t-ideal. While D[X] is a UFD whenever D is, this need not be the case for a weakly factorial domain. We have the following result. Recall that D is a generalized Krull domain if D is weakly Krull and for each P ∈ X (1) (D), DP is a valuation domain. Theorem 2.3. For an integral domain D the following conditions are equivalent. (1 ) D[X] is weakly factorial. (2 ) D is a weakly factorial GCD domain. (3 ) D is a weakly factorial generalized Krull domain. (4 ) D is a generalized Krull domain and a GCD domain. (5 ) D is weakly factorial and if p and q are non v-coprime primary elements then p|q or q|p. Proof. (1)⇐⇒(2) [9, Theorem 17]. The equivalence of (2)–(4) is [9, Theorem 20]. (2)⇐⇒(5) [9, Theorem 18] and its proof. It is interesting to observe that D[X] is an atomic weakly factorial domain if and only if D[X] and hence D is a UFD. This follows from the previous theorem and the well-known fact that an atomic GCD domain is a UFD.

3

Independent Locally-Finite Intersections of Localizations

In Section 2 we saw that factorization into primary elements was intimately connected to the locally finite representation D = ∩DP ∈X (1) (D) DP . Now of course an integral domain need not have such a representation. In this section we extend the results of Section 2 to more general intersections of localizations. Much of this comes from Anderson and Zafrullah [16]. We begin with some definitions given there.

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Let D be an integral domain with quotient field K. A family F = {Pi }i∈I of nonzero prime ideals of D is called a defining family of primes for D if D = ∩i∈I DPi . If, further, every nonzero nonunit of D belongs to at most finitely many members of F , F is of finite character, and if no two members of F contain a common nonzero prime ideal, F is independent. An integral domain is independent of finite character F (or an F -IFC domain) if it has a defining family F of prime ideals that is independent and of finite character. Let us note that the h-local domains of Matlis [26] (where F = Max(D)), Noetherian domains whose grade-one primes are of height one, Krull domains, generalized Krull domains of Ribenboim [30], independent rings of Krull type [22], and weakly Krull domains [10] are all F -IFC domains with well-defined family of primes. Let F be a defining family of primes for D. Then F induces the star-operation ∗F on D by A∗F = ∩P ∈F ADP for A ∈ F (D). Moreover, if F has finite character, so does ∗F [1, Theorem 1]. We call an integral ideal A of D unidirectional if A is contained in a unique member of F . The following four theorems come from [16]. Theorem 3.1. ([16, Theorem 2.1] ) Let D be an integral domain and let F be a defining family of primes for D. Then the following are equivalent. (1 ) D is an F -IFC domain. (2 ) For every nonzero nonunit x of D, (x) is a ∗F -product of finitely many unidirectional ∗F -ideals of D. (3 ) Every nonzero prime ideal of D contains a nonzero element x such that (x) is a ∗F -product of finitely many unidirectional ∗F -ideals. If P1 , · · · , Pn are the members of F containing (x) in (2), then the factorization alluded to is (x) = (xDP1 ∩ D) ∩ · · · ∩ (xDPn ∩ D) = ((xDP1 ∩ D) · · · (xDPn ∩ D))

∗F

.

Theorem 3.2. ([16, Theorem 3.3] ) Let F be a defining family of mutually incomparable primes for D such that ∗F is of finite character. Then the following are equivalent. (1 ) F is independent of finite character. (2 ) Every nonzero prime ideal of D contains an element x such that (x) is a ∗F product of unidirectional ∗F -ideals. (3 ) Every nonzero prime ideal of D contains a unidirectional ∗F -invertible ∗F -ideal. (4 ) For P ∈ F and 0 = x ∈ P , xDP ∩ D is ∗F -invertible and unidirectional. (5 ) F is independent and for any nonzero ideal A of D, A∗F is of finite type whenever ADP is finitely generated for all P in F . Theorem 3.3. ([16, Corollary 3.5]) Let F be a defining family for D consisting of incomparable primes. Then the following are equivalent. (1 ) F is independent of finite character and every ∗F -invertible ∗F -ideal is principal. (2 ) Every nonzero nonunit x of D may be written in the form x = x1 · · · xn where each xi generates a unidirectional ideal. (3 ) For each nonzero nonunit x in D and for each P in F containing x, xDP ∩ D is principal and unidirectional. (4 ) The natural map G(D) → ΠP ∈F G(DP ) has image ⊕P ∈F G(DP ).

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The factorization in (2) becomes unique (up to order and associates) once the xi ’s belonging to the same unidirectional ideal are combined. If P1 , · · · , Pn are the primes of F containing x and if xDPi ∩ D = (xi ), then this factorization is x = ux1 · · · xn for some unit u. Now suppose that ∗ is a star-operation on D. Now D = ∩P ∈∗s -Max(D) DP , so F = ∗s -Max(D) is a defining family for D. Here the star-operation ∗sF is given by A∗sF = ∩P ∈F ADP . But by [7], ∩P ∈F ADP = A∗w = {x ∈ K|xI ⊆ A for some I ∈ f (D) with I ∗ = D}. Also, ∗sF = ∗w has finite character, A is ∗sF -invertible if and only if A is ∗s -invertible and Cl∗sF (D) = Cl∗s (D). So we have the following result. Theorem 3.4. Let ∗ be a finite character star-operation on the integral domain D. Then the following conditions are equivalent. (1 ) ∗-Max(D) is independent of finite character and Cl∗ (D) = 0. (2 ) Every nonzero nonunit x of D may be written in form x = x1 · · · xn where each xi is contained in a unique member of ∗-Max(D). (3 ) Every nonzero nonunit x of D may be written uniquely (up to order and associates) in the form x = x1 · · · xn where each xi is contained in a unique Pi ∈ ∗Max(D) and for i = j, Pi = Pj , or equivalently, xi and xj are ∗-coprime. (4 ) For each nonzero nonunit x in D and for each P ∈ ∗-Max(D), xDP ∩ D is principal and unidirectional. (5 ) The natural map G(D) → ΠP ∈∗- Max(D) G(DP ) has image ⊕P ∈∗- Max(D) G(DP ) (and hence is an order isomorphism where ⊕G(DP ) has the cardinal order ). (6 ) ∗-Max(D) is independent, Cl∗ (D) = 0, and for each nonzero ideal A of D, A∗w is of finite type whenever ADP is finitely generated for all P ∈ ∗-Max(D). Of special interest is the case where ∗ is the t-operation. Here ∗w = tw is the w-operation introduced by Wang and McCasland [20]: Aw = {x ∈ K|xI ⊆ D for some finitely generated ideal I with I −1 = D}. Let P be a prime ideal of a domain D. Then an ideal A of D is P -pure if AP ∩ D = A (we usually assume A ⊆ P ) and A is t-pure if A is P -pure for some P ∈ t-Max(D) [10]. For A t-invertible (with At = D), At is t-pure if and only if A is contained in a unique maximal t-ideal [11, Lemma 2.1]. A nonzero nonunit x ∈ D is t-pure if (x) is t-pure, or equivalently, if x is contained in a unique maximal t-ideal. Following [11] we call an integral domain D t-pure if every nonzero nonunit of D is t-pure and we call D semi-t-pure if every nonzero nonunit of D is a finite product of t-pure elements. Evidently D is t-pure if and only if D is quasilocal with maximal ideal a t-ideal. Observe that if 0 = (q) is P -primary, then (q) is P -pure and t-pure and q is t-pure. We have the following t-version of Theorem 3.4. Theorem 3.5. For an integral domain D, the following conditions are equivalent. (1 ) t-Max(D) is independent of finite character and Clt (D) = 0. (2 ) D is semi-t-pure, i.e., every nonzero nonunit of D is a finite product of t-pure elements. (3 ) Every nonzero nonunit x of D may be written uniquely in the form x = x1 · · · xn where each xi is t-pure and for i = j, xi and xj are v-coprime. (4 ) For 0 = x ∈ D and x ∈ P ∈ t-Max(D), xDP ∩ D is principal and t-pure. (5 ) The natural map G(D) → ΠP ∈t- Max(D) G(DP ) has image ⊕P ∈t- Max(D) G(DP ) (and hence is an order ismorphism where ⊕G(DP ) has the cardinal order ).

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(6 ) t-Max(D) is independent, Clt (D) = 0, and for each nonzero ideal A of D, Aw is of finite type whenever ADP is finitely generated for all P ∈ t-Max(D). (7 ) For each P ∈ t-Max(D), D − P is a splitting set, or equivalently, < D − P >= {s1 s−1 2 U (D)|s1 s2 ∈ D − P } is a cardinal summand of G(D). (8 ) For each nonempty subset F ⊆ t-Max(D), S = D − UP ∈F P is a splitting set, or equivalently < S > is a cardinal summand of G(D). (9 ) D is an F -IFC domain, for each Q ∈ F, DQ is t-pure, and Clt (D) = 0. Proof. The equivalence (1)–(6) follows from Theorem 3.4. The equivalence of (1), (7) and (8) follows from [16, Theorem 2.14]. And the equivalence of (1)–(3) and (8) is given in [11, Theorem 3.4]. Observe that Theorem 2.1 (D weakly factorial) is just the case X (1) (D) = tMax(D). We next give three more generalizations of prime powers that turn out to be equivalent to being t-pure in GCD domains. Let h be a nonzero nonunit of an integral domain D. Then h is rigid [31] (homogeneous, strongly homogeneous [11]) if for x, y ∈ D, x, y|h (with [x, y] = 1, (x, y)v = D) implies x|y or y|x (x or y is a unit). And D is (semi-) rigid if every nonzero nonunit of D is (a product of) rigid (elements). Similar definitions for (semi-) homogeneous domains and (semi-) strongly homogeneous domains may be given. It is easy to see that an atom or prime power is rigid, that a rigid element is strongly homogeneous, and that a strongly homogeneous or t-pure (and hence primary) element is homogeneous. Evidently an integral domain is rigid if and only if it is a valuation domain. Thus any one-dimensional quasilocal domain that is not a valuation domain will have a nonzero primary element that is not rigid. To study factorization in a nonatomic setting, P.M. Cohn [19] defined a nonzero nonunit h ∈ D to be primal if h|xy implies h = h1 h2 where h1 |x and h2 |y and to be completely primal if each of its nonunit factors is primal. (See [15] for a survey of completely primal elements.) Thus an atom is (completely) primal if and only if it is prime. An integral domain in which every nonzero nonunit is (completely) primal is called a pre-Schreier domain [34] and an integrally closed pre-Schreier domain is a Schreier domain. A GCD domain is a Schreier domain [19]. We have the following result. Theorem 3.6. ([11, Theorem 2.3, Corollary 2.4]) Let h be a completely primal element of an integral domain D. Then the following are equivalent. (1 ) h is homogeneous. (2 ) h is strongly homogeneous. (3 ) h is t-pure. If further D is a GCD domain, we have (1)–(3) equivalent to (4 ) h is rigid. It is well-known that Clt (D) = 0 for a pre-Schreier domain D [34, Corollary 3.7]. Combining this with Theorem 3.5 and Theorem 3.6 we get our next result. Theorem 3.7. ([11, Corollary 3.7]) For an integral domain D, the following are equivalent. (1 ) D is a pre-Schreier semi-homogeneous integral domain.

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(2 ) Every nonzero nonunit of D may be written uniquely as a product of mutually coprime completely primal homogeneous elements. (3 ) D is pre-Schreier and t-Max(D) is independent of finite character. (4 ) D is a F -IFC domain, for each Q ∈ F, DQ is a homogeneous pre-Schreier domain, and Clt (D) = 0. Combining Theorem 3.5 and the second part of Theorem 3.6 we get the following result which should be compared to Theorem 2.3. Recall that an integral domain D is an independent ring of Krull type if t-Max(D) is independent of finite character and DP is a valuation domain for each P ∈ t-Max(D). The implication (1)=⇒(3) was first given in [31] while the implication (3)=⇒(1) was given in [32]. Theorem 3.8. ([11, Corollary 3.8]) For an integral domain D, the following conditions are equivalent. (1 ) D is a semi-rigid GCD domain. (2 ) D is a semi-t-pure GCD domain. (3 ) D is a GCD domain that is an independent ring of Krull type. (4 ) D is an independent ring of Krull type with Clt (D) = 0. We end this section with the following result. Recall that two ideals I and J of a ring are condensed if IJ = {ij|i ∈ I, j ∈ J}. Theorem 3.9. ([8, Theorem 2.1] ) Let D be an atomic domain. Then the following statements are equivalent. (1 ) Every pair of comaximal ideals of D is condensed. (2 ) Every pair of distinct maximal ideals of D is condensed. (3 ) Each atom of D is unidirectional. (4 ) D is h-local with Pic(D) = 0. (5 ) Every nonzero nonunit of D is a product of unidirectional elements. (6 ) For each nonzero nonunit x ∈ D and each maximal ideal M containing x, xDM ∩ D is a principal unidirectional ideal. As before, the factorization into unidirectional elements given in (5) is unique up to units and associates once elements contained in the same maximal ideal are combined. Note that (4)–(6) are still equivalent if we drop the hypothesis that D is atomic. Indeed these three statements are just three of the equivalences of Theorem 3.4 where ∗ is the d-operation.

4

Comaximal Factorizations and Generalizations

Recently McAdam and Swan [27] studied comaximal factorizations of elements and ideals in integral domains and rings. In this section we discuss their comaximal factorizations of elements in integral domains, give a generalization to star-operations, and compare it to results from Sections 2 and 3. We begin with some definitions from their chapter. Let D be an integral domain. A nonzero nonunit b of D is pseudo-irreducible (resp. pseudo-prime) if b = cd with c and d comaximal (i.e., (c, d) = D) implies c or d is a unit (resp. b|cd with c and d comaximal implies that b|c or b|d). Clearly b pseudo-prime implies that b is pseudo-irreducible. Note that b is pseudo-prime

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if and only if D/ (b) is indecomposable [27, Lemma 3.1]. For a nonzero nonunit b of D we call b = b1 · · · bm a (complete) comaximal factorization of b if the bi are pairwise comaximal nonunit (pseudo-irreducible) elements. Evidently a comaximal factorization b = b1 · · · bm is complete if and only if it has no proper refinements that are also comaximal factorizations of b. Then D is a comaximal factorization domain (CFD) if every nonzero nonunit of D has a complete comaximal factorization and D is a unique comaximal factorization domain (UCFD) if D is a CFD in which complete comaximal factorizations are unique up to order and associates. They showed [27, Lemma 1.1] that an integral domain D is a CFD if either (i) each nonzero nonunit of D has only finitely many minimal primes or (ii) each nonzero nonunit of D is contained in only finitely many maximal ideals. Here (i) insures that a Noetherian domain is a CFD. To this list of CFDs we can add: D has a defining family of primes of finite character. The proof is similar to the proof given for (i). A key concept in their work is the notion  of an S-ideal. A nonzero ideal I of  D is called an S-ideal if I = (a, c) = a2 , c for some a,  c2 ∈ I. The relation to comaximal factorizations is the observation that (a, c) = a , c if and only if there is an element b ∈ D with (a, b) = D and c|ab [27, Lemma 1.2]. They proved that S-ideals are invertible and that any two-generated invertible ideal is isomorphic to an S-ideal [27, Lemma 1.5]. They gave the following characterization of UCFDs (we have added (3)). Theorem 4.1. ([27, Theorem 1.7] ) For an integral domain D the following conditions are equivalent. (1 ) D is a UCFD. (2 ) D is a CFD and every pseudo-irreducible element of D is pseudo-prime. (3 ) Every nonzero nonunit of D has a comaximal factorization into pseudo-prime elements. (4 ) Every two-generated invertible ideal of D is principal. (5 ) Every S-ideal of D is principal. It should be noted that in the previous theorem we can not add P ic(D) = 0 as the following example from Section 4 of [27] shows (but see Theorem 4.2 below).   Let An be the subring of Bn = R[x0 , · · · , xn ]/ x20 + · · · + x2n − 1 , the ring of realvalued polynomial functions on the n-sphere S n , consisting of all even functions. Then for n ≥ 2, An is a regular domain that is a UCFD, but P ic(An ) = 0. Now a UFD D is a UCFD, indeed every nonzero nonunit of D is contained in only finitely many minimal prime ideals and P ic(D) = 0. For exactly the same reason, a weakly factorial domain is a UCFD. Note that the domains given by Theorem 3.4 for the case where ∗ is the d-operation are UCFDs. Indeed, these are precisely the UCFDs in which every pseudo-irreducible element is contained in a unique maximal ideal. In the case of a one-dimensional UFD (that is, a PID) a pseudo-prime element is just a primary element and a complete comaximal factorization is the same thing as a factorization into v-coprime prime powers or primary elements. Actually this result extends to any one-dimensional domain. Theorem 4.2. For a one-dimensional integral domain D the following are equivalent. (1 ) D is weakly factorial.

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(2 ) Every nonzero nonunit of D is contained in only finitely many maximal ideals and P ic(D) = 0. (3 ) D is a UCFD. Proof. (1)⇐⇒(2) Example 2.2 (2). (2)=⇒(3) Since every nonzero nonunit is contained in only finitely many minimal primes, D is a CFD. By Theorem 4.1 D is a UCFD. (3)=⇒(1) Let d ∈ D be pseudo-prime.  Hence D/ (d) is indecomposable and  (d) is indecomposable. Since D/ (d) is zero-dimensional and reduced, thus D/  (d) is von Neumann regular and hence a field since it is indecomposable. Thus D/  (d) is a maximal ideal and hence (d) is a primary ideal. Thus every nonzero nonunit of D is a product of pseudo-prime elements or equivalently a product of primary elements. So D is weakly factorial. Also, a quasilocal domain (D, M ) is a UCFD. Indeed, for 0 = a ∈ M , a = a is the unique comaximal factorization of a ! One should note that while a UFD is a UCFD, the complete comaximal factorization of an element need not be the prime-power factorization. Indeed in K[X, Y ], K a field, XY is pseudo-irreducible. We next generalize comaximal factorizations to coprime and ∗-comaximal factorizations where ∗ is a finite character star operation on D. A nonzero nonunit a ∈ D is ∗-pseudo-irreducible (resp. ∗-pseudo-prime) if for b, c ∈ D with (b, c)∗ = D, a = bc (resp. a|bc) implies b or c is a unit (resp. a|b or a|c). Certainly a ∗-pseudoprime element is ∗-pseudo-irreducible. A factorization b = b1 · · · bm into nonunits is a (complete) ∗-comaximal factorization if for i = j (bi , bj )∗ = D (and each bi is ∗-pseudo-irreducible). Evidently b = b1 · · · bm is a complete ∗-comaximal factorization if and only if it has no proper refinements that are ∗-comaximal factorizations. Finally, D is a ∗-CFD if each nonzero nonunit of D has a complete ∗-comaximal factorization and a ∗-CFD is a ∗-UCFD if each complete ∗-comaximal factorization is unique up to units and associates. Note that if we take ∗ to be the d-operation (Ad = A) we get the comaximal factorizations of McAdam and Swan. We can also define coprime factorizations by replacing (a, b)∗ = D by [a, b] = 1, but we must be careful (see next paragraph). Define a nonzero nonunit a ∈ D to be [ ] -pseudo-irreducible (resp. [ ]-pseudoprime) if a = a1 · · · an (resp. a|a1 · · · an ) where [ai , aj ] = 1 for i = j implies some ai is a unit (a|ai for some i). A (complete) coprime factorization is a factorization b = b1 · · · bm where each bi is a nonunit and [bi , bj ] = 1 for i = j (with each bi [ ]-pseudo-irreducible). Evidently a coprime factorization is complete if and only if it has no proper refinement which is again a coprime factorization. The domain D is a [ ]-CFD if each nonzero nonunit has a complete coprime factorization and a [ ]-CFD is a [ ]-UCFD if the complete coprime factorization of an element is unique up to units and associates. Note that if a is ∗-pseudo-irreducible and a = a1 · · · an where (ai , aj )∗ = D for i = j, then some ai is a unit. Indeed, this follows from the fact that if (a, b)∗ = (a, c)∗ = D, then (a, bc)∗ = D. For we can write a = a1 (a2 · · · an ) where (a1 , a2 · · · an )∗ = D. However, [a, b] = [a, c] = 1 does not imply [a, bc] = 1. (In fact, a domain D satisfies this property (called the P P -property) if and only if D satisfies GL: the product of two primitive polynomials is primitive. And it is well-known that an atomic domain satisfying GL is a UFD. See [12] for details.) Consider the element X 3 of Q+XR[X]. Then X 3 is not [ ]-pseudo-irreducible since

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√ 1 3 X 3 = X( 3 2X)( √ does not have a 3 X) is a coprime factorization. However, X 2 3 coprime factorization into two elements. For if X = ab, then we can take a = uX and b = u−1 X 2 where u ∈ R. But [uX, u−1X 2 ] = uX. Note that if ∗1 and ∗2 are finite character star-operations on D with ∗1 ≤ ∗2 (i.e., A∗1 ⊆ A∗2 for each A ∈ F (D)), then a ∈ D ∗2 -pseudo-irreducible implies a is ∗1 -pseudo-irreducible and that a [ ]-pseudo-irreducible implies that a is t-pseudoirreducible. Similar statements hold for the corresponding types of pseudo-prime elements. However, for comaximal factorizations we have the reverse situation. If ∗1 ≤ ∗2 , then a ∗1 -comaximal factorization is a ∗2 -comaximal factorization. Thus the relationship between ∗1 -UCFDs and ∗2 -UCFDs is somewhat murky. As in the case of CFDs, a domain D is a ∗-CFD if each nonzero nonunit is either (i) contained in only finitely many minimal primes or (ii) contained in only finitely many maximal ∗-ideals. Also, it is not hard to show that a domain satisfying ACCP is a ∗-CFD. We have the following partial analog of Theorem 4.1. Theorem 4.3. Let ∗ be a finite character star-operation on an integral domain D. Consider the following four conditions. (1 ) D is a ∗-CFD and each two-generated ∗-invertible ∗-ideal is principal. (2 ) D is a ∗-CFD and each ∗-pseudo-irreducible element is ∗-pseudo-prime. (3 ) Every nonzero nonunit of D has a ∗-comaximal factorization into ∗-pseudoprime elements. (4 ) D is a ∗-UCFD. Then (1 )=⇒(2 )⇐⇒(3 )=⇒(4 ). Proof. The proof of (1)=⇒(2)=⇒(4) is the star operation version of [27, Lemma 1.3]. We give the details for the readers not familiar with star-operations. (1)=⇒(2) Let a be ∗-pseudo-irreducible. Suppose that a|bc where (b, c)∗ = D. Then ((a, b), (a, c))∗ = D, so (a, b)∗ ∩ (a, c)∗ = ((a, b)(a, c))∗ . Hence (a) ⊆ (a, b)∗ ∩ (a, c)∗ = ((a, b)(a, c))∗ ⊆ (a); so (a) = ((a, b)(a, c))∗ . So (a, b) and (a, c) are ∗-invertible. Thus (a, b)∗ = (d) and (a, c)∗ = (e) for some d, e ∈ D. So (a) = (d)(e) and hence a = ude for some unit u. But then a ∗-pseudo-irreducible and (d, c)∗ = D gives ud or e is a unit. If ud is a unit, (a) = (e) and c ∈ (a, c)∗ = (e) = (a). Likewise e a unit gives c ∈ (a). So a is ∗-pseudo-prime. (2)=⇒(3) Clear. (3)=⇒(2) Let a be ∗-pseudo-irreducible. Then a has a ∗-comaximal factorization a = a1 · · · an where each ai is ∗-pseudo-prime. Since a is ∗-pseudo-irreducible, n = 1, that is, a = a1 is ∗-pseudo-prime. (2)=⇒(4) Suppose that x1 · · · xn = y1 · · · ym are two complete ∗-comaximal factorizations. By hypothesis, each xi , yj is ∗-pseudo-prime. Now x1 ∗-pseudo-prime and x1 |y1 · · · ym implies x1 |yi for some i which we can take to be 1. Likewise yi |xj for some j. Then x1 |xj and since (x1 , xj )∗ = D for j = 1, we must have j = 1. Thus x1 and y1 are associates. Cancelling x1 gives x2 · · · xn = (uy2 ) · · · ym for some unit u. By induction n = m and after re-ordering xi and yi are associates. Note that if D is a domain satisfying the equivalent conditions of Theorem 3.4, then D is a ∗-UCFD. Indeed, these are precisely the ∗-UCFDs in which each ∗-pseudo-irreducible element is contained in a unique maximal ∗-ideal. We do not know if (4)=⇒(1), even in the case of the t-operation. However, in the onedimensional case we have the following result.

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Theorem 4.4. For a one-dimensional domain D and a finite character staroperation ∗ on D the following are equivalent. (1 ) D is weakly factorial. (2 ) D is a ∗-UCFD. Proof. This follows from Theorem 4.2 and the fact that for D one-dimensional, ∗-Max(D) = Max(D). However, we can not add D is a [ ]-UCFD to Theorem 4.4. For let D = K + XL[[X]] where K ⊆ L are fields. Now D is always a [ ]-CFD, but D is a [ ]-UCFD if and only if K = L, i.e., D is a UFD. Now up to associates the nonunits (atoms) of D have the form uX n where u ∈ L∗ and n ≥ 1 (n = 1). Let u, v ∈ L∗ . Then uX and vX are associates ⇐⇒ uv −1 ∈ K and for n > 1, uX n = (vX)(uv −1 X n−1 ). Hence [uX n1 , vX n2 ] = 1 ⇐⇒ n1 = n2 = 1 and uv −1 ∈ / K. Let a be a nonzero nonunit of D. Then either a is [ ]-pseudo-irreducible or a = a1 · · · an where each ai is a nonunit and [ai , aj ] = 1 for i = j. In the latter case [ai , aj ] = 1 gives that each ai has order one and hence is irreducible and thus [ ]-pseudo-irreducible. Thus D is a [ ]-CFD. Suppose that K  L and let u ∈ L − K. Then uX 2 = uX · X = u2 X · u−1 X = u(u + 1)X · (u + 1)−1 X. Now the first two complete coprime factorizations are distinct unless u2 ∈ K while the first and the third complete coprime factorizations are distinct unless u(u + 1) ∈ K. But then uX 2 having a unique complete coprime factorization gives that u = u(u + 1) − u2 ∈ K, a contradiction.

5

Generalizations of Prime Powers

In this section we summarize the various generalizations of prime powers given in Sections 2–4 and give several new ones (see Figure 1). For each of these prime power generalizations, we consider the domains whose nonzero nonunits are a product of such elements (see Figure 2). The pattern runs as follows. Suppose that D is an integral domain and that P is a property generalizing prime power. Then D could be called a P-domain if each nonzero nonunit of D has a property P and a semi-P-domain if each nonzero nonunit of D is a finite product of elements with property P. In several cases we have seen that D is a semi-P-domain if and only if D is an F -IFC domain where for each P ∈ F, DP is a P-domain. For example, this is the case where property P is “prime power”, “primary”, or “t-pure”. In [6] a nonzero nonunit q was defined to be a prime quantum if q satisfies Q1 : For every nonunit r|q, there exists a natural number n with q|rn , Q2 : For every natural number n, if r|q n and s|q n , then r|s or s|r, and Q3 : For every natural number n, each element t with t|q has the property that if t|ab, then t = t1 t2 where t1 |a and t2 |b. Thus Q2 says that each power of q is rigid and Q3 says that q is completely primal (since a product of completly primal elements is completely primal). A prime power pn is a prime quantum and a prime quantum q is primary. An integral domain D is a generalized unique factorization domain (GUFD) if every nonzero nonunit of D is a product of prime quanta. Clearly a UFD is a GUFD and a GUFD is weakly factorial and a GCD domain [6]. M. Zafrullah [33] considered yet another generalization of a prime power. He  defined a nonzero nonunit x ∈ D to be a packet if (x) is prime, that is, there is a

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unique minimal prime over (x). He studied GCD domains, called unique representations domains (URD ), with the property that every nonzero nonunit is a product of packets. In a URD, each nonzero nonunit has only finitely many primes minimal over it. Thus by Theorem 4.1, a URD is a UCFD. For a study of domains with this and related properties, see [25]. The various generalizations of a prime element are indicated in Figure 1.

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prime    =  irreducible B B

prime power





B BN rigid 





Z Z Z ~ Z





prime quantum A





A AA U





homogeneous  A

completely primal A

A AAU

strongly homogeneous A

-

primary Q Q
Q s Q


primal

packet

t-pure

A A U A

t-pseudo-irreducible

-

Figure 1.

A AAU

pseudo-irreducible

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Corresponding to Figure 1, we have Figure 2 showing the various generalizations of UFDs. Again, the only implications are the obvious ones.

atomic domain 

? semi-rigid domain 



semi-strongly-homogeneous domain A

A AA U

semi-homogeneous domain 

UFD

?

GCD domain :   GUFD @ @ R @ A A Schreier domain AAU weakly factorial domain




?




pre-Schreier domain

semi-t-pure domain H HH HH HH j H ? t-UCFD

Figure 2.

UCFD

Bibliography [1] D.D. Anderson, Star operations induced by overrings, Comm. Algebra 16 (1988), 2535–2553. [2] D.D. Anderson, Extensions of unique factorization: A survey, Advances in Commutative Ring Theory (Fez, 1997), 31-53, Lecture Notes in Pure and Appl. Math., 205, Dekker, New York, 1999. [3] D.D. Anderson, D.F. Anderson, and M. Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra 69 (1990), 1–19. [4] D.D. Anderson, D.F. Anderson, and M. Zafrullah, Splitting the t-class group, J. Pure Appl. Algebra 74 (1991), 17–37. [5] D.D. Anderson, D.F. Anderson, and M. Zafrullah, Atomic domains in which almost all atoms are prime, Comm. Algebra 20 (1992), 1447–1462. [6] D.D. Anderson, D.F. Anderson, and M. Zafrullah, A generalization of unique factorization, Boll. Un. Mat. Ital. A(7) 9 (1995), 401–413. [7] D.D. Anderson and S.J. Cook, Two star-operations and their induced lattices, Comm. Algebra 28 (2000), 2461–2475. [8] D.D. Anderson and T. Dumitrescu, Condensed domains, Canad. Math. Bull. 46 (2003), 3–13. [9] D. D. Anderson and L.A. Mahaney, On primary factorizations, J. Pure Appl. Algebra 54 (1988), 141–154. [10] D.D. Anderson, J. L. Mott, and M. Zafrullah, Finite character representations for integral domains, Boll. Un. Mat. Ital. B(7) 6 (1992), 613–630. [11] D.D. Anderson, J.L. Mott, and M. Zafrullah, Unique factorization in nonatomic integral domains, Boll. Unione Mat. Ital. Sez B Artic Ric. Mat. (8) 2 (1999), 341–352. [12] D.D. Anderson and R.O. Quinterro, Some generalizations of GCD-domains, Factorization in Integral Domains, 189–195, Lecture Notes in Pure and Appl. Math., 189, Dekker, New York, 1997. [13] D.D. Anderson and M. Zafrullah, Weakly factorial domains and groups of divisibility, Proc. Amer. Math. Soc. 109 (1990), 907–913. 19

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[14] D.D. Anderson and M. Zafrullah, On a theorem of Kaplansky, Boll. Un. Mat. Ital. A(7) 8 (1994), 397–402. [15] D.D. Anderson and M. Zafrullah, P.M. Cohn’s completely primal elements, Zero Dimensional Commutative Rings, 115–123, Lecture Notes in Pure and Appl. Math., 171, Dekker, New York, 1995. [16] D.D. Anderson and M. Zafrullah, Independent locally finite intersections of localizations, Houston J. Math. 25 (1999), 433–452. [17] D.D. Anderson and M. Zafrullah, Splitting sets in integral domains, Proc. Amer. Math. Soc. 129 (2001), 2209–2217. [18] D.F. Anderson, The class group and local class group of an integral domain, 33-55, Non-Noetherian Commutative Ring Theory, Mathematics and Its Applications, vol. 520, Kluwer Academic Publishers, Dordrect/Boston/London, 2000. [19] P.M. Cohn, Bezout rings and their subrings, Proc. Cambridge Philos. Soc. 64 (1968), 251–264. [20] Wang Fanggui and R.L. McCasland, On w-modules over strong Mori domains, Comm. Algebra 25 (1997), 1285–1306. [21] R. Gilmer, Multiplicative Ideal Theory, Queen’s Papers in Pure and Appl. Math. 90, Queen’s University, Kingston, Ontario, 1992. [22] M. Griffin, Rings of Krull type, J. Reine Angew. Math. 229 (1968), 1–17. [23] F. Halter-Koch, Ideal Systems - An Introduction to Multiplicative Ideal Theory, Pure and Applied Mathematics, vol. 211, Dekker, New York, 1998. [24] P. Jaffard, Les Syst`emes d’Ideaux, Travaux et Recherches Math´ematiques, vol IV, Dunod, Paris, 1960. [25] P. Malcolmson and F. Okoh, Minimal prime ideals and generalizations of factorial domains, preprint. [26] E. Matlis, Torsion-free Modules, The University of Chicago Press, ChicagoLondon, 1972. [27] S. McAdam and R.G. Swan, Unique comaximal factorization, J. Algebra (to appear). [28] J.L. Mott, Convex directed subgroups of the group of divisibility, Canad. J. Math. 26 (1974), 532–542. [29] J.L Mott and M. Schexnayder, Exact sequences of semi-value groups, J. Reine Angew. Math. 283/284 (1976), 388–401. [30] P. Ribenboim, Anneaux normaux, r´eels `a caract`ere fini, Summa Brasil Math. 3 (1956), 213–253. [31] M. Zafrullah, Semirigid GCD domains, Manuscripta Math. 17 (1975), 55–66.

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[32] M. Zafrullah, Rigid elements in GCD domains, J. Natur. Sci. Math. 17 (1977), 7–14. [33] M. Zafrullah, Unique representation domains, J. Natur. Sci. Math. 18 (1978), 19–29. [34] M. Zafrullah, On a property of pre-Schreier domains, Comm. Algebra 15 (1987), 1895–1920. [35] M. Zafrullah, Putting t-invertibility to use, 429-457, Non-Noetherian Commutative Ring Theory, Mathematics and Its Applications, vol. 520, Kluwer Academic Publishers, Dordrect/Boston/London, 2000.

Chapter 2

Divisibility Properties in Graded Integral Domains by

David F. Anderson Abstract Let R be a graded integral domain. In this chapter, we study the extent to which conditions on the homogeneous elements or the homogeneous ideals of R determine divisibility properties of R.

1

Introduction

There are many divisibility properties one can consider on an integral domain R. They range from chain conditions and closedness properties to lengths of factorizations and class groups. If, in addition, R is a graded ring, then in some cases we need only consider homogeneous information. Here we investigate to what extent conditions on the monoid S of nonzero homogeneous elements of R or the homogeneous ideals of R determine divisibility properties of R. It is somewhat surprising that in some cases divisibility properties of R are completely determined by S; this depends on the divisibility property considered and how R is graded. Things behave best when R is a Z+ -graded integral domain or a semigroup ring. Many standard ring-theoretic constructions such as polynomial rings, semigroup rings, and the A + XB[X] construction yield graded integral domains. These constructions illustrate the diversity in graded integral domains, and they have been very useful in constructing examples. We are also interested in determining when these various classes of graded integral domains satisfy certain divisibility conditions. The literature on these constructions is vast; so we will not needlessly duplicate it. For more on semigroup rings, we refer the reader to [54], and for the A + XB[X] and the related D + XDS [X] constructions, we refer the reader to [74], [44], and [89]. This chapter has six sections including the introduction. In the second section, we review some basic facts about graded integral domains and the homogeneous quotient field of a graded integral domain. In the third section, we discuss several 22

Divisibility Properties in Graded Integral Domains

23

divisibility properties on a graded integral domain and investigate the extent to which conditions on the homogeneous elements or homogeneous ideals of R determine divisibility properties of R. In the fourth section, we look at some factorization properties in several special classes of graded integral domains, with emphasis on semigroup rings and graded subrings of polynomial rings. In the fifth section, we discuss the Picard group and (t-)class group of a graded integral domain. In the final section, we give several examples using graded integral domains, usually semigroup rings. Of course, there are many divisibility properties we do not discuss. We have chosen those divisibility properties we have worked on and are most interested in. Hopefully, we have supplied enough additional references for the interested reader to pursue other properties. Our notation is standard. We will usually denote an integral domain by either D or R, a field by K, a monoid by Γ, and a torsionfree abelian group by G. Given an integral domain R, we will denote its quotient field and group of units by qf (R) and U (R), respectively. By the dimension of a ring, we will always mean Krull dimension. Usually, X, Y, Z, W , or {Xα } will denote indeterminates. The nonzero elements of a subset A of a ring R will be denoted by A∗ . Given a ring D  and an additive semigroup Γ, the semigroup ring will be denoted by D[Γ] = { as X s | as ∈ D, s ∈ Γ } with X s X t = X s+t . As usual, N, Z, Q, R, C, and Z/nZ will denote the positive integers, integers, rational numbers, real numbers, complex numbers, and the integers modulo n, respectively. For a partially ordered abelian group G, let G+ = { x ∈ G | x ≥ 0 }. References for any undefined concepts or notation are [53], [54], [49], [80], [64].

2

Graded Domains and the Homogeneous Quotient Field

In this section, we review some basic properties of graded integral domains which will be used throughout this article. Let R = R0 ⊕ R1 ⊕ · · · be a Z+ -graded integral domain. That is, each Rn is an additive subgroup of R and Ri Rj ⊆ Ri+j for all i, j ∈ Z+ . We will say that a nonzero x ∈ Rn is homogeneous of degree n, and deg(0) = 0. Thus each x ∈ R may be written uniquely as x = x1 + · · · + xn with deg(xi ) = mi and m1 < . . . < mn . In this case, R0 is a subring of R, and (R0 )∗ is a saturated multiplicatively closed subset of R. Thus U (R) = U (R0 ). Also, the set S = { 0 = x ∈ R | x is homogeneous } = ∪n∈Z+ (Rn )∗ of nonzero homogeneous elements of R is a saturated multiplicatively closed subset of R. Then RS is a Z-graded integral domain via deg(a/s) = deg(a) − deg(s) for all a, s ∈ S. We will call RS the homogeneous quotient field of R; each nonzero homogeneous element of R is a unit. Note that (RS )0 is a field, and moreover, RS = (RS )0 [t, t−1 ] for any homogeneous t ∈ RS of smallest positive degree (such a t is transcendental over (RS )0 ), and hence RS is a PID. The simplest example of a Z+ -graded integral domain is the polynomial ring R = D[X] over an integral domain D with each Rn = DX n . More generally, let {Dn }n∈Z+ be a family of subrings of an integral domain D with Dn ⊆ Dn+1 for all n ∈ Z+ . Then R = ⊕n∈Z+ Dn X n is a Z+ -graded integral domain with each Rn = Dn X n . Special cases of this construction which have received considerable

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Anderson

attention, often for constructing examples, are the A + XB[X] construction and the D + XDS [X] construction. Let Γ be a commutative, cancellative monoid, written additively, with the property that nx = ny for x, y ∈ Γ and n a positive integer implies that x = y. We will call such a monoid a torsionless grading monoid. In this case, Γ is a submonoid of its quotient group, the torsionfree abelian group Γ = {a − b | a, b ∈ Γ}. (Thus a torsionless grading monoid is just a submonoid of a torsionfree abelian group.) The group of units of Γ is U (Γ) = Γ ∩ −Γ. Note that a cancellative monoid can be totally ordered if and only if it is torsionless. Also, the semigroup ring D[Γ] is an integral domain if and only if D is an integral domain and Γ is a torsionless grading monoid. General references for torsionless grading monoids are [54] and [80]. Let Γ be a torsionless grading monoid (which we will always assume to be totally ordered). In this article, we will study Γ-graded integral domains. That is, R = ⊕α∈Γ Rα , where each Rα is an additive subgroup of R and Rα Rβ ⊆ Rα+β for all α, β ∈ Γ. We say that a nonzero x ∈ Rα is homogeneous of deg(x) = α, and we will usually make the harmless assumption that each Rα is nonzero. In this case, R0 is a subring of R, and thus (R0 )∗ is multiplicatively closed, but (R0 )∗ is saturated if and only if Γ ∩ −Γ = 0. Let S = ∪α∈Γ (Rα )∗ be the saturated multiplicatively closed set of nonzero homogeneous elements of R. Note that U (R) ⊆ S. The homogeneous quotient field RS of R is Γ-graded with (RS )α = { a/s | a, s ∈ S with deg(a) − deg(s) = α } ∪ {0} for each α ∈ Γ. The best example of a Γ-graded integral domain is the semigroup ring R = D[Γ] over an integral domain D, graded by deg(aX α ) = α for all a ∈ D∗ and α ∈ Γ. For R = D[Γ], its set of nonzero homogeneous elements is S = { aX α | a ∈ D∗ , α ∈ Γ }, and its homogeneous quotient field RS is just the group ring K[G], where K = qf (D) and G = Γ. So, when we say a “graded integral domain R”, we mean an integral domain R graded by a torsionless grading monoid Γ. A general reference for graded integral domains is [80]. We have seen that RS = (RS )0 [Z] when R is a Z+ - or Z-graded integral domain, and in this case, RS is a UFD (in fact, a PID). However, in general RS need be neither a UFD nor a group ring over Γ. The UFD case is easy. Just let R = K[Q] for any field K; then RS = R is not a UFD because ACCP fails since (1 − X) ⊆ (1 − X 1/2 ) ⊆ (1 − X 1/4 ) ⊆ . . . is a properly ascending chain of principal ideals of R (see Theorem 2.3(a) below). We next show that RS need not be a group ring. Example 2.1. Let D be an integral domain, Γ a torsionless grading monoid, and T ⊆ Γ such that [T ] = Γ, where [T ] is the submonoid generated by T . Define R = D[{Xα }α∈T ]. Then R is Γ-graded with deg(Xαn11 · · · Xαnrr ) = n1 α1 + · · · + nr αr . In fact, R = D[Γ ], where Γ is the free abelian monoid on T. Let D be a UFD and T = Γ = Q+ . Then R, and hence RS , is a UFD. If RS were a group ring over Q, then necessarily RS ∼ = (RS )0 [Q]. But the group ring (RS )0 [Q] is not a UFD since ACCP fails. So R is not isomorphic to a group ring over Q. However, the homogeneous quotient field RS is a twisted group ring. Let K = (RS )0 and G = Γ. For each α ∈ G, pick a nonzero Yα ∈ (RS )α . Then (RS )α = KYα and Yα Yβ = γ(α, β)Yα+β for some nonzero γ(α, β) ∈ K. Thus RS = (RS )γ0 [G] is a twisted group ring (see [34] for more details). But, we may not always be able to choose the Yα ’s so that each γ(α, β) = 1. Some cases when RS is a group ring or

Divisibility Properties in Graded Integral Domains

25

a UFD are given in [12]. For example, RS is a UFD if Γ satisfies ACC on cyclic subgroups [12, Proposition 3.5] (note that the converse fails by Example 2.1). Although RS need not be a UFD, it is “close” to being a UFD. In fact, by our next result, RS is a UFD if and only if it is a Krull domain, if and only if it satisfies ACCP. Theorem 2.2. Let R be a graded integral domain and S its set of nonzero homogeneous elements. Then RS is a completely integrally closed GCD-domain. Proof. The ring RS is completely integrally closed by [12, Proposition 3.2] and is a GCD-domain by [12, Proposition 3.3]. For future reference, we record the following result for group rings. The semigroup ring case will be discussed in the next section. The condition that G has ACC on cyclic subgroups is equivalent to each rank-one subgroup of G being free (cyclic) [12, Lemma 2.5] or each nonzero element of G having type (0, 0, 0, . . . ) (see [51, Section 85]). We will always use the ACC on cyclic subgroups condition. Theorem 2.3. Let D be an integral domain and G a torsionfree abelian group. (a) D[G] is a UFD if and only if D is a UFD and G satisfies ACC on cyclic subgroups. (b) D[G] is a Krull domain if and only if D is a Krull domain and G satisfies ACC on cyclic subgroups. Proof. Part (a) is due to Gilmer and Parker [57, Theorem 7.13], and part (b) is due to Matsuda [75, Proposition 3.3]. They each used the type (0, 0, 0, . . .) condition on G. Following Cohn [42], we say that an extension A ⊆ B of integral domains is inert if whenever xy ∈ A for x, y ∈ B, then x = ru and y = su−1 for some r, s ∈ A and u ∈ U (B). Clearly R0 ⊆ R = ⊕α∈Γ Rα is inert if and only if Rα contains a unit for each α ∈ Γ ∩ −Γ. For example, this happens if Γ ∩ −Γ = 0, R0 is a field, or R = R0 [Γ]. In particular, if R is Z+ -graded, then R0 ⊆ R is always an inert extension. We will see that things usually behave better when R0 ⊆ R is an inert extension. An integral ideal I of R = ⊕α∈Γ Rα is homogeneous if I = ⊕α∈Γ (I ∩ Rα ); equivalently, if I is generated by homogeneous elements. A fractional ideal I of R is homogeneous if sI is an integral homogeneous ideal of R for some s ∈ S (thus I ⊆ RS ). For x = x1 + · · · + xn ∈ RS with each xi ∈ (RS )αi and α1 < . . . < αn , we define the content of x to be the homogeneous ideal C(x) = (x1 , . . . , xn ). Thus a fractional ideal I ⊆ RS is homogeneous if and only if C(x) ⊆ I for each x ∈ I. For x, y ∈ R, we have C(x)n C(xy) = C(x)n+1 C(y) for some integer n ≥ 0 [79]. A subring A of R is a graded subring if A = ⊕α∈Γ (A ∩ Rα ). In a similar manner, we define a graded overring of R to be a graded subring of RS containing R.

3

Divisibility Properties of Graded Domains

Divisibility properties of an integral domain R are often equivalent to the corresponding divisibility properties in the multiplicative monoid R∗ . Some of these equivalences are obvious (R is a UFD iff R∗ is a factorial monoid); others are more

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difficult (R is a Krull domain iff R∗ is a Krull monoid). In fact, nowadays one often studies divisibility properties in the more general context of a commutative, cancellative monoid (for example, see [64] or [40], and the next paragraph). In this section, we investigate to what extent divisibility properties of R are actually determined by the corresponding divisibility properties on the monoid S of nonzero homogeneous elements of R, or by conditions on the homogeneous ideals of R. One can also ask how divisibility properties of R0 relate to those of R. In other words, how do divisibility properties on the submonoids (R0 )∗ and S of R∗ relate to divisibility properties of R (or R∗ )? Let H be a commutative, cancellative (multiplicative) monoid with quotient group H = { a/b | a, b ∈ H }. Ring-theoretic properties can often be translated to monoid properties. This is obvious if it is a purely multiplicative property, and we usually won’t give the explicit definition in this case. For example, we say that H is integrally closed (resp., completely integrally closed) if xn ∈ H (resp., hxn ∈ H for some h ∈ H) for x ∈ H and some (resp., all) n ∈ N implies x ∈ H. Also, H is a Krull monoid if H is completely integrally closed and H satisfies ACC on integral v-ideals (for other equivalent conditions, see [41], [40], or [64]). Recall that a torsionless grading monoid Γ is written additively; so here integrally closed would mean that nx ∈ Γ for some n ∈ N and x ∈ Γ implies x ∈ Γ. Let R be an integral domain with quotient field K. Then its group of divisibility is the abelian group G(R) = K ∗ /U (R) partially ordered by xU (R) ≤ yU (R) if and only if y/x ∈ R. Divisibility properties of R are often reflected in its group of divisibility. For example, R is a GCD-domain (resp., valuation domain) if and only if G(R) is lattice ordered (resp., totally ordered). In a similar manner, for a graded integral domain R, we define its homogeneous group of divisibility to be HG(R) = U (RS )/U (R). It is a partially ordered subgroup of G(R). We first give two examples to show that divisibility properties of R need not be determined by S. We call a graded integral domain R a graded valuation domain if for x, y ∈ S, then either x|y or y|x (cf. [66]); equivalently, HG(R) is a totally ordered abelian group. Certainly RS is always a graded valuation domain, and any polynomial ring over a field is a graded valuation domain, but neither is a valuation domain. So a condition on the nonzero homogeneous elements of R need not force that given condition on all the nonzero elements of R. Secondly, for the group ring R = K[G] over a field K, its set of nonzero homogeneous elements S = { aX α | a ∈ K ∗ , α ∈ G } ∼ = K ∗ × G is certainly a very nice monoid and satisfies almost any conceivable divisibility property. For example, S is always a factorial monoid, but R need not be a UFD (cf. Theorem 2.3(a) and Theorem 3.3). A given integral domain R may be graded in many different ways (cf. [3]); so R0 and S depend on the given grading of R. Our next example illustrates this problem. Example 3.1. Let R = Q[X, Y, Z, W ]. Then R is a 4-dimensional Noetherian UFD. We consider three very different gradings on R. (a) R is Z+ -graded in the usual way with degX = degY = degZ = degW = 1. In this case, R0 = Q is a UFD and R0 ⊆ R is an inert extension. (b) R is (Z+ )4 -graded with degX = (1, 0, 0, 0), degY = (0, 1, 0, 0), degZ = (0, 0, 1, 0), and degW = (0, 0, 0, 1). In this case, R = Q[(Z+ )4 ] is a semigroup ring, R0 = Q is a UFD, and R0 ⊆ R is an inert extension. (c) R is Z-graded with degX = degY = 1 and degZ = degW = −1. In this

Divisibility Properties in Graded Integral Domains

27

case, R0 = Q[XZ, XW, Y Z, Y W ] is not a UFD. In fact, R0 is a Krull domain with Cl(R0 ) = Z [49, Proposition 14.8]. Note that R0 ⊆ R is not an inert extension. As in [1], we define a graded integral domain R to be a graded GCD-domain if each pair of nonzero homogeneous elements of R has a (necessarily homogeneous) GCD; equivalently, S is a GCD-monoid, or HG(R) is a lattice ordered abelian group. Example 3.1(c) shows that R a graded GCD-domain does not imply that R0 is a GCD-domain. However, if R0 ⊆ R is an inert extension and R is a graded GCD-domain, then it is easily proved that R0 is a GCD-domain. The key fact that gives the equivalence in the next theorem is that RS is a GCD-domain. Theorem 3.2. Let R be a graded integral domain and S its set of nonzero homogeneous elements. Then R is a GCD-domain if and only if S is a GCD-monoid. Thus R is a GCD-domain if and only if R is a graded GCD-domain. Proof. This is proved in [1, Theorem 3.4] using [57, Theorem 3.1]. In a similar manner, we define a graded integral domain R to be a graded UFD if each homogeneous nonzero nonunit of R is a product of (necessarily homogeneous) prime elements of R; equivalently, S is a factorial monoid, or HG(R) is orderisomorphic to a free abelian group with the usual product order. We have already seen that a graded UFD need not be a UFD since RS , which is trivially a graded UFD, need not be a UFD. Example 3.1(c) also shows that R0 need not be a UFD when R is a graded UFD. However, R0 is a UFD when R is a graded UFD and R0 ⊆ R is an inert extension. We next give the analog of Theorem 3.2 for UFDs. But in this case, and the next, the graded and nongraded conditions are not equivalent. Theorem 3.3. Let R be a graded integral domain and S its set of nonzero homogeneous elements. Then R is a UFD if and only if S is a factorial monoid and RS is a UFD. In particular, if R is Z+ - or Z-graded, then R is a UFD if and only if R is a graded UFD. Proof. This is proved in [1, Theorem 4.4]. The “in particular” statement follows since RS ∼ = (RS )0 [Z] is a UFD (PID) when R is Z+ - or Z-graded. This case was first proved in [10, Theorem 5]. We next consider the Krull domain case. We define a graded integral domain R to be a graded Krull domain if R is completely intregrally closed with respect to homogeneous elements of RS (i.e., a/b almost integral over R for a, b ∈ S implies a/b ∈ R) and R satisfies ACC on homogeneous integral v-ideals; equivalently, S is a Krull monoid. However, unlike the previous two cases, if R is a graded Krull domain, then R0 is also a Krull domain [1, Corollary 5.9]. Theorem 3.4. Let R be a graded integral domain and S its set of nonzero homogeneous elements. Then R is a Krull domain if and only if S is a Krull monoid and RS is a Krull domain (UFD). In particular, if R is Z+ - or Z-graded, then R is a Krull domain if and only if R is a graded Krull domain. Proof. This is proved in [1, Theorem 5.8]. We have already observed that RS is a Krull domain if and only if it is a UFD. The “in particular” statement follows since RS is a UFD (PID) in this case.

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We next recall several closedness properties on an integral domain R. For a positive integer n, we say that R is n-root closed if xn ∈ R for x ∈ qf (R) implies x ∈ R, and that R is root closed if R is n-root closed for each positive integer n. Also, R is seminormal if x2 , x3 ∈ R for x ∈ qf (R) implies x ∈ R. An integrally closed domain is root closed, and an n-root closed (for some n ≥ 2) domain is seminormal, but neither converse holds (see Example 6.7). The seminormal property will be useful in Section 5 when we discuss Picard groups. By a result of Swan [84], R is seminormal if and only if whenever x2 = y 3 for x, y ∈ R, then x = z 3 and y = z 2 for some z ∈ R. In the obvious manner, we can define a (n-)root closed or seminormal monoid (for seminormality, we can use either equivalent condition). Note that for monoids, the definitions of integrally closed and root closed are the same. The n-root closed version of the next theorem is left to the reader. Theorem 3.5. Let R be a graded integral domain and S its set of nonzero homogeneous elements. Then the following statements are equivalent. (a) R is root closed (resp., seminormal). (b) R is root closed (resp., seminormal) with respect to homogeneous elements of RS , i.e., if xn ∈ R for some positive integer n (resp., x2 , x3 ∈ R) for a homogeneous x ∈ RS , then x ∈ R. (c) S is a root closed (resp., seminormal) monoid. Proof. The root closed (resp., seminormal) part is given in [16, Theorem 2.4] (resp., [2, Theorem 6.1]). Theorem 3.6. Let R be a graded integral domain and S its set of nonzero homogeneous elements. (a) R is integrally closed (resp., completely integrally closed) if and only if R is integrally closed (resp., completely integrally closed) with respect to homogeneous elements of RS , i.e., if a homogeneous x ∈ RS is integral (resp., almost integral) over R, then x ∈ R. (b) R is completely integrally closed if and only if S is a completely integrally closed monoid. Proof. (a) The integrally closed (resp., completely integrally closed) case is given in [1, Theorem 5.4] (resp., [1, Theorem 5.2]). The integrally closed case also follows from [66, Theorem 2.10]. (b) This follows directly from part (a). The above seven properties P all satisfy: the graded integral domain R is a Pdomain if and only if R is a “graded P-domain” and RS is a P-domain. Note that RS is always a completely integrally closed GCD-domain, and hence is integrally closed, root closed, and seminormal. Also, as noted above, RS is a UFD if and only if it is a Krull domain. This equivalence also holds for π-domains, PVMDs, and G-GCD domains (see [1] and [2]). Note that RS is a valuation domain if and only if R is trivially graded, i.e., R = R0 , so RS = qf (R). Thus this property also has the above form. We next give the semigroup ring analogs of Theorem 2.3 which were promised in the previous section. These are all fairly immediate consequences of the previous theorems.

Divisibility Properties in Graded Integral Domains

29

Theorem 3.7. Let D be an integral domain and Γ a torsionless grading monoid with group of units U (Γ) = Γ ∩ −Γ. (a) D[Γ] is a GCD-domain if and only if D is a GCD-domain and Γ is a GCDmonoid. (b) D[Γ] is a UFD if and only if D is a UFD, Γ is a factorial monoid, and U (Γ) satisfies ACC on cyclic subgroups. (c) D[Γ] is a Krull domain if and only if D is a Krull domain, Γ is a Krull monoid, and U (Γ) satisfies ACC on cyclic subgroups. (d ) D[Γ] is integrally closed if and only if D is integrally closed and Γ is a root closed monoid. (e) D[Γ] is completely integrally closed if and only if D is completely integrally closed and Γ is a completely integrally closed monoid. (f ) D[Γ] is (n-)root closed if and only if D is (n-)root closed and Γ is a (n-)root closed monoid. (g) D[Γ] is seminormal if and only if D is seminormal and Γ is a seminormal monoid. Proof. We prove parts (f) and (g); the proofs of the others may all be found in [54]. If D[Γ] is root closed (resp., seminormal), then clearly D and Γ are each root closed (resp., seminormal). Conversely, to show that D[Γ] is root closed (resp., seminormal), by Theorem 3.5 we need only check that (aX α )n ∈ D[Γ] (resp., (aX α )2 , (aX α )3 ∈ D[Γ]) for aX α ∈ qf (D)[Γ] implies aX α ∈ D[Γ]. But this is clear if D and Γ are each root closed (resp., seminormal). Remark 3.8. (a) By Theorem 3.7, If D is an integrally closed domain, then R = D[Γ] is integrally closed if and only if it is root closed. (b) We give a more concrete description of factorial and Krull monoids. A factorial monoid has the form G ⊕ F+ and a Krull monoid has the form G ⊕ T , where G is a torsionfree abelian group, F = ⊕Zα is a free abelian group with the usual product order, and T is a submonoid of F such that T = T  ∩ F+ . Thus a UFD semigroup ring is just a polynomial ring over a UFD group ring, and a Krull domain semigroup ring is just a subring of a polynomial ring over a Krull group ring generated by monomials (cf. [1, pages 204 and 209], [41, Proposition 1], and [57, Lemma 7.15]). (c) By Theorem 3.7 (with Γ = Z+ ), we have that D[X] is a P-domain if and only if D is a P-domain. Thus it seems natural to ask if a graded integral domain R satisfies property P if and only if R0 satisfies property P and RS satisfies property P. However, this fails for all seven properties in Theorem 3.7. Let R = Z + ZX + X 2 Q[X] be considered as a graded subring of Q[X]. Here R0 = Z and RS = Q[X, X −1 ] satisfy all seven properties, but R does not satisfy any of them.

4

Factorization Properties in Graded Domains

In this section, we consider several divisibility properties related to factorizations for an integral domain R (they all translate in the obvious way to monoids). We say that R is atomic if each nonzero nonunit of R is a product of a finite number of irreducible elements (atoms) of R. We say that R is a bounded factorization domain (BFD) if R is atomic and for each nonzero nonunit x of R there is a bound

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on the length of factorizations of x into irreducible elements. We say that R is a half-factorial domain (HFD) if R is atomic and each factorization of a nonzero nonunit of R into a product of irreducible elements has the same length. Finally, R is a finite factorization domain (FFD) if each nonzero nonunit of R has only a finite number of factorizations up to order and associates. These properties, and several others, have been studied systematically in [6], [7], and [9]. We have the following implications (no implication is reversible, see [6] and Examples 6.1-6.4). UFD ⇓ FFD

⇒ HF D ⇓ ⇒ BF D

⇒ ACCP

⇒ atomic

It is well-known that R[X] satisfies ACCP if and only if R satisfies ACCP. Also, R[X] is a BFD (resp., FFD) if and only if R is a BFD (resp., FFD) [6, Proposition 2.5 (resp., Proposition 5.3)]. Clearly R is atomic (resp., an HFD) if R[X] is atomic (resp., an HFD). However, R an atomic integral domain does not imply that R[X] is atomic [83]. For R a Krull domain, R[X] is an HFD if and only if |Cl(R)| ≤ 2 [90, Theorem 2.4]. Also, R = R + Y C[Y ] is an HFD (see Theorem 4.3(a)), but R[X] is not an HFD [7, Example 5.4]. In fact, if R[X] is an HFD, then R is necessarily integrally closed [45, Theorem 2.2]. For more on HFDs, see the survey article [39]. Several divisibility properties are studied for group rings and semigroup rings in [54] and [70]. We mention three for group rings. Theorem 4.1. Let D be an integral domain and G a torsionfree abelian group. Then D[G] satisfies ACCP (resp., is a BFD, FFD) if and only if D satisfies ACCP (resp., is a BFD, FFD) and G satisfies ACC on cyclic subgroups. Proof. The ACCP case is proved in [54, Theorem 14.17], and the BFD and FFD cases are from [70, Theorem 3.1] and [70, Theorem 4.1], respectively. We next consider Z+ -graded integral domains of the form R = ⊕n∈Z+ Dn X n , where {Dn }n∈Z+ is a family of subrings of an integral domain D with Dn ⊆ Dn+1 for all n ∈ Z+ . Various divisibility and factorization properties of R have been studied in [65], [58], [68], and [35]. For example, if D0 is a field, then R is atomic, satisfies ACCP, and is a BFD; but R need not be an HFD or an FFD (see Theorems 4.2 and 4.3). However, R is completely integrally closed if and only if Dn = D0 for all n ∈ Z+ and D0 is completely integrally closed. Thus R is a UFD (resp., Krull domain) only in the trivial case when R = D0 [X] and D0 is a UFD (resp., Krull domain). The other closedness properties we have studied all force Dn = D1 for all n ∈ N. Thus we will concentrate on the special case of the A + XB[X] construction. There has also been considerable work on the related A + XB[[X]] construction. In fact, most papers treat both the A + XB[X] and A + XB[[X]] constructions simultaneously. We will just give a few results for the A + XB[X] construction; for many more see [7], [27], [32], [33], [38], [44], [47], [48], [58], [59], [60], [61], [65] [68] [69], [74], [87], [88],and [89]. For the definition of splitting set in part (d), see the next section. Theorem 4.2. Let A ⊆ B be an extension of integral domains and R = A+XB[X]. (a) R is completely integrally closed (resp., a UFD, Krull domain) if and only if A = B and A is completely integrally closed (resp., a UFD, Krull domain).

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(b) R is integrally closed (resp., root closed, seminormal) if and only if B is integrally closed (resp., root closed, seminormal) and A is integrally closed in qf (B) (resp., root closed in qf(B), seminormal). (c) R satisfies ACCP if and only if U (B) ∩ A = U (A) and every ascending chain b1 B ⊆ b2 B ⊆ . . . with each bn ∈ B nonzero and bn /bn+1 ∈ A terminates. (d ) R is a GCD-domain if and only if A is a GCD-domain and B = AS for S a splitting multiplicative set of A. (e) R is a BFD if and only if U (B) ∩ A = U (A) and for each nonzero nonunit b ∈ B, there is an N (b) ∈ N such that whenever b = b1 · · · bn with each bi ∈ B a nonunit, then at most N (b) of the bi ’s are in A. (f ) R is an FFD if and only if B is an FFD and U (B)/U (A) is finite. Proof. We have already mentioned part (a). Part (b) is clear by comments in the previous section. Part (c) is from [32, Proposition 1.1]; part (d) is from [33, Theorem 2.10]; part (e) is from [32, Proposition 2.1]; and part (f) is from [32, Proposition 3.1] and uses facts from [11]. The most popular problem seems to be to determine conditions for A + XB[X] to be an HFD. Partial results have appeared in many of the papers listed above. Probably the simplest example of an HFD which is not a UFD, is k + XK[X] for any proper extension k  K of fields [7, Theorem 5.3]. Below we give three conditions for A + XB[X] to be an HFD. A surprising consequence of part (b) is that A + XB[X] can be an HFD without B[X] being an HFD. (Note that A is an HFD if A + XB[X] is an HFD.) Theorem 4.3. Let A ⊆ B be an extension of integral domains and R = A+XB[X]. (a) If B is a field, then R is an HFD if and only if A is a field. (b) If A is a field, then R is an HFD if and only if B is integrally closed. (c) If B is a UFD, U (B) ∩ A = U (A), and each irreducible element of A is also irreducible in B, then R is an HFD. Proof. Part (a) is from [7, Theorem 5.3], and part (b) is from [47, Theorem 2.1]. For part (c), see [59, Proposition 1.8] and [32, Proposition 5.4]. For an atomic integral domain R, the elasticity of R is ρ(R) = sup{ m/n | x1 · · · xm = y1 · · · yn for irreducible xi , yj ∈ R }. (In a similar manner, one can define the elasticity ρ(H) of an atomic monoid H.) Then 1 ≤ ρ(R) ≤ ∞, and ρ(R) = 1 if and only if R is an HFD. Not much more can be said since for each real number r ≥ 1 or r = ∞, there is a Dedekind domain R with torsion class group and ρ(R) = r [4, Theorem 3.2]. This concept is due to Valenza [85]. For a recent survey article on the elasticity, see [20]. Let R be a graded integral domain with S its monoid of nonzero homogeneous elements. It is clear that ρ(R0 ) ≤ ρ(R) if R0 ⊆ R is an inert extension and that ρ(S) ≤ ρ(R). The next example shows that a “graded HFD” need not be an HFD and that we may have ρ(R0 ) > ρ(R). Example 4.4. Let R = Q[X, Y, Z, W ] be Z-graded with degX = degY = 1 and degZ = degW = −1 as in Example 3.1(c). Here R is a UFD, and hence an HFD. In this case, R0 = Q[XZ, XW, Y Z, Y W ] = Q[Γ], where Γ = (1, 0, 1, 0), (1, 0, 0, 1), (0, 1, 1, 0), (0, 1, 0, 1) ⊆ (Z+ )4 , is not a UFD. We observed in Example 3.1(c) that

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R0 is a Krull domain with Cl(R0 ) = Z. Thus ρ(R0 ) = ∞ since each nonzero divisor class of Cl(Z[Γ]), and hence Cl(R0 ), contains a height-one prime ideal [71, Theorem 7](also see Theorem 4.5(c)). Hence 1 = ρ(R) < ρ(R0 ) = ∞. Note that R0 ⊆ R is not an inert extension. Let S0 be the set of nonzero homogeneous elements of R0 . Then S0 ∼ = Q∗ × Γ. Clearly Γ is an HFD monoid, and thus so is S0 ; but 1 = ρ(S0 ) < ρ(R0 ) = ∞. Hence R0 is a “graded HFD”, but not an HFD. The elasticity of certain families of graded integral domains has been studied extensively. We will just mention two cases for graded subrings of a polynomial ring, and one for Krull semigroup rings. For a finite abelian group G, we define the Davenport constant of G, denoted by D(G), to be the least positive integer d such that for each sequence S ⊆ G with |S| = d, some nonempty subsequence of S has sum 0 (D(G) = ∞ if G is infinite). Then D(G) ≤ |G| and D(Z/nZ) = n. A special case of our next result is given in Example 6.4. For other related results, see [28], [31], [36], [63], and [72] for semigroup rings, and [35], [58], [59], [60], [61], and [72] for the A + XB[X] and related constructions. Theorem 4.5. (a) Let K be a field and R = K[X n , X n+1 , . . . , X 2n−1 ]. Then ρ(R) = (nD(P ic(R)) + 3n − 2)/2n. (b) For n ≥ 2, let K0 ⊆ K1 ⊆ . . . ⊆ Kn−1 be an ascending sequence of subfields of a field K with Kn−1  K, and let R = K0 + K1 X + · · · + Kn−1 X n−1 + X n K[X]. Then ρ(R) = (2n + D(P ic(R)) − 1)/2. (c) Let D be a UFD with proper quotient field K and D[Γ] a Krull domain which is not a UFD. Then ρ(D[Γ]) = D(Cl(K[Γ]))/2. Proof. Part (a) follows from [31, Theorem 2.4] and [35, Remark 4.6(e)]; part (b) is proved in [35, Theorem 4.4 and Remark 4.6(e)]; and part (c) is proved in [72, Corollary 2.3].

5

The Picard Group and Class Group of a Graded Domain

In this section, we investigate the Picard group and (t-)class group of a graded integral domain. We first recall a few definitions. Let R be an (not necessarily graded) integral domain with quotient field K. For a nonzero fractional ideal I of R, let I −1 = { x ∈ K | xI ⊆ R }, Iv = (I −1 )−1 , and It = ∪{ Jv | 0 = J ⊆ I is finitely generated }. A nonzero fractional ideal I of R is a v-ideal (resp., t-ideal ) if Iv = I (resp., It = I), and I has finite type if I = Jv with J ⊆ I finitely generated. A fractional ideal I of R is t-invertible if (II −1 )t = R. A t-invertible t-ideal necessarily has finite type. Let T (R) be the group of t-invertible t-ideals of R under the t-multiplication I ∗ J = (IJ)t , and let P rin(R) be its subgroup of nonzero principal fractional ideals. Then the (t-)class group of R is the abelian group Cl(R) = T (R)/P rin(R). Let Inv(R) ⊆ T (R) be the subgroup of invertible ideals of R. Then the Picard group or ideal class group of R, P ic(R) = Inv(R)/P rin(R), is a subgroup of Cl(R). If R is a Krull domain, then Cl(R) is just the usual divisor class group of R; while if R is a Pr¨ ufer domain or a one-dimensional integral domain, then Cl(R) = P ic(R). The class group is important because ring-theoretic properties of R are often reflected

Divisibility Properties in Graded Integral Domains

33

in group-theoretic properties of Cl(R). For example, if R is a Krull domain (resp., Pr¨ ufer domain), then Cl(R) = 0 if and only if R is a UFD (resp., B´ezout domain). For more on the divisor class group of a Krull domain, see [49]; for a recent survey article on the t-class group, see [22]. Given an extension of integral domains A ⊆ B, we write P ic(A) = P ic(B) (resp., Cl(A) = Cl(B)) to mean that the inclusion map induces an isomorphism P ic(A) −→ P ic(B) (resp., Cl(A) −→ Cl(B)) given by [I] → [IB], where [I] denotes the class of I in P ic(A) or Cl(A). Note that this map is always defined for P ic and is defined for Cl when B is a flat A-module. A first natural question for polynomial rings is: when does Cl(D) = Cl(D[X]) or P ic(D) = P ic(D[X])? In this case, we have nice answers. Theorem 5.1. Let D be an integral domain. (a) P ic(D) = P ic(D[X]) if and only if D is seminormal. (b) Cl(D) = Cl(D[X]) if and only if D is integrally closed. Proof. Part (a) is due to Gilmer and Heitmann [56, Theorem 1.6], and part (b) is due to Gabelli [52, Theorem 3.6]. We next give the semigroup ring analog of Theorem 5.1 for the Picard group; the class group case will be discussed later. Recall that an integral domain D is called quasinormal if P ic(D) = P ic(D[X, X −1 ]) ( = P ic(D[Z])). An integrally closed domain is always quasinormal, and a quasinormal integral domain is seminormal, but neither converse holds. An interesting consequence of the next theorem is that P ic(D) = P ic(D[Γ]) for all seminormal torsionless grading monoids Γ if and only if P ic(D) = P ic(D[Z]). Theorem 5.2. Let D be an intergal domain and Γ a torsionless grading monoid with group of units U (Γ) = Γ ∩ −Γ. (a) P ic(D) = P ic(D[Γ]) if and only if D[Γ] is seminormal and P ic(D) = P ic(D[U (Γ)]). (b) If U (Γ) = 0, then P ic(D) = P ic(D[Γ]) if and only if D is seminormal and Γ is seminormal. (c) If U (Γ) = 0, then P ic(D) = P ic(D[Γ]) if and only if D is quasinormal and Γ is seminormal. Proof. This is from [18, Corollary]. Part (b) follows directly from part (a) and Theorem 3.7(g). The key fact needed for part (c) is that D is quasinormal if and only if P ic(D) = P ic(D[Zn ]) for all positive integers n (see [86]), and hence if and only if P ic(D) = P ic(D[G]) for all torsionfree abelian groups G. Let R be a graded integral domain. If I is a nonzero homogeneous fractional ideal of R, then I −1 , Iv , and It are also homogeneous fractional ideals of R. Thus the abelian groups HP rin(R) ⊆ HInv(R) ⊆ HT (R) can be defined in the natural way, and we can define the homogeneous analogs of P ic(R) and Cl(R) by HP ic(R) = HInv(R)/HP rin(R) and HCl(R) = HT (R)/HP rin(R), the homogeneous Picard group and homogeneous (t-)class group, respectively. Thus HCl(R) = Cl(R) if and only if for each I ∈ T (R), I = xJ for some x ∈ RS and J ∈ HT (R). A similar result holds for HP ic(R).

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It is well-known that HCl(R) = Cl(R) if R is a graded Krull domain (see [49, Proposition 10.2] for the Z+ -graded case and [12, Theorem 4.2] for the general Γgraded case) since Cl(R) is generated by the classes of the height-one homogeneous prime ideals of R. We next give some conditions to have HP ic(R) = P ic(R) and HCl(R) = Cl(R). We first give some relevant definitions from [2] and [15]. Recall from Section 3 that a graded integral domain R is integrally closed (resp., seminormal) if and only if R is integrally closed (resp., seminormal) with respect to homogeneous elements of RS . With this motivation, we define R to be almost normal (resp., almost seminormal) if R is integrally closed (resp., seminormal) with respect to homogeneous elements of RS of nonzero degree. Thus R is integrally closed (resp., seminormal) if and only if R is almost normal (resp., almost seminormal) and R0 is integrally closed (resp., seminormal) in (RS )0 . Moreover, if R contains a (homogeneous) unit of nonzero degree, then R is integrally closed (resp., seminormal) if and only if R is almost normal (resp., almost seminormal) [2, Theorem 3.7(1)]. Also, note that D[Γ] is almost normal (resp., almost seminormal) if and only if it is integrally closed (resp., seminormal). Based on earlier work of Querre [82] for polynomial rings, a careful study of invertible ideals and v-ideals in graded integral domains was initiated in [2]. In [37], these ideas were extended to t-invertible t-ideals. We next give a sampling of results from [2]. Theorem 5.3. Let R be a graded integral domain and S its set of nonzero homogeneous elements. Then the following statements are equivalent. (a) Each integral v-ideal of R which contains a nonzero homogeneous element of R is homogeneous. (b) C(xy)v = (C(x)C(y))v for all nonzero x, y ∈ R. (c) xRS ∩ R = xC(x)−1 for all nonzero x ∈ R. (d ) If I is an integral v-ideal of R of finite type, then I = xJ for some x ∈ RS and homogeneous v-ideal J of R of finite type. Proof. This is part of [2, Theorem 3.2]. The finite type hypothesis may be deleted in part (d) above if IRS is a principal ideal of RS [2, Proposition 3.3]. In particular, this holds if R is Z+ - or Z-graded. However, an example given in [76] shows that it can not be deleted in general. We next examine the content condition in Theorem 5.3(b) more carefully. In [78], an example of a Z-graded integral domain is given which is almost normal, but does not satisfy the content condition. Thus the inert extension hypothesis can not be deleted in Theorem 5.4(c). Theorem 5.4. Let R be a graded integral domain. (a) If R is integrally closed, then C(xy)v = (C(x)C(y))v for all nonzero x, y ∈ R. (b) If C(xy)v = (C(x)C(y))v for all nonzero x, y ∈ R, then R is almost normal. (c) If R0 ⊆ R is an inert extension, then C(xy)v = (C(x)C(y))v for all nonzero x, y ∈ R if and only if R is almost normal. Proof. Parts (a) and (b) are proved in [2, Theorem 3.5], and part (c) is proved in [2, Theorem 3.7(2)].

Divisibility Properties in Graded Integral Domains

35

We next discuss the relationship between P ic(R) and HP ic(R). An example given in [77] shows that the inert extension hypothesis in Theorem 5.5(b) can not be eliminated. Theorem 5.5. Let R be a graded integral domain. (a) If HP ic(R) = P ic(R), then R is almost seminormal. Suppose, in addition, that R0 ⊆ R is an inert extension. (b) P ic(R0 ) = HP ic(R). (c) If R is almost normal, then P ic(R0 ) = P ic(R). Proof. Part (a) is [2, Theorem 4.3(3)], part (b) is [2, Theorem 5.4], and part (c) is [2, Theorem 5.5]. Our next goal is to determine when Cl(D) = Cl(D[Γ]). The key fact is the next result, which is the t-analog of Theorems 5.3 and 5.4. Theorem 5.6. Let R be a graded integral domain such that R0 ⊆ R is an inert extension. Then HCl(R) = Cl(R) if and only if R is almost normal. Proof. This is proved in [37, Theorem 1.1]. Corollary 5.7. Let R be a Z+ -graded integral domain. Then HCl(R) = Cl(R) if and only if R is almost normal. Proof. This is [37, Corollary 1.8]. It follows directly from Theorem 5.6 since in this case R0 ⊆ R is an inert extension. An example of a Z-graded almost normal integral domain R with HCl(R) = Cl(R) is given in [37, Example 1.11]. Thus the inert extension hypothesis is needed in Theorem 5.6. However, in Theorem 5.6 it is sufficient to assume that R contains a (homogeneous) unit of nonzero degree [37, Remark 1.7]. If D[Γ] is a Krull domain, then Cl(D[Γ]) = Cl(D)⊕Cl(K[Γ]), where K = qf (D), and Cl(K[Γ]) is independent of the field K [12, Proposition 7.2]. In fact, one can define the divisor class group Cl(Γ) of the Krull monoid Γ, and it turns out that Cl(Γ) ∼ = Cl(K[Γ]) (see [41] and [54, Section 16]). So it seems natural to ask what happens in the general semigroup ring case. In [67], it is shown that Cl(D) = Cl(D[G]) when G is a torsionfree abelian group which satisfies ACC on cyclic subgroups, and in [73] it is shown that Cl(D[Γ]) = Cl(D) ⊕ Cl(Γ) when D[Γ] is a PVMD and Γ satisfies ACC on cyclic subgroups. For integrally closed semigroup rings, a very satisfactory analog of the Krull domain result is given in [37]. Theorem 5.8. Let D be an integral domain with quotient field K and Γ a torsionless grading monoid. (a) HCl(D[Γ]) = Cl(D[Γ]) if and only if D[Γ] is integrally closed. (b) HCl(D[Γ]) = Cl(D) ⊕ HCl(K[Γ]). Proof. Part (a) follows from Theorem 5.6; it is in [37, Proposition 2.2]. Part (b) is in [37, Theorem 2.6]. Corollary 5.9. Let D be an integral domain with quotient field K and Γ a torsionless grading monoid. If D[Γ] is integrally closed, then Cl(D[Γ]) = Cl(D)⊕Cl(K[Γ]).

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Proof. This is proved in [37, Corollary 2.8]. It is an immediate consequence of Theorems 5.6 and 5.8. In analogy with integral domains, one can define the (t-)class group Cl(Γ) for any torsionless grading monoid Γ to be the group of t-invertible t-ideals of Γ under t-multiplication modulo its subgroup of principal fractional ideals. By [37, Theorem 2.9], HCl(K[Γ]) ∼ = Cl(Γ) for any field K, and thus Cl(K[Γ]) ∼ = Cl(Γ) when Γ is integrally closed [37, Corollary 2.10]. Hence Cl(D[Γ]) = Cl(D) ⊕ Cl(Γ) when D[Γ] is integrally closed [37, Corollary 2.11]. Theorem 5.10. Let D be an integral domain and Γ a torsionless grading monoid. Then Cl(D) = Cl(D[Γ]) if and only if D[Γ] is integrally closed and Cl(Γ) = 0. Proof. This is from [37, Theorem 2.12] and follows immediately from the previous results. Example 5.11. (a) When Γ is a Krull monoid and K is a field, then Cl(K[Γ]) = Cl(Γ) can be calculated. If Γ ⊆ F+ with Γ = Γ ∩ F+ (here, F = ⊕Zα is a free abelian group with the usual product order and prα is the natural projection map, cf. Remark 3.8(b)) and the prα |Γ ’s are distinct essential valuations of Γ, then Cl(Γ) ∼ = F/Γ ([41, Theorem 2], [54, Section 16], and [13]). For some specific calculations, see Example 6.6. (b) If R = D[Γ] is a Krull domain, then Cl(D) = Cl(D[Γ]) if and only if Cl(Γ) = 0, if and only if Γ is a factorial monoid. This need not be the case in general. For example, R = K[Q+ ] is a B´ezout domain for any field K [54, Theorem 13.5]; so Cl(R) = 0. Thus Cl(Q+ ) = 0, but Q+ is not a factorial monoid. (c) However, even for Krull domains, we may have Cl(D) ∼ = Cl(D[Γ]) with Cl(Γ) nonzero. For example, let k be any field, Γ = (1, 0, 1, 0), (1, 0, 0, 1), (0, 1, 1, 0), (0, 1, 0, 1) ⊆ (Z+ )4 , D = k[⊕n∈Z+ Γ], and K = qf (D). Then Cl(D) ∼ = ⊕n∈Z+ Z [49, Proposition 14.8], and thus Cl(D[Γ]) = Cl(D) ⊕ Cl(K[Γ]) = Cl(D) ⊕ Z ∼ = Cl(D). (d) Another consequence of Theorem 5.10 is that Cl(D) = Cl(D[G]) for a torsionfree abelian group G if and only if D is integrally closed. However, we always have Cl(D[F+ ]) = Cl(D[F ]) for any free abelian group F with the usual product order [8, Example 4.5]. Next one should try to compute Cl(D[Γ]) in general. As a step in this direction, we have the following result from [25]. Recall that a numerical semigroup Γ is a submonoid of Z+ with Z+ \Γ finite. So for Γ a numerical semigroup, D[Γ] is a subring of D[X] generated by monomials over D with quotient field K(X) and D[Γ] ⊆ D[X] integral. Theorem 5.12. Let D be an integral domain with quotient field K and Γ a numerical semigroup. Then Cl(D[Γ]) = Cl(D[X]) ⊕ P ic(K[Γ]). In particular, if D is integrally closed, then Cl(D[Γ]) = Cl(D) ⊕ P ic(K[Γ]). Proof. Let N = { X α | α ∈ Γ } and T = D∗ ; each is a multiplicatively closed subset of D[Γ]. In [25, Theorem 5], it is shown that the natural homomorphism Cl(D[Γ]) −→ Cl(D[Γ]N ) ⊕ Cl(D[Γ]T ) = Cl(D[X, X −1 ]) ⊕ Cl(K[Γ]), given by [I] → ([IN ], [IT ]), is an isomorphism. Since Cl(D[X]) = Cl(D[X, X −1 ]) for any integral domain D by Example 5.11(d) and Cl(K[Γ]) = P ic(K[Γ]) because K[Γ] is onedimensional, the result follows. For the “in particular” statement, use Theorem 5.1(b).

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The above result can be iterated to hold when Γ = Γ1 × · · · × Γn is a finite direct product of numerical semigroups Γi , 1 ≤ i ≤ n. Also, note that P ic(K[Γ]) can be calculated (for example, by using the Mayer-Viertoris exact sequence for (U, P ic)). It seems reasonable to ask if Cl(D[Γ]) = Cl(D[Γ]) ⊕ Cl(K[Γ]) for any integral domain D and torsionless grading monoid Γ. We next consider the natural homomorphism φ : P ic(R0 ) −→ P ic(R) given by φ([I]) = [IR]. When R is a Z+ -graded integral domain, φ is always injective (in fact, splits since P ic is a functor), and we have a nice answer for when φ is surjective. Theorem 5.13. Let R be a Z+ -graded integral domain. Then P ic(R0 ) = P ic(R) if and only if R is almost seminormal. Proof. This is proved in [15, Theorem 1]. It is easy to see that imφ ⊆ HP ic(R); so we have φ : P ic(R0 ) −→ HP ic(R) ⊆ P ic(R). We have already seen in Theorem 5.5 that φ is injective and imφ = HP ic(R) when R0 ⊆ R is an inert extension, and that P ic(R0 ) = P ic(R) if, in addition, R is almost normal. In [77], an example of a Z-graded integral domain R is given with imφ = HP ic(R). For some other results, mainly in the Z-graded case, see [18], [81], and [2]. However, kerφ is easy to describe. Theorem 5.14. Let R be a Γ-graded integral domain, and let φ : P ic(R0 ) −→ P ic(R) be the natural homomorphism. Define Γ1 = { α ∈ U (Γ) | Rα R−α = R0 } and Γ2 = { α ∈ Γ | Rα ∩ U (R) = ∅ }. Then Γ2 ⊆ Γ1 are subgroups of U (Γ). (a) kerφ ∼ = Γ1 /Γ2 . (b) Let 0 −→ A −→ B −→ C −→ 0 be a short exact sequence of abelian groups. Then there is a graded integral domain R with P ic(R0 ) ∼ = B, P ic(R) ∼ = C, ∼ and kerφ = A. Proof. These are proved in [19]. The analog of Theorem 5.13 for the class group fails, even for Krull domains. A Z+ -graded Krull domain R may have Cl(R0 ) = 0 and Cl(R) = 0 (see Example 6.6). For other results on the class group of a graded Krull domain, see [12] and [49]. We end this section with some results about P ic(A + XB[X]) and Cl(A + XB[X]). Note that A + XB[X] is almost normal (resp., almost seminormal) if and only if B is integrally closed (resp., seminormal). Also, the natural homomorphism φ : Cl(A) −→ Cl(A + XB[X]) is always injective when B is a flat A-module [29, Lemma 4.2]. However, the overring hypothesis is needed in Theorem 5.15(c) for φ to be surjective [29, Corollary 4.12]. For related results on Cl(A + XB[X]), see [29] and [30]. Theorem 5.15. Let A ⊆ B be an extension of integral domains, R = A + XB[X], and S a multiplicatively closed subset of A. (a) P ic(A) = P ic(R) if and only if B is seminormal. (b) HCl(A) = HCl(R) if and only if B is integrally closed. (c) If B is an integrally closed flat overring of A, then Cl(A) = Cl(R). In particular, Cl(A) = Cl(A + XAS [X]) if A is integrally closed.

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Proof. Part(a) is a special case of Theorem 5.13. Part (b) is from [30, Corollary 1.2], and is a special case of Corollary 5.7. Part (c) is from [29, Theorem 4.4]. The “in particular” statement is clear since AS is always a flat overring of A. Let S be a saturated multiplicatively closed subset of an integral domain D. Then N = { 0 = x ∈ D | xD ∩ sD = xsD for all s ∈ S } is a saturated multiplicatively closed subset of D called the m-complement of S. We call S a splitting set if SN = D∗ . If S is a splitting set, then Cl(D) = Cl(DS ) ⊕ Cl(DN ) [8, Corollary 3.8]. For other applications of splitting sets, see [8] and [24]. If S is a splitting set, then we can explicitly calculate Cl(D + XDS [X]) and P ic(D + XDS [X]). Theorem 5.16. Let S be a splitting multiplicative subset of an integral domain D with m-complement N . Then Cl(D + XDS [X]) = Cl(DS [X]) ⊕ Cl(DN ) and P ic(D + XDS [X]) = P ic(DS [X]) ⊕ P ic(DN ). Proof. This is proved in [24, Theorem 4.3].

6

Examples

In this final section, we have collected some of our favorite examples illustrating various divisibility properties. In each case, we use a semigroup ring construction. In Examples 6.4, 6.6, and 6.7, the semigroup rings are subrings of a polynomial ring generated by monomials, and hence are also Z+ -graded. Example 6.1. An integral domain with no atoms. Let R = Z/2Z[Q+ ]. Then R is a one-dimensional integral domain with f = X α1 +· · ·+X αn = (X α1 /2 +· · ·+X αn/2 )2 for each f ∈ R; so no element of R is irreducible. Integral domains with no atoms have been investigated in [46]. Example 6.2. An atomic integral domain which does not satisfy ACCP ([62]). Let R = K[Γ]M , where K is a field, Γ = 1/3, 1/(2·5), 1/(22·7), . . . , 1/(2k ·pk ), . . . ⊆ Q+ (here p0 = 3, p1 = 5, . . . are the odd primes), and M = { f ∈ K[Γ] | f (0) = 0 }, a maximal ideal of K[Γ]. Then R is a one-dimensional quasilocal integral domain. In [62], it is shown that R is atomic. However, R does not satisfy ACCP since XR ⊂ X 1/2 R ⊂ X 1/4 R ⊂ . . . is a properly ascending chain of principal ideals of R. Example 6.3. An integral domain which satisfies ACCP, but is not a BFD ([6, Example 2.1]). Let R = K[Γ], where K is a field and Γ = 1/2, 1/3, 1/5, . . . , 1/pk , . . . ⊆ Q+ (here pk is the kth prime). Then R is a one-dimensional integral domain which satisfies ACCP. However, X = (X 1/p )p and X 1/p is an atom for each prime p; so R is not a BFD. An integral domain which satisfies ACCP, but is not a BFD ([6, Example 2.1]). Let R = K[Γ], where K is a field and Γ = (1/2, 1/3, 1/5, . . . , 1/pk , . . .) ⊆ Q+ (here pk is the kth prime). Then R is a one-dimensional integral domain which satisfies ACCP. However, X = (X 1/p )p and X 1/p is an atom for each prime p; so R is not a BFD. Example 6.4. An atomic integral domain which is not an HFD. Let R = Z/2Z[X 2 , X 3 ]. Then R is a one-dimensional Noetherian integral domain which is not seminormal. Clearly ρ(R) ≥ 3/2 since (X 2 )3 = (X 3 )2 . In fact, for any field K, we have ρ(K[X 2 , X 3 ]) = (D(K) + 2)/2 [28, Theorem 2.4]. Thus ρ(R) = 2. Moreover,

Divisibility Properties in Graded Integral Domains

39

f 2 = g 3 h for irreducible f = X 3 (1 + X + X 2 ), g = X 2 , and h = 1 + X 2 + X 4 ∈ R. As an application of Theorem 5.12, for any integral domain D with quotient field K, we have Cl(D[X 2 , X 3 ]) = Cl(D[X]) ⊕ K since P ic(K[X 2 , X 3 ]) = K as additive abelian groups. Example 6.5. A two-dimensional non-Noetherian UFD ([55]). Let R = K[G], where K is a field and G is a torsionfree rank-two infinite abelian group which is not free, but each rank-one subgroup is free. Such abelian groups exist (see [51, Section 88]). Then dim(D) = rank(G) = 2, and R is a UFD by Theorem 2.3(a). Clearly R is not Noetherian since G is not finitely generated. For each integer n ≥ 2, R[X1 , . . . , Xn−2 ] is an n-dimensional non-Noetherian UFD. Note that a one-dimensional UFD is a PID, and thus Noetherian. Example 6.6. Krull domains with specified class group. Let K be a field and G an abelian group. One can construct a Krull domain R = K[Γ] with Cl(R) = G [41, Corollary 2]. In fact, we may localize R to obtain a quasilocal Krull domain A with Cl(A) = G. We show this explicity when G = Z/n1 Z ⊕ · · · ⊕ Z/nr Z ⊕ Zn is finitely generated. Let R = K[{ Xini , Xi Yi , Yini | 1 ≤ i ≤ r }, { Sj Uj , Sj Vj , Tj Uj , Tj Vj | 1 ≤ j ≤ n }], a subring of T = K[{ Xi , Yi | 1 ≤ i ≤ r }, { Sj , Tj , Uj , Vj | 1 ≤ j ≤ n }] generated by monomials. Then Cl(A) = Cl(R) = G, where A = RM with M = R ∩ ({Xi , Yi }, {Sj , Tj , Uj , Vj })T [13, Example 9 and Theorem 10]. Recall that for any abelian group G, there is a Dedekind domain D with Cl(D) = G [43]. Example 6.7. Independence of n-root closed and seminormal conditions ([16]). For an integral domain R, define C(R) = { n ∈ N | R is n-root closed }. Then C(R) is a multiplicative submonoid of N such that mn ∈ C(R) if and only if m, n ∈ C(R); so C(R) is generated by primes. We show that for any multiplicative submonoid T of N generated by primes, there is an integral domain R with C(R) = T . If T = N, let R be any root closed integral domain. Otherwise, let Q = {pi }i∈I be the set of primes not in T . Define R = Q[{ Xipi | i ∈ I }, { XimYjn | m ≥ 0, n ≥ 1, and i, j ∈ I }], a subring of Q[{ Xi , Yi | i ∈ I }] generated by monomials. In [16, Theorem 2.7], it is shown that C(R) = T . Note that if T = {1}, then R is seminormal, but not n-root closed for any integer n ≥ 2. Other constructions for R with C(R) = T are discussed in [21, Section 6].

Bibliography [1] D. D. Anderson and D. F. Anderson, Divisibility properties in graded domains, Canad. J. Math. 34 (1982), 196–215. [2] D. D. Anderson and D. F. Anderson, Divisorial ideals and invertible ideals in a graded integral domain, J. Pure Appl. Algebra 76 (1982), 549–569. [3] D. D. Anderson and D. F. Anderson, Grading integral domains, Comm. Algebra 11 (1983), 1–19. [4] D. D. Anderson and D. F. Anderson, Elasticity of factorizations in integral domains, J. Pure Appl. Algebra 80 (1992), 217–235. [5] D. D. Anderson and D. F. Anderson, The ring R[X, r/X], Zero-dimensional Commutative Rings, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 171 (1995), 95–113. [6] D. D. Anderson, D. F. Anderson, and M. Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra 69 (1990), 1–19. [7] D. D. Anderson, D. F. Anderson, and M. Zafrullah, Rings between D[X] and K[X], Houston J. Math. 17 (1991), 109–129. [8] D. D. Anderson, D. F. Anderson, and M. Zafrullah, Splitting the t-class group, J. Pure Appl. Algebra 74 (1991), 17–37. [9] D. D. Anderson, D. F. Anderson, and M. Zafrullah, Factorization in integral domains, II, J. Algebra 152 (1992), 78–93. [10] D. D. Anderson and J. Matijevic, Graded π-rings, Canad. J. Math. 31 (1979), 449–457. [11] D. D. Anderson and B. Mullins, Finite factorization domains, Proc. Amer. Math. Soc. 124 (1996), 389–396. [12] D. F. Anderson, Graded Krull domains, Comm. Algebra 7 (1979), 79–106. [13] D. F. Anderson, The divisor class group of a semigroup ring, Comm. Algebra 8 (1980), 467–476. [14] D. F. Anderson, Seminormal graded rings, J. Pure Appl. Algebra 21 (1981), 1–7. 40

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[15] D. F. Anderson, Seminormal graded rings, II, J. Pure Appl. Algebra 23 (1982), 221–226. [16] D. F. Anderson, Root closure in integral domains, J. Algebra 79 (1982), 51–59. [17] D. F. Anderson, The Picard group of a monoid domain, J. Algebra 115 (1988), 342–351. [18] D. F. Anderson, The Picard group of a monoid domain, II, Arch. Math. 55 (1990), 143–145. [19] D. F. Anderson, The kernel of P ic(R0 ) −→ P ic(R) for R a graded domain, C. R. Math. Rep. Acad. Sci. Canada 13 (1991), 248–252. [20] D. F. Anderson, Elasticity of factorizations in integral domains, a survey, Factorization in Integral Domains, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 189 (1997), 1–29. [21] D. F. Anderson, Root closure in commutative rings, a survey, Advances in Commutative Ring Theory, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 205 (1999), 55–71. [22] D. F. Anderson, The class group and local class group of an integral domain, Non-Noetherian Ring Theory, Math. Appl., Kluwer Acad. Publ., Dordrecht 520 (2000), 33–55. [23] D. F. Anderson and G. W. Chang, The class group of D[X 2 , X 3 ], Internat. J. Commutative Rings 2 (2003), 1–7. [24] D. F. Anderson and G. W. Chang, The class group of integral domains, J. Algebra 264 (2003), 535–552. [25] D. F. Anderson and G. W. Chang, The class group of D[Γ] for D an integral domain and Γ a numerical semigroup, Comm. Algebra 32 (2004), 787–792. [26] D. F. Anderson, G. W. Chang, and J. Park, D[X 2 , X 3 ] over an integral domain D, Commutative Ring Theory and Applications, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 231 (2002), 1–14. [27] D. F. Anderson, G. W. Chang, and J. Park, Weakly Krull and related domains of the form D + M, A + XB[X], and A + X 2 B[X], Rocky Mountain J. Math., to appear. [28] D. F. Anderson, S. Chapman, F. Inman, and W. Smith, Factorization in K[X 2 , X 3 ], Math. Arch. 61 (1993), 521–528. [29] D. F. Anderson, S. El Baghdadi, and S. Kabbaj, On the class group of A + XB[X] domains, Advances in Commutative Ring Theory, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 205 (1999), 73–85. [30] D. F. Anderson, S. El Baghdadi, and S. Kabbaj, The homogeneous class group of A + XB[X] domains, Internat. J. Commutative Rings 1 (2002), 11–25.

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[31] D. F. Anderson and S. Jenkins, Factorization in K[X n , X n+1 , . . . , X 2n−1 ], Comm. Algebra 23 (1995), 2561–2576. [32] D. F. Anderson and D. Nour El Abidine, Factorization in integral domains, III, J. Pure Appl. Algebra 135 (1999), 107–127. [33] D. F. Anderson and D. Nour El Abidine, The A + B[X] and A + XB[[X]] constructions from GCD-domains, J. Pure Appl. Algebra 159 (2001), 15–24. [34] D. F. Anderson and J. Ohm, Valuations and semi-valuations of graded domains, Math. Ann. 256 (1981), 145–156. [35] D. F. Anderson and J. Park, Factorization in subrings of K[X] or K[[X]], Factorization in Integral Domains, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 189 (1997), 227–241. [36] D. F. Anderson and C. Scherpenisse, Factorization in K[S], Commutative Ring Theory, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 185 (1997), 45–56. [37] S. El Baghdadi, L. Izelgue, and S. Kabbaj, On the class group of a graded domain, J. Pure Appl. Algebra 171 (2002), 171–184. [38] V. Barucci, L. Izelgue, and S. Kabbaj, Some factorization properties of A + XB[X] domains, Commutative Ring Theory, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 185 (1997), 69–78. [39] S. T. Chapman and J. Coykendall, Half-factorial domains, a survey, NonNoetherian Ring Theory, Math. Appl., Kluwer Acad. Publ., Dordrecht 520 (2000), 97–115. [40] S. T. Chapman and A. Geroldinger, Krull domains and monoids, their sets of lengths, and associated combinatorial problems, Factorization in Integral Domains, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 189 (1997), 73–112. [41] L. G. Chouinard, Krull semigroups and divisor class groups, Canad. J. Math. 33 (1981), 1459–1468. [42] P. M. Cohn, Bezout rings and their subrings, Proc. Cambridge Philos. Soc. 64 (1968), 1459–1464. [43] L. Claborn, Every abelian group is a class group, Pacific J. Math. 18 (1966), 219–222. [44] D. L. Costa, J. L. Mott, and M. Zafrullah, The construction D + XDS [X], J. Algebra J. 53 (1978), 423–439. [45] J. Coykendall, A characterization of polynomial rings with the half-factorial property, Factorization in Integral Domains, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 189 (1997), 291–294. [46] J. Coykendall, D. E. Dobbs, and B. Mullins, On domains with no atoms, Comm. Algebra 27 (1999), 5813–5831.

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[47] J. Coykendall, T. Dumitrescu, and M. Zafrullah, The half-factorial property and domains of the form A + XB[X], preprint. [48] T. Dumitrescu, S. O. Ibrahim Al-Salihi, N. Radu, and T. Shah, Some factorization properties of composite domains A + XB[X] and A + XB[[X]], Comm. Algebra 28 (2000), 1125–1139. [49] R. M. Fossum, The Divisor Class Group of a Krull Domain, Springer, New York, 1973. [50] L. Fuchs, Infinite Abelian Groups, vol. I, Academic Press, New York, 1970. [51] L. Fuchs, Infinite Abelian Groups, vol. II, Academic Press, New York, 1973. [52] S. Gabelli, On divisorial ideals in polynomial rings over Mori domains, Comm. Algebra 15 (1987), 2349–2370. [53] R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972. [54] R. Gilmer, Commutative Semigroup Rings, Chicago Lecture Notes in Mathematics, University of Chicago Press, Chicago, 1984. [55] R. Gilmer, A two-dimensional non-Noetherian factorial ring, Proc. Amer. Math. Soc. 44 (1974), 25–30. [56] R. Gilmer and R. Heitmann, On P ic(R[X]) for R seminormal, J. Pure Appl. Algebra 16 (1980), 251–257. [57] R. Gilmer and T. Parker, Divisibility properties in semigroup rings, Michigan Math. J. 21 (1974), 65–86. [58] N. Gonzalez, Factorisation pour les polynˆomes sur une suite croissante d’anneaux, Th`ese de doctorat de l’Universit´e de Droit, d’Economie et des Sciences d’Aix Marseille, 1997. [59] N. Gonzalez, Elasticity of A + XB[X] domains, J. Pure Appl. Algebra 138 (1999), 119–137. [60] N. Gonzallez, Elasticity and ramification, Comm. Algebra 27 (1999), 1729– 1736. [61] N. Gonzallez, S. Pellerin, and R. Robert, Elasticity of A + XI[X] domains where A is a UFD, J. Pure Appl. Algebra 160 (2001), 183–194. [62] A. Grams, Atomic domains and the ascending chain condition, Proc. Cambridge Philos. Soc. 75 (1974), 321–329. [63] F. Halter-Koch, Finitely generated monoids, finitely primary monoids, and factorization properties of integral domains, Factorization in Integral Domains, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 189 (1997), 31–72. [64] F. Halter-Koch, Ideal Systems, Marcel Dekker, New York, 1998.

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[83] M. Roitman, Polynomial extensions of atomic domains, J. Pure Appl. Algebra 87 (1993), 187–199. [84] R. G. Swan, On seminormality, J. Algebra 67 (1980), 210–229. [85] R. J. Valenza, Elasticity of factorizations in number fields, J. Number Theory 36 (1990), 212–218. [86] C. Weibel, P ic is a contracted functor, Invent. Math. 103 (1991), 351–377. [87] M. Zafrullah, The D+XDS [X] construction from GCD-domains, J. Pure Appl. Algebra 50 (1988), 93–107. [88] M. Zafrullah, Various facets of rings between D[X] and K[X], Commutative Ring Theory and Applications, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 231 (2002), 445–460. [89] M. Zafrullah, Various facets of rings between D[X] and K[X], Comm. Algebra 31 (2003), 2497–2540. [90] A. Zaks, Half-factorial domains, Israel. J. Math. 137 (1993), 281–302.

Chapter 3

Extensions of Half-Factorial Domains: A Survey by

Jim Coykendall Abstract A half-factorial domain (HFD) is an atomic domain, R, with the property that if one has the irreducible factorizations in R π1 π2 · · · πn = ξ 1 ξ 2 · · · ξ m then n = m. Half-factorial domains were implicitly studied in the case of rings of algebraic integers in a 1960 paper of L. Carlitz [11] and were subsequently abstracted and studied by A. Zaks. Since their inception as a generalization of the classical notion of unique factorization domains, half-factorial domains have been the subject of much interest in commutative algebra. This chapter will give a survey of some recent advances in the study of half-factorial domains with the emphasis on advances in the understanding of ring extension behavior of half-factorial domains.

1

Introduction and A Bit of History

The purpose of this chapter is to give an expository overview of some of the recent work in the study of half-factorial domains and their extension rings. This chapter is written as an expansion on an invited 45-minute address presented in Chapel Hill in October of 2003. The author would like to begin by expressing his gratitude to Professors Scott Chapman and Bill Smith for their kind invitation and hospitality. The concept of “half-factorial domain” was introduced implicitly in 1960 in a striking paper of L. Carlitz [11]. The paper was written as an answer to a challenge of Narkiewicz who asked for a purely arithmetic characterization of rings of integers in terms of the class number. Of course, it was well-known that a ring of algebraic integers, and more generally, a Dedekind domain, has unique factorization if and only if its class number is one. Given this interplay between the concepts of unique 46

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47

factorization and class number (and class group), Narkiewicz’ line of questioning was most natural. In 1960, Carlitz’ partial answer to this question appeared in the Proceedings of the American Mathematical Society. In our opinion, this result is one of the most succinct and beautiful results in the study of factorization in rings of integers. It is also, from at least one point of view, the paper that is the ancestor of much of the current study in the general theory of factorization in general integral domains. Additionally, this paper is one of the most cited papers in factorization theory, and the arguments utilized in the proof have made a “factorization industry” possible for large classes of domains. The proofs in the paper underscore the “right” direction in which to proceed in the more general case. The main result of this seminal paper is that if R is a ring of algebraic integers then one has unique length of factorizations if and only if the class number does not exceed 2. Because of the importance of this result and the fundamental nature of its proof, we will state this result (and demonstrate its proof) more formally later. In the interim, we will remind the reader of the “nicest” class of domains from a factorization point of view and give the current definition of a half-factorial domain as a generalization. This not only gives the historical line of reasoning, but also makes the statement of Carlitz’ result much cleaner. The result of Carlitz’ 1960 paper remained mostly untouched for about 15-20 years until Zaks generalized this “unique length of factorization” property. He published a couple of papers [47, 48] on the subject and coined the terminology “half-factorial domain.” The paper [48] is a longer work with a number of important theorems on the subject of the half-factorial property. It is interesting to note that what seems to be the current accepted definition of a half-factorial domain is actually different from the one in Zaks’ paper. About 10 years later the real gold rush began with a paper by D. D. Anderson, D. F. Anderson, and M. Zafrullah [2]. This paper (and its sequel [3]) is a blueprint for much of the work that is taking place in modern factorization theory, and the scope of the paper far exceeded the scope of the half-factorial property. Historically, this marks the time when much activity began in the factorization theatre. In this chapter, we will concentrate on work that has been done in the theory of half-factorial domains and their extension rings. Because of the large number of results obtained in various aspects of factorization, this chapter will be far from complete. The interested reader would be well served to consult the volume [1] for a very good collection of papers on the half-factorial property, its generalizations, and factorization in general. In this chapter we use the following notation conventions: 1) R is (unless otherwise stated) an atomic domain with quotient field K. 2) RS is the localization of R at a multiplicative set S. 3) R is the integral closure of R. 4) R is the complete integral closure of R. 5) U (R) is the unit group of R. 6) Irr(R) is the set of irreducible elements of R.

48

Coykendall 7) Cl(R) is the class group of R. 8) R[x] and R[[x]] are the rings of polynomial and formal power series over R, respectively. 9) D(G) is the Davenport constant of the finite abelian group G.

10) d(K) is the discriminant of the algebraic number field K. 11) Sn is the symmetric group on n letters. Other notations are standard.

2

Some Preliminary Results and Definitions

Of course, the genesis of factorization theory is motivated by the domains where factorization is extremely well-behaved. For the purposes of this chapter, we will assert some control over the “badness” of the factorization behavior. That is, unless otherwise stated, we will assume that our domains are atomic (i.e., every nonzero, nonunit element of R can be written as a product of irreducible elements). We now recall the definition of unique factorization domains. Definition 2.1. A unique factorization domain (UFD) is an atomic integral domain, R, such that given the irreducible factorizations π1 π2 · · · πn = ξ1 ξ2 · · · ξm we have that: a) n = m and b) ∃ σ ∈ Sn such that for each 1 ≤ i ≤ n, πi = ui ξσ(i) for some ui ∈ U (R). There are a number of equivalent definitions of UFDs that are useful in different settings. In particular, a UFD may be more succinctly defined as the class of domains where every nonzero nonunit can be expressed as a product of prime elements. From the ideal theoretic point of view, UFDs can also be characterized as the class of domains where every nonzero prime ideal contains a nonzero principal prime [32]. Unique factorization domains possess a large number of very nice properties and we will compare and contrast some of these with the properties of half-factorial domains. The class of half-factorial domains (HFDs) is, from one point of view, the nicest generalization of the class of UFDs with respect to factorization. Definition 2.2. An (atomic) domain is said to be a half-factorial domain (HFD) if given two irreducible factorizations π1 π2 · · · πn = ξ1 ξ2 · · · ξm then n = m.

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So, in a certain sense, an HFD is just a UFD with “half the axioms.” It is worth noting that Zaks’ original definition omitted the “atomic” assumption (and in Zaks’ paper he quickly specialized to atomic cases). Of course, it is natural to ask if the HFD condition without the atomic assumption automatically forces atomicity, but this is not the case as we will see in the next example. Without the “atomic” assumption, there is some pathology that we wish to avoid; for this reason (and since today’s common definition also includes the assumption “atomic”), we will always assume that HFD implies atomic unless otherwise indicated. In the following example from [25], we will produce a nonatomic domain that has the property that any two equal irreducible factorizations have the same length. Example 2.3. This example will give a nonatomic domain with the property that if π1 π2 · · · πn = ξ1 ξ2 · · · ξm with πi , ξj ∈ Irr(R), 1 ≤ i ≤ n, 1 ≤ j ≤ m, then n = m. The example is a domain with a unique nonprime irreducible element (up to associates) and was constructed in [25]. In this domain, the uniqueness of the irreducible assures that any two irreducible factorizations are of the same length (in fact, unique). The ease of this observation belies the delicate nature of the construction of a domain with a unique nonprime irreducible (for atomic domains with finitely many irreducibles, the so-called CK domains, see [4]). Speaking very loosely, we will outline the major steps to the construction of such a domain. Begin with a ring R that possesses a nonprime π ∈ Irr(R). One then constructs a new domain, R1 , by indexing all the irreducibles in R that are not associated to π (say our relevant irreducibles are {ξi }i∈I with I some indexing set) and building a larger domain that “forces” each ξi to become reducible. In particular, we let R1 = R[{xi }][

ξi ]i∈I , xi

with each xi a polynomial indeterminate. It is then shown that in R1 , π remains a nonprime irreducible and each ξi is reducible in R1 by construction. Of course, there may be many “new irreducibles”, but one again isolates π and constructs R2 analogously. Continuing in this fashion, we obtain a tower of rings ∞

R1 ⊆ R2 ⊆ R3 ⊆ · · ·

and we define T := i=1 Ri . It can be shown that in T , π is the unique nonprime irreducible (intuitively, uniqueness comes from the fact that every possible irreducible not associated to π is “destroyed” in the next stage of the construction). This domain has the claimed property, and it is interesting to note that this construction shows that any domain with a nonprime irreducible element can be embedded in such a domain. We remark that the general theme of this construction is very much in the spirit of the construction performed by M. Roitman in [42] where it was shown that a polynomial extension of an atomic domain is not necessarily atomic.

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With these definitions in hand, we now have the tools to succinctly state the classical result of Carlitz. We give a slightly more general statement in the spirit of the general utility of Carlitz’ proof and in the spirit of much of the research that has taken place concerning the interplay of the class number and factorization. Theorem 2.4. Let D be a Dedekind domain. If |Cl(D)| ≤ 2 then D is an HFD. If Cl(D) is torsion and D has the property that there is a prime ideal in every class, then the converse also holds. Proof. Of course, it is well-known that if Cl(D) is trivial, then D is a UFD (and hence an HFD), so we will concentrate on the case where |Cl(D)| = 2. Assume that we have the irreducible factorizations π1 π2 · · · πn = ξ1 ξ2 · · · ξm . Since a prime factor appears on the left side if and only if it appears on the right, we will reduce to the case where the factorizations above consist entirely of nonprime irreducible elements. Since |Cl(D)| = 2 and each irreducible is a nonprime we can factor (in terms of prime ideals) (i)

(i)

(i)

(i)

πi = P1 P2 and

ξi = Q1 Q2 . We now have the ideal factorization (1)

(1)

(2)

(2)

(n)

(n)

P1 P2 P1 P2 · · · P1 P2

(1)

(1)

(2)

(2)

(m)

(m)

= Q1 Q2 Q1 Q2 · · · Q1 Q2 .

Since prime ideal factorizations are unique, we have that 2n = 2m and hence n = m. On the other hand, assume that D is a Dedekind HFD and Cl(D) has a prime in every class. Since Cl(D) is torsion, every class has finite order. We first assume the existence of a class of order n > 2 and select a prime ideal P in this class. By construction, Pn is principal and generated by some α ∈ Irr(D). We now choose a prime ideal Q in the class of P−1 and note that PQ is principal and generated by some π ∈ Irr(D) and additionally Qn is principal and generated by the irreducible β. We now consider the ideal factorizations (PQ)n = (Pn )(Qn ) which give the elemental factorizations π n = uαβ for some u ∈ U (D). Since n > 2, D is not an HFD. The only case left to consider is the case where Cl(D) is of exponent 2 and |Cl(D)| > 2. In this case, Cl(D) must contain a subgroup isomorphic to the Klein 4-group Z2 ⊕ Z2 . In a fashion similar to the above, we choose prime ideals P in the class corresponding to (0, 1), Q in the class corresponding to (1, 0) and R in the class corresponding to (1,1). It is easy to see that there are irreducibles α, β, γ,

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and ξ such that P2 = (α), Q2 = (β), R2 = (γ), and PQR = (ξ). The ideal factorizations (PQR)2 = (P2 )(Q2 )(R2 ) give rise to the elemental factorizations ξ 2 = uαβγ for some u ∈ U (R) and once again D is not an HFD. A couple of remarks are worth noting here. First, the original Carlitz result was couched strictly in terms of algebraic number theory. His result was that a ring of algebraic integers, R, is an HFD if and only if the class number of R does not exceed 2. But the seeds of the original proof pointed the way towards much more generality. The second remark is that although the above theorem is slightly more general than the original statement of Carlitz, it is not as general as possible either. There has been much work done in more general cases and the interested reader is encouraged to consult [5, 6, 14].

3

“Good” Properties of UFDs versus HFDs

As a generalization of UFDs, one would like to understand what, if any, “nice” behaviors of UFDs are shared by HFDs. UFDs possess a number of nice properties (especially in the case of localizations and polynomial extensions). Some of these classical results are that if R is a UFD then so is R[x], and the localization RS . Both of these central results for UFDs can be obtained from the following nice ideal-theoretic characterization of UFDs. Theorem 3.1. A domain R is a UFD if and only if every nonzero prime ideal in R contains a (nonzero) prime element. So a UFD is the “multi-dimensional” generalization of a principal ideal domain (PID). Generally speaking, HFDs do not have a readily apparent ideal theoretic characterization. In the case of rings of algebraic integers (and more generally, Dedekind and Krull domains with certain restrictions on the class group), HFDs do have a nice ideal-theoretic characterization (evidenced by Carlitz’ theorem, for one). A sweeping ideal-theoretic characterization for general HFDs seems to be elusive (if indeed such a characterization exists), and we cannot resist the following question. Question 3.2. Find a general ideal-theoretic characterization of HFDs in the spirit of the characterization of UFDs, if it exists. Below we produce a table that records some of the classical results about UFDs and compares them with what is known about the analog property for HFDs. It should be explicitly stated that the entries “no” in the table mean, more precisely, “not always.”

52

Coykendall Table of Inherited Factorization Properties R is a: UFD HFD

R[x] Yes No

R[[x]] No No

R Yes No

R Yes No

RS Yes No

We make a couple of remarks about the table. First it should be noted that if R is a UFD then the (complete) integral closure of R coincides with R, so this result is trivial (but it does indeed invite the question as to whether the (complete) integral closure of an HFD is still an HFD). Also, although it is possible for a power series ring over a UFD to lose unique factorization [43], if R is a one-dimensional UFD (i.e., a PID) then R[[x1 , x2 , · · · , xn ]] is a UFD for all n. If the reader hopes for the half-factorial property to be preserved in standard extensions, then this table will look quite depressing. However, we remark that there are indeed positive results with (in some cases) rather minimal restrictions placed on the HFD. We will look at some of the positive results and some of the pathologies in the next sections.

4

Localizations

In our view, one of the most disturbing results in the table from the previous section is that if R is an HFD, then it is not necessarily true that RS is an HFD (where S is a multiplicatively closed subset of R). The proof that the UFD property holds in localizations is very elementary (the characterization of UFD via every nonzero prime contains a nonzero principal prime is very useful in this regard). Of course, one would not expect general overrings to preserve the half-factorial property since unique factorization is not preserved in general overrings. A quick example of this is the ring K[x, y] where K is a field. This ring is a UFD (and hence an HFD), but the overring K[x, xy , xy2 , xy3 , · · · ] is not even atomic. In this section we take a brief look at some examples, positive (and negative) results, and generalizations of HFDs in the case of localizations and more general overrings. We will begin by giving an example of an HFD which possesses a localization which does not have the half-factorial property. This example and other very nice results in this spirit may be found in the papers [7], [8], and [9]. Example 4.1. We begin by noting that one can construct a Dedekind domain, D, with class group Z6 such that the prime ideals of D are restricted to the classes corresponding to {1, 2, 3} ⊆ Z6 (a more general theorem on the construction of such Dedekind domains can be found in [27]). As it turns out, such a Dedekind domain is necessarily an HFD. We now consider the set Γ = {Q|Q is a prime ideal of D with [Q] = 1 or [Q] = 2}.

 It is easy to see that if P is a prime ideal such that [P ] = 3 then P ⊆ Q∈Γ Q. Let t ∈ P \ ∪Q∈Γ Q, T = {1, t, t2 , · · · } and form the localization R = DT . We make the following observations. 1. R is a Dedekind domain.

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2. Cl(R) ∼ = Cl(D)/(ker(τ )) where τ is the natural map from Cl(D) −→ Cl(R) defined by τ : [I] −→ [IR]. It follows ([7]) that Cl(R) ∼ = Z6 /Z2 ∼ = Z3 and the primes of R are in the nontrivial classes of Z3 . With this reduction, the technique of the Carlitz proof can be applied to show that R is not an HFD. This example motivates the following definitions (once again from [7] and [8]). Definition 4.2. Let R be an integral domain. We say that R is a a) Locally half-factorial domain (LHFD) if every localization of R is an HFD. b) Strong half-factorial domain (SHFD) if every overring of R is an HFD. Of course, an SHFD is an LHFD, but the converse does not hold in general. Here are some nontrivial examples of these types of domains. Example 4.3. Let R be a Dedekind domain with Cl(R) = Zp for a prime integer p, such that all primes of R are contained in the class {1}. Then it can be shown that any (nontrivial) localization RS has trivial class group and hence is a PID. The upshot is that R is a LHFD, and since R has torsion class group, every overring is a localization and hence R is SHFD. We note that this example can be tweaked to the case where Cl(R) is not torsion to give an example of an LHFD that is not an SHFD (the key in this augmentation of the example is that if the class group is not torsion, then there are overrings which are not localizations). Theorem 4.4. Let D be a Dedekind domain satisfying one of the following. a) |Cl(D)| ≤ 5. b) Cl(D) = Zpn . c) Cl(D) = ⊕ki=1 Z2 . Then the following statements are equivalent. 1. D is an HFD. 2. D is a SHFD. 3. D is a LHFD. We conclude this brief section with a couple of observations. The first is that the SHFD property is very strong in the sense that if R is an SHFD then the Krull dimension of R is no more than 1 (the introductory example at the beginning of the section of the overring pair K[x, y] ⊆ K[x, xy , xy2 , xy3 , · · · ] illustrates very plainly the hazards of domains where the Krull dimension is 2 or more). Finally, if G is a finite abelian group NOT of the form a), b), or c) in the theorem, then there exists a Dedekind domain D with Cl(D) = G such that D is not an HFD but every proper overring is an HFD. Some interesting generalizations of HFDs (e.g., “congruence” HFDs and kHFDs) are studied in [15].

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Polynomial Extensions

One of the most famous and useful properties of UFDs is the fact that unique factorization in a coefficient ring R makes a smooth transition to the polynomial ring R[x]. This result, originally due to Gauss, makes polynomial computations (over a well-behaved coefficient ring) quite tractable. Theorem 5.1. If R is a UFD, then R[x] is a UFD. The standard proof of this theorem uses the notion of “content” of a polynomial (recall that the content of a polynomial f ∈ R[x] is the ideal of R generated by its coefficients), but a more streamlined proof can be given using the ideal-theoretic characterization of UFDs (any nonzero prime contains a nonzero prime element). This theorem is a classic and begs the analogous question for HFDs. In [48], Zaks proved the following generalization of this theorem to a special class of HFDs. Theorem 5.2. If R is a Krull domain, then R[x] is an HFD ⇐⇒ |Cl(R)| ≤ 2. Unfortunately, Zaks’ result does not generalize to all HFDs as the following example will show. √ √ √ Example 5.3. Consider the order R := Z[ −3] ⊆ Z[ ω] where ω = 1+ 2 −3 . The √ fact that R is an HFD is easily shown (for example, see [48]). Intuitively, Z[ −3] is an HFD because the integral √ closure, Z[ω], is a UFD and a norm argument shows that every irreducible in Z[ −3] is prime in Z[ω]. Now consider the factorization (2x − 2ω)(2x − 2ω) = (2)(2)(x2 − x + 1) where ω is the complex conjugate of ω. Clearly, the right hand side of the above equation has at least three irreducible factors (in fact, precisely three), and so if both of the factors on the left are ir√ reducible, then Z[ −3][x] is not an HFD. And indeed this is the case, as any irreducible factorization of 2x − 2ω (respectively 2x − 2ω) must consist of a constant √ in Z[ −3] of norm 2 or 4, and a degree one polynomial factor. It is easy to check that there is no such factorization. Although the above example demonstrates the impossibility of a full generalization of Gauss’ theorem to general HFDs, it does provide some insight. A slight tweaking of this example to a slightly more general setting allows us more to glean the following theorem [17]. Theorem 5.4. If R[x] is an HFD, then R is an integrally closed HFD. The proof of this theorem may be found in [17], but the example above holds almost the entire content of this theorem. The interested reader is encouraged to “play” with the√above example; it is easily seen that the failure of the half-factorial property in Z[ −3] is a direct consequence of the fact that the coefficient ring is not integrally closed. As a corollary to the previous theorem (coupled with the results from [48]), one can classify all Noetherian polynomial HFDs.

Extensions of Half-Factorial Domains: A Survey

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Corollary 5.5. Let R be Noetherian. The following conditions are equivalent. a) R[x] is an HFD. b) R[x1 , x2 , · · · , xn ] is an HFD for all n ≥ 1. c) R is a Krull domain with |Cl(R)| ≤ 2. Proof. It is known from [48] that if R is a Krull domain, then R[x] is an HFD if and only if |Cl(R)| ≤ 2, which gives c) implies a). For a) implies c), we assume that R[x] is an HFD. This means that R is integrally closed, and since R is Noetherian, R is a Krull domain. We apply Zaks’ result again to obtain c), and hence a) and c) are equivalent. Certainly b) implies a) and to get the converse, we merely note that if R is a Krull domain, then so is R[x1 , · · · , xn ] and what is more, the class number is stable upon the adjunction of any finite number of indeterminates.

6

Power Series Extensions

Power series extensions are oftentimes more problematic than polynomial extensions. Many classical results that hold (in general commutative algebra) sometimes fail wildly in the setting of power series. Passing to a completion (even an x–adic one) is sometimes a bit tricky and some nice properties may be lost. For the sake of perspective, a striking example of this phenomenon is in dimension theory. A classical result for polynomials is that if the (Krull) dimension of a ring (dim(R)) is finite (say dim(R) = n), then so is the dimension of R[x] (in particular, if dim(R) = n then n + 1 ≤ dim(R[x]) ≤ 2n + 1). This is wildly untrue (and in fact, from the non-Noetherian point of view, usually untrue) in the case of power series rings. In fact, there are 0−dimensional rings whose power series extensions are infinite dimensional. Of course, it should be noted that there are instances in which the behavior of formal power series is at least as nice as the analog behavior in polynomials. One example where this occurs is in the case of the passage of the unit group of a ring to polynomials and power series. It is well-known that U (R) = U (R[x]) and that the set of units in R[[x]] is the set of power series f (x) ∈ R[[x]] such that f (0) ∈ U (R). An even more striking example of good power series behavior involves (semi-) quasi-local rings. It is a central result that R is quasi-local (resp., semi-quasi-local) if and only if R[[x]] is quasi-local (resp. semi-quasi-local). The analogous result for polynomials is not true, since the ring R[x] is never semi-quasi-local. But such nice behavior of power series rings relative to the polynomial case is the exception and not the rule in practice. From a factorization point of view we can find bad behavior as well. For example, there are UFDs which have non-UFD power series extensions [43]. As has been pointed out earlier, if R[x] is an HFD, then R is integrally closed. Since for the polynomial case, the coefficient ring being integrally closed is necessary for R[x] to have a chance at the half-factorial property, intuitively one would (perhaps) expect that R[[x]] being an HFD would demand at least this much. In light of this “intuition” let us revisit an earlier example.

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√ √ √ Example 6.1. Consider the order R := Z[ −3] ⊆ Z[ ω] where ω = 1+ 2 −3 . As noted before, R is an HFD with R[x] failing to be an HFD. We again look at the factorization in Z[x] that demonstrated the loss of the half-factorial property:

(2x − 2ω)(2x − 2ω) = (2)(2)(x2 − x + 1). A close inspection of this factorization shows that, in contrast to the Z[x] case, this does not deny the half-factorial property in Z[[x]]. The reason for this is that the element x2 + x + 1, although an irreducible in Z[x], is a unit in Z[[x]]. Hence up to units, the factorizations on both sides of the above equation are of length 2. In fact, in the example above, R[[x]] is, in fact, an HFD. Much more general situations are considered in [22]. We will outline a proof that the above example is a (nonintegrally closed) example of an HFD such that R[[x]] is an HFD. A more thorough treatment of this phenomenon can be found in [22]. Theorem 6.2. Let R be a 1−dimensional domain with integral closure R and conductor I. Also suppose that every nonzero coset of R/I can be written in the form u + I with u ∈ U (R). Then R is a UFD implies that R[[x]] is an HFD. Before beginning this proof, we note that, in particular, the hypotheses apply to √ the ring√Z[ −3]. Indeed, the integral closure of this ring is (the UFD) Z[ω] where √ ω = 1+ 2 −3 and the conductor of Z[ω] to Z[ −3] is the ideal 2Z[ω] = P. Since this conductor ideal is prime, the quotient ring Z[ω]/2Z[ω] is isomorphic to F4 , the field of 4 elements. It is easy to see that the nonzero cosets can be written in the form 1 + P, ω + P, and ω 2 + P. We will now give an outline of the proof of the theorem. Proof. We claim that every irreducible element of R[[x]] is again irreducible in R[[x]]. If not, then we factor an irreducible f ∈ R[[x]] as gh with g, h ∈ R[[x]]. First note that if both h and g are in I[[x]], then this is a direct contradiction. We begin by assuming that neither g nor h is an element of I[[x]]; we write g(x) = a0 + a1 x + · · · + ak−1 xk−1 + xk (bk + bk+1 x + · · · ) where bk is the first term not in the conductor I. We write bk = u1 + Ig with u1 a unit of R and Ig ∈ I. Collapsing notation we write g = g + xk (ug + Ig ) where ug is a unit power series with constant coefficient u1 ∈ U (R) and g ∈ I[[x]]. In a similar fashion, we write h = h + xm (uh + Ih ) with uh a unit power series with constant coefficient v1 ∈ U (R), h ∈ I[[x]], and Ih ∈ I. Note that gh =gh + gxm (uh + Ih ) + hxk (ug + Ig )+ + xk+m (ug uh + ug Ih + uh Ig + Ig Ih ).

Extensions of Half-Factorial Domains: A Survey

57

Since h, g, Ig , and Ih are in I[[x]] and g and h are in R[[x]], we obtain that ug uh is an element of R[[x]]. As a consequence we note that since g = g + xk (ug + Ig ) we have that uh g = uh g + xk (uh ug + uh Ig ) and so uh g ∈ R[[x]]. Similarly, we obtain that ug h ∈ R[[x]]. We now note that f = gh = (uh g)(ug h)(uh ug )−1 and we have a contradiction. The final case to consider is the case when precisely one of the factors, g or h, is an element of I[[x]]. We will assume without loss of generality that g = g is the factor in I[[x]]. In this case, uh g ∈ R[[x]] and we have f = gh = (uh g)(u−1 h h) which is again a contradiction. Now that we have established that every irreducible in R[[x]] is irreducible in R[[x]], we observe that if we have two irreducible factorizations in R[[x]] f1 f2 · · · fn = g1 g2 · · · gm then each fi , gj is irreducible in R[[x]] which is a UFD (since R is a 1 dimensional UFD and hence a PID). Hence n = m. We cannot resist highlighting what we believe to be an interesting implication of this example. Corollary 6.3. There exist HFDs R such that the half-factorial property is lost in R[x] and regained in R[[x]]. √ Proof. If one considers the ring R := Z[ −3], then this is an HFD such that R[x] is not an HFD, since R is not integrally closed. Nonetheless, the above result shows that R[[x]] is indeed an HFD. Here is a final observation along these lines. Corollary 6.4. If R[[x]] is a UFD, then R and R[x] are UFDs. If R[[x]] is an HFD, then R is an HFD, but R[x] is not necessarily an HFD. Proof. First we note that any irreducible element of R remains irreducible in R[[x]]. Indeed,  if π ∈ Irr(R) and π  = f (x)g(x) with f (x), g(x) ∈ R[[x]] then we write ∞ i i f (x) = ∞ i=0 ai x and g(x) = i=0 bi x and factor ∞ ∞ π=( ai xi )( bi xi ). i=0

i=0

It is immediate that we get the factorization

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π = a0 b 0 and since π ∈ Irr(R), either a0 or b0 is a unit in R, forcing either f (x) or g(x) to be a unit in R[[x]]. This establishes our claim. Given this claim, we consider the following factorization in R π1 π2 · · · πn = ξ1 ξ2 · · · ξm with each πi , ξj ∈ Irr(R). If R[[x]] is an HFD, then all of the above irreducible elements remain irreducible in R[[x]] and since R[[x]] is an HFD, we must have n = m and R is an HFD. Additionally, if R[[x]] is a UFD, then every irreducible element of R[[x]] is prime, and hence each πi and ξj is a prime element of R[[x]] (and it is easy to see that the elements are therefore prime as elements of R). Hence R is a UFD. We now have that if R[[x]] is a UFD (respectively HFD) then R is a UFD (respectively HFD). The fact that if R[[x]] is a UFD implies that R[x] is a UFD follows from the previously mentioned result of Gauss, and the absence of the analogous result for HFDs is demonstrated by the above example. We remark here that one might notice that there is a similar result for the case of (semi)quasi-local rings. That is, if R[[x]] is (semi)quasi-local then R is (semi)quasi-local, but R[x] is not. One thing to contrast, however, is that R[x] is never (semi)quasi-local. As was noted before, the above technique is used with some success in [22] in a more general setting. In the previous section, a classification of all Noetherian polynomial HFDs was given, but a general classification for the power series case (even for Noetherian rings) remains elusive. In hindsight, this might not be too surprising since a good characterization of when the UFD property is preserved in power series extensions is not known. One of the best theorems in this sense is the result that states that if R is a PID then R[[x1 , x2 , · · · , xn ]] is a UFD for all n ≥ 1. The class of PIDs is precisely the class of one-dimensional UFDs and it should be noted that the classical example of a UFD, R, such that R[[x]] is not a UFD is a two-dimensional Noetherian domain. This demonstrates, in a certain sense, that one does not have to travel far from the class of PIDs to find examples of bad factorization behavior in power series extensions.

7

Extensions of the Form A + xB[x] and A + xI[x]

A key source of examples in commutative algebra is the so-called “D + M” construction. Recently the D + M construction has shown that its utility extends to the study of factorization (often as a rich source of “bad” factorization behavior). A useful illustration of this is the ring Z + xQ[x]. This ring is one of the simplest examples of the previous statement. It is nonatomic, since the element x is a nonunit that possesses no finite factorization into irreducibles. To see this, we note that if x = π1 π2 · · · πn is an irreducible factorization of x in Z + xQ[x], then a degree argument, coupled with the fact that Q[x] is a PID, shows that precisely one of the irreducible factors

Extensions of Half-Factorial Domains: A Survey

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(say π1 without loss of generality) must be an associate of x as an element of Q[x]. We write π1 = qx with q ∈ Q and note that π1 = 2( q2 )x, which contradicts the irreducibility of π1 in Z + xQ[x]. The constructions A + xB[x] and A + xI[x], specializations of the standard “D + M” construction, are gold mines for interesting (especially nonunique) factorizations [10] (note that it is assumed that A ⊆ B are domains and in the second case that I ⊆ A is an ideal). Indeed, there is no way that an A + xB[x] construction can ever be a UFD unless A = B and A is a UFD (in fact, unless A = B and A is completely integrally closed, A + xB[x] is not completely integrally closed). Also note that if I is a proper nonzero ideal of A, then x is almost integral over A + xI[x] and hence A + xI[x] is not a UFD either. For the A + xI[x] construction, assuming that A is an UFD is a very natural starting point. With respect to this condition, Gonzalez, Pellerin and Roberts proved the following nice theorem [30]. Theorem 7.1. Let A be a UFD and R := A + xI[x] with I a proper nonzero ideal of A. Then R is an HFD if and only if I is a prime ideal of A. Actually much more is shown in this paper. They show, in fact, that R is atomic and that ρ(R) is finite if and only if I = P1 · · · Pk with the Pi ’s being noncomparable primes with at most one of them nonprincipal (and if this is the case then ρ(R) = k). This result has also been extended to the case where A is a Dedekind domain. It has been shown that if |Cl(A)| = ∞ then ρ(R) = ∞. In the case where |Cl(A)| < ∞ and I is a product of k distinct principal primes, we have the following from [37]. Theorem 7.2. Let A be a Dedekind domain with finite class group and let I ⊆ A , be the product of k distinct principal primes. Then ρ(A + xI[x]) = k + D(Cl(A)) 2 where D(Cl(A)) is the Davenport constant of Cl(A). Here we declare that the Davenport constant is 0 if the class group is trivial. We now present some recent advances in the spirit of the work of Gonzalez, Pellerin, and Roberts. This next result can be found in [23] and is an extension of some of the work of Kim [33]. Theorem 7.3. If K is a field and B is any domain containing K then K + xB[x] is an HFD if and only if B is integrally closed. We now look at the A + xI[x] construction. Question 7.4. If K ⊆ B is an extension of domains with K a field and B a UFD, and P ⊆ B is a prime ideal, then is K + xP[x] an HFD? To help in this, we record the following result [23]. Theorem 7.5. Let A ⊆ B be an extension of domains with A a UFD and let I ⊆ B be a proper ideal. If I contains an irreducible element π ∈ B such that π 2 has no nontrivial factor in A, then A + xI[x] is not an HFD. Proof. (πx2 )(π 2 x) = (πx)(πx)(πx).

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Corollary 7.6. Let K ⊆ B be an extension of domains with K a field and let I ⊆ B be a proper ideal. If I contains an irreducible element of B, then K + xI[x] is not an HFD. This answers the question raised above. Once again, we see the usefulness of the ideal theoretic characterization of UFD. This time, the fact that the prime ideal xI[x] ⊆ K + xI[x] contains no prime element helps us to do something “bad”.

8

Integral Closure

As UFDs are (completely) integrally closed, a very natural question is “are HFDs integrally closed?” Earlier examples quickly show that√this is not the case; for example, we have seen the concrete example of the ring Z[ −3] which is a nonintegrally closed HFD. Along this line of thought, a natural next question is, “is the integral closure of an HFD still an HFD?” This question was answered in the negative by the present author in [21]. We will outline this result (as the construction is somewhat intricate). But we will note that, in some sense, this counterexample is a bit unsatisfactory. More specifically, the reason why the integral closure of the HFD presented in [21] fails to have the half-factorial property is that it fails to be atomic. From the point of view of the current accepted definition of HFD, this example is a bona fide counterexample, but only because of the loss of atomicity. As one will see from the outline of the construction, it turns out that any two irreducible factorizations in the integral closure of this domain are of the same length. (So from the point of view of the original discarded definition of Zaks, the integral closure of this HFD is again an HFD.) We will outline this example in a couple of steps and give the briefest idea of how it works. This construction will deliver a specific example of an HFD whose integral closure does not have the half-factorial property, but it should be noted that this construction can be generalized (for example, the choice of the value group and residue fields are only for computational convenience). 1. We begin with the semigroup ring F2 [x; Q+ ], and we denote the maximal ideal generated by the “monomials” by M (i.e., M is the ideal generated by the set {xα }α∈Q+ ). 2. We now form the localization R := F2 [x; Q+ ]M . This localization is a 1− dimensional nondiscrete valuation domain with value group Q and residue field F2 . 3. Abusing the notation, we let M ⊆ R be the maximal ideal and consider R1 := F2 + tM[t] where t is a polynomial indeterminate. It is worth noting at / R1 . It is this step that the element t ∈ / R1 ; additionally for all α ∈ Q+ , xα ∈ also worth noting that R1 is integrally closed. Indeed, its integral closure is certainly contained in F2 + tR[t] since R[t] is integrally closed and F2 + tR[t] is integrally closed in R[t] (the integral closure of F2 in R is F2 ). It is now easily seen that no element of (F2 + tR[t]) \ (F2 + tM[t]) is integral over F2 + tM[t]. 4. In this step, we simplify by localizing at the ideal tM[t]. The effect of this is to get rid of irreducible “polynomials” that may crop up in the ring R1 and

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clarifies factorization behavior. We let R2 := (F2 + tM[t])tM[t] and note that this is an HFD. Indeed, if π ∈ R2 is irreducible, then π ∈ tM[t]R2 , and this observation allows us to conclude that every irreducible in R2 can be written in the form u(xα1 t + 2 xα2 t2 + · · · + n xαn tn )  with each u a unit in R2 , i ∈ U (R2 ) {0}, and each αi ∈ Q+ . At this step it is fairly straightforward to verify that R2 is an HFD. Indeed, a general nonzero nonunit in R2 assumes the form (with notation as above) u(xα0 tm + 1 xα1 tm+1 + · · · + k xαk tm+k ). In this form it is easy to see via a “degree argument” that the above element has at most m irreducible factors (since no rational power of x is an element of R2 ). The fact that if m > 1, the above element actually has a factorization with m elements follows from the fact that the original valuation domain from earlier in this construction was nondiscrete. Note that if α = min(α0 , α1 , · · · , αk ), then we can factor the above element u(xα0 tm + 1 xα1 tm+1 + · · · + k xαk tm+k ) α

= u(x 2(m−1) t)m−1 (xα0 − 2 t + 1 xα1 − 2 t2 + · · · + k xαk − 2 tk+1 ). α

α

α

Of course this element may have many factorizations, but the above shows that each has length m. 5. The problem that we encounter now is that the domain from the previous step, R2 , is integrally closed, as it is a localization of the integrally closed domain, R1 . In particular, the strategy is to make the elements {xα }α∈Q+ integral and to ensure that the integral closure of our target domain is not atomic. To continue our construction we wish to adjoin elements to R2 that make this happen (while not destroying the half-factorial property). With this in mind, we consider R3 := (F2 + tM[t])tM[t] [x + t], obtained by adjoining the element x + t of the quotient field of R2 to R2 . In the ring R3 there are two prime ideals worth noting. The first is the prime ideal P = (x + t) (it is fairly easy to show that x + t is a prime element of R3 ) and second is the prime ideal Γ which is the extension of the maximal ideal tM[t]R2 to our new domain R3 . 6. We now wish to, once again, simplify for the purposes of factorization. We let R4 := (F2 + tM[t])tM[t] [x + t](P S Γ)c . By construction, the only prime ideals of R4 are (the extensions of) P and Γ. A simple computation shows that the only nonunits of R4 are of the form (x + t)n h where h can be thought of as an element of R2 (up to units). If we consider any irreducible factorization of (x + t)n h, we note first that since x + t is a prime element of R4 , we can ignore the effect of the (x + t)n factors. An argument similar to the proof that any nonunit in R2 has the half-factorial property (by counting the multiplicity of t) applies here and so R4 is an HFD.

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Coykendall 7. To see that the integral closure of R4 is not an HFD (more specifically, not atomic) note that 1 1 tx1+ n xn = tx 1

and so x n is in the quotient field of R4 . Also note that 1

1

(x n )2n + (x + t)(x n )n + xt = 0 1

and so x n is a root of the monic polynomial Y 2n + Y n (x + t) + xt ∈ R4 [Y ]. It is easy to see that the existence of the factorizations 1

x = (x n )n for all n deny the atomicity of the integral closure of R4 . Purists (those who would like to follow the original concept of HFDs from Zaks) will be understandably displeased with this example, for, in a certain sense, it is cheating. Along lines similar to the above, it can be shown that the integral closure of R4 contains every (positive) rational power of x and every positive integral power of t and hence is a localization of a polynomial ring over a (nondiscrete) valuation domain. As a consequence, it can be shown that any two irreducible factorizations of the same element are of equal length. The question still remains as to whether or not the integral closure of an HFD is again an HFD if the integral closure remains atomic.

9

Orders in Rings of Integers

Despite the title of this chapter, we take a step back and look at the dual notion of underrings of half-factorial domains (here, by “underring” of a ring T , we mean a ring R ⊆ T such that R and T share the same quotient field). Much of the study here is motivated by questions concerning orders in rings of algebraic integers. We briefly recall that an order in a ring of integers is a ring R such that the integral closure R is the full ring of algebraic integers. The study of the factorization behavior is well-motivated by questions in algebraic number theory and is important in a number of applications including representation of integers by quadratic forms. For an interesting example of the ramifications of the half-factorial property with respect to the representation of integers by quadratic forms applied to finite geometry, see [24] for an example. Much of the theory of factorization has its roots (and is still motivated by) questions from number theory. In particular, Carlitz’ paper was motivated by a number-theoretic question of Narkiewicz. Carlitz’ result gave an interpretation of the half-factorial property in terms of the class group, which is a subject that has been of interest in algebraic number theory for a very long time. In the spirit of factorization and the class group, here is a very famous open conjecture due to Gauss that has resisted solution for quite some time. Conjecture 9.1. There is an infinite number of real quadratic UFDs.

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Despite the apparent difficulty of Gauss’ conjecture, we offer another conjecture along these lines. Conjecture 9.2. There is an infinite number of real quadratic HFDs that are not UFDs. The curious reader may at this juncture wonder about the status of the questions for the imaginary case. In light of the fact that the Carlitz result allows us to completely determine the half-factorial property via the class number (with class number one being a UFD and class number 2 being a non-trivial HFD), we present a couple of results that answer this question. The following result is due to H. Stark [44]. Theorem 9.3. Let R be the ring of integers of an imaginary quadratic field. Then |Cl(R)| = 1 if and only if the field discriminant is −3, −4, −7, −8, −11, −19, −43, −67, or −163. The proof of the fact that the rings of integers corresponding to the field discriminants listed above are all UFDs is fairly straightforward. The converse of the theorem was only completely solved in the late 1960s, when Stark showed that there was no tenth imaginary quadratic field. A result from analytic number theory states that there are only finitely many totally complex abelian extensions of Q of given degree and class number ([35]). This makes the existence of only finitely many imaginary quadratic HFDs immediate. Due to another result of Stark [45] we can list all imaginary quadratic HFDs that are rings of integers. Theorem 9.4. Let R be the ring of integers of an imaginary quadratic field. Then |Cl(R)| = 2 if and only if the field discriminant is −15, −20, −24, −35, −40, −51, −52, −88, −91, −115, −123, −148, −187, −232, −235, −267, −403, or −427. It is interesting to note that before this list was complete, an initial upper bound was set for the size of the discriminant of imaginary quadratic fields K with the class number equal to 2. This bound was |d(K)| < 101030 . Stark’s paper [45] found all of the fields, and it is interesting to note that, in fact, |d(K)| ≤ 427. There is a striking difference to consider when looking for HFDs that occur as orders in rings of algebraic integers, for unlike UFDs, HFDs do not have to be integrally closed. UFDs that occur as orders must occur as the full ring of algebraic integers (and in some sense, this may make them easier to find, since the tools generally used in this vein become more unwieldy when one leaves the assumption “integrally closed” behind). The upshot of all this is that, in searching for HFDs in rings of integers, we do not have to restrict to the full ring of integers; we can consider proper orders. When considering which orders may contain HFDs, the following theorem is useful. It demonstrates that one can restrict the search for HFDs to orders inside rings of integers which are HFDs. It is worth noting that the following theorem is a positive answer to the question, “Is the integral closure of an HFD again an HFD?” for number theorists. Theorem 9.5. Let R be an order in a ring of algebraic integers with integral closure R. If R is an HFD, then R is an HFD.

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If the search for HFDs in imaginary quadratic fields is augmented to include the nonintegrally closed case, we are done quickly. A proof of the following theorem using norms can be found in [19]. √ Theorem 9.6. The ring Z[ −3] is the unique nonintegrally closed imaginary quadratic HFD. Since we know that any nonintegrally closed HFD that is an order in an imaginary quadratic field must appear as a subring of one of the 27 imaginary HFDs (the 9 UFDs and 18 nontrivial HFDs listed above), we could prove this theorem by restricting to searching for subrings of these 27 rings of integers. It is of interest to note, however, that the above theorem was known before it was known that HFDs as orders must be subrings of integrally closed HFDs (see [19] for a direct result or [31] for at least an implicit one). The proof of this theorem in [19] utilizes the norm, and since in the case of imaginary quadratic fields, the norm is positive definite, the proof is rather quick and clean (it basically depends only upon representing certain integers via the norm form). But it should be noted that although perhaps elegant in this case, generalizing this technique to more general fields (even real quadratic fields) is fraught with difficulties because of the difficulty determining integer representations when the form may take on negative values (and it also should be pointed out that even for relatively small degree fields, the norm form becomes quite large and hard to handle very quickly). Despite the computational difficulties inherent in actually applying the norm, the above discussion and results beg the question as to how much information can be obtained by looking at factorizations from the point of view of norms. We begin by recalling a concept known as elasticity. Definition 9.7. Let S be an atomic set (that is, every element of S can be factored into irreducible elements of S). We define the elasticity of S to be n |α1 α2 · · · αn = β1 β2 · · · βm } m where αi , βj are irreducible elements of S for all i, j. ρ(S) = sup{

Basically, elasticity is a global measure of how long ratios of differing (yet equal) factorizations in S can be. The definition that we give slightly generalizes the definition of elasticity of a domain given in [46] and studied in (among others) [12, 28, 29, 30]. Here is a theorem that has appeared in a paper by Valenza [46] and was improved by Narkiewicz [36]. Theorem 9.8. Let R be a ring of algebraic integers. Then ρ(R) = D 2 where D is the Davenport constant of the class group of R if R is not a UFD (and if R is a UFD, then ρ(R) = 1). We remind the reader that the Davenport constant is the invariant of a finite abelian group G that is the smallest positive integer, n, such that any sequence of n (not necessarily distinct) elements of G has a subsequence that sums to 0. The Davenport constant has been studied rather extensively in the realm of pure group theory and has become a topic of much interest in commutative algebra because of its natural connection with factorization. There are many remaining open questions relating to the Davenport constant (e.g., “Is there a closed form representation of

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the Davenport constant?”). The answers to many of these questions would have far-reaching ramifications in factorization theory. For a more in-depth look at the theory of the Davenport constant and some of its applications, see [12, 13, 26]. We now produce a theorem that is useful in determining factorization behavior in rings of integers. This theorem is a hybrid of a couple of theorems that can be found in [16] and [20]. Theorem 9.9. Let F/Q be a Galois extension, R the ring of algebraic integers of F , and S the multiplicative set of integral norms from R to Z. Then R is an HFD if and only if ρ(S) = 1, and additionally, R is a UFD if and only if S has unique factorization into irreducibles. It should be noted that, in general, for Galois extensions we have that ρ(R) ≥ ρ(S). Perhaps this is not too surprising since one would expect that the set of norms would lose some of the factorization information that is contained in R. It is a subtle point that although ρ(R) ≥ ρ(S) in general, one cannot lose “too much” information in the sense that ρ(R) = 1 if and only if ρ(S) = 1. We would also like to highlight that the “Galois” assumption is very necessary here. In the non-Galois case, an example of a non-Galois extension with normal closure having Galois group S5 can be produced to show the failure of this theorem in general. Mainly because of its more complicated unit structure, the real case cannot be disposed of as easily as the imaginary case. We supply the following conjecture. Conjecture 9.10. There are infinitely many real quadratic HFDs (even inside √ Z[ 2]). This result makes use of the “boundary map” which we define below. Definition 9.11. Let R be an HFD with quotient field K. We define the boundary map ∂R : K \ {0} −→ Z via ∂R (α) = n − k ···πn with each πi , ξj ∈ Irr(R). In the where α ∈ K \ {0} is of the form α = πξ11πξ22···ξ k case that R = K we declare that ∂R ≡ 0.

The boundary map is a homomorphism from the multiplicative group K \ {0} to the additive group Z. The definition of the boundary map assumes that the domain, R, is an HFD and this is a necessary (and sufficient) condition for the boundary map to be well-defined. In Zaks’ paper [48] a function called the length function was defined. A length function is a function from the nonzero nonunits of R to the natural numbers such that f (st) = f (s) + f (t) for all s, t nonzero, nonunit elements of R. Of course, HFDs are precisely the domains which admit a well-defined length function such that f (x) = 1 for all irreducible elements x ∈ R. The boundary map may be thought of as a natural generalization of this length function to a homomorphism from the set of nonzero elements of the quotient field of R to the integers. In [18], the boundary map proved to be a useful tool in showing that, in the case of orders in rings of algebraic integers, the integral closure of an HFD is again an HFD. Since the time of the paper [18], more work has been done on exploring

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properties of the boundary map and attempts have been made, with some success, to glean properties of overring behavior of HFDs. One recent avenue of interest is the so-called “boundary valuation domains” introduced by J. Maney. Definition 9.12. Let R be an HFD with quotient field K and boundary map ∂R . R is called a boundary valuation domain if for all α ∈ K such that ∂R (α) = 0, either α or α−1 is an element of R. This definition was motivated by an attempt to understand the overring behavior of HFDs in the spirit of “ordinary” valuation domains. A good example of a boundary valuation domain is any Noetherian valuation domain, and a more general example is given by the construction F + xK[[x]], where F  K are fields. In [34], many interesting properties of the boundary valuation domains are explored, and insights into the overring behavior of HFDs are obtained. Additionally, boundary valuation domains are completely characterized using groups of divisibility. Recently, Maney has also shown that the class of boundary valuation domains coincides with the class of atomic pseudo-valuation domains. We close this section by pointing to some interesting recent work along these lines. In a fairly recent sequence of papers [38, 39, 40, 41] M. Picavet-L’Hermitte has done some striking work (often utilizing t-closedness, seminormality, and the concept of weak factoriality) that has advanced the understanding of the factorization behavior in orders of rings of integers. Finally, we leave the reader with a list of questions that the present author would like to see answered. This list of questions is by no means exhaustive (and is motivated by the present author’s current tastes). There are myriad interesting open questions with respect to the structure and behavior of HFDs (and certainly many, many more in factorization theory in general). It seems clear that despite recent vigorous progress, factorization theory has very far to go. Questions: 1. If R[x] is an HFD, does it follow that R[x, y] is an HFD? (Note: it is known that the answer to this is “yes” if R is Noetherian.) 2. If R[[x]] is an HFD, then is R[[x, y]] an HFD? (Note: this one might be quite difficult; the analogous question for UFDs has been open for nearly 40 years [43].) 3. If the integral closure of an HFD is atomic, then is it an HFD? 4. Find a general ideal-theoretic characterization of HFDs (if it can be done). 5. If R is an HFD, find general conditions for which R[[x]] is an HFD. 6. Classify all (non-Noetherian) polynomial HFDs. 7. Under what conditions is an affine ring an HFD? √ 8. Is there an infinite number of (real) quadratic HFDs (inside Z[ 2])? 9. Find a closed-form representation of the Davenport constant (if one exists).

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10. Characterize the relationship between the elasticity of an order in a ring of integers and the elasticity of its set of norms. (Note: this is only partially known even in the case that the order is a full ring of integers and its quotient field is Galois over Q.)

Bibliography [1] D.D. Anderson, Factorization in Integral Domains, Lecture Notes in Pure and Applied Mathematics, 189 (1997), Marcel Dekker, New York. [2] D.D. Anderson, D.F. Anderson and M. Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra 69 (1990), 1–19. [3] D.D. Anderson, D.F. Anderson and M. Zafrullah, Factorization in integral domains, II, J. Algebra 152 (1992), 78–93. [4] D.D. Anderson and J.L. Mott, Cohen-Kaplansky domains: integral domains with a finite number of irreducible elements, J. Algebra 148 (1992), 17–41. [5] D.F. Anderson, S.T. Chapman and W.W. Smith, Some factorization properties of Krull domains with infinite cyclic divisor class group, J. Pure Appl. Algebra 96 (1994), 97–112. [6] D.F. Anderson, S.T. Chapman and W.W. Smith, On Krull half-factorial domains with infinite cyclic divisor class group, Houston J. Math. 20 (1994), 561–570. [7] D.F. Anderson, S.T. Chapman and W.W. Smith, Overrings of half-factorial domains, Canad. Math. Bul. 37 (1994), 437–442. [8] D.F. Anderson, S.T. Chapman and W.W. Smith, Overrings of half-factorial domains, II, Comm. Algebra 23 (1995), 3961–3976. [9] D.F. Anderson and J. Park, Locally half-factorial domains, Houston J. Math. 23 (1997), 617–630. [10] V. Barucci, L. Izelgue and S.E. Kabbaj, Some factorization properties of A + XB[X] domains, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 185 (1997), 69–78. [11] L. Carlitz, A characterization of algebraic number fields with class number two. Proc. Amer. Math. Soc. 11 (1960), 391–392. [12] S.T. Chapman, On the Davenport constant, the cross number and their application in factorization theory, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 171 (1995), 167–190. 68

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[13] S.T. Chapman and A. Geroldinger, Krull domains and monoids, their sets of lengths and associated combinatorial problems, Lecture Notes in Pure and Applied Mathematics, Marcel-Dekker, 189 (1997), 73–112. [14] S.T. Chapman and W.W. Smith, Factorization in Dedekind domains with finite class group, Israel J. Math 71 (1990), 65–95. [15] S.T. Chapman and W.W. Smith, On the HFD, CHFD and k-HFD properties in Dedekind domains, Comm. Algebra 20 (1992), 1955–1987. [16] J. Coykendall, Normsets and determination of unique factorization in rings of algebraic integers, Proc. Amer. Math. Soc. 124 (1996), no. 6, 1727–1732. [17] J. Coykendall, A characterization of polynomial rings with the half-factorial property, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 189 (1997), 291–294. [18] J. Coykendall, The half-factorial property in integral extensions, Comm. Algebra 27 (1999), 3153–3159. [19] J. Coykendall, Half-factorial domains in quadratic fields, J. Algebra 235 (2001), 417–430. [20] J. Coykendall, Elasticity properties preserved in the normset, J. Number Theory 94 (2002), no. 2, 213–218. [21] J. Coykendall, On the integral closure of a half-factorial domain, J. Pure Appl. Algebra 180 (2003), 25–34. [22] J. Coykendall, The half-factorial property in power series rings, preprint. [23] J. Coykendall, T. Dumitrescu and M. Zafrullah, The half-factorial property and domains of the form A + xB[x], submitted. [24] J. Coykendall and J. Dover, Sets with few intersection numbers from Singer subgroup orbits, European J. Combin. 22 (2001), no. 4, 455–464. [25] J. Coykendall and M. Zafrullah, AP-domains and unique factorization, J. Pure Appl. Algebra 189 (2004), 27–35. [26] A. Geroldinger and R. Schneider, On Davenport’s Constant, J. Combin. Theory Ser. A 61 (1992), 147–152. [27] R. Gilmer, W. Heinzer and W.W. Smith, On the distribution of prime ideals within the ideal class group, Houston J. Math. 22 (1996), 51–59. [28] N. Gonzalez, Elasticity and ramification, Comm. Algebra 27 (1999), 1729–1736. [29] N. Gonzalez, Elasticity of A + XB[X] domains, J. Pure Appl. Algebra 138 (1999), 119–137. [30] N. Gonzalez, S. Pellerin and R. Robert, Elasticity of A+ XI[X] domains where A is a UFD, J. Pure Appl. Algebra 160 (2001), no. 2-3, 183–194.

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[31] F. Halter-Koch, Factorization of Algebraic Integers, Ber. Math. Stat. Sektion im Forschungszentrum 191 (1983). [32] I. Kaplansky. Commutative Rings. University of Chicago Press, 1974. [33] H. Kim, Examples of half-factorial domains, Canad. Math. Bull. 43 (2000), no. 3, 362–367. [34] J. Maney, Boundary valuation domains, J. Algebra 273 (2004), 373–383. [35] W. Narkiewicz, Elementary and analytic theory of algebraic numbers, SpringerVerlag, 1990. [36] W. Narkiewicz, A note on elasticity of factorizations, J. Number Theory 51 (1995), no. 1, 46–47. [37] S. Pellerin and R. Robert, Elasticity of A + XI[X] domains where A is a Dedekind domain, J. Pure Appl. Algebra 176 (2002), no. 2-3, 195–212. [38] M. Picavet-L’Hermitte, Some remarks on half-factorial orders, Rend. Circ. Mat. Palermo 52 (2003), no. 2, 297–307. [39] M. Picavet-L’Hermitte, Weakly factorial quadratic orders, Arab. J. Sci. Eng. Sect. C Theme Issues 26 (2001), no. 1, 171–186. [40] M. Picavet-L’Hermitte, Factorization in some orders with a PID as integral closure, Algebraic number theory and Diophantine analysis (Graz, 1998), 365– 390, de Gruyter, Berlin, 2000. [41] M. Picavet-L’Hermitte, Seminormality and t-closedness of algebraic orders, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 205 (1999), 521–540. [42] M. Roitman, Polynomial extensions of atomic domains, J. Pure Appl. Algebra 87 (1993), 187–199. [43] P. Samuel, On unique factorization domains, Illinois J. Math. 5 (1961), 1–17. [44] H.M. Stark, A complete determination of the complex quadratic fields of classnumber one, Michigan Math. J. 14 (1967), 1–27. [45] H.M. Stark, On complex quadratic fields with class-number two, Math. Comp. 29 (1975), 289–302. [46] R.J. Valenza, Elasticity of factorization in number fields, J. Number Theory 36 (1990), no. 2, 212–218. [47] A. Zaks, Half-factorial domains, Bull. Amer. Math. Soc. 82 (1976), 721–723. [48] A. Zaks, Half-factorial domains, Israel J. of Math. 37(1980), 281-302.

Chapter 4

C-Monoids and Congruence Monoids in Krull Domains by

Franz Halter-Koch1 Abstract Only recently, several types of auxiliary monoids were used for the investigation of the arithmetic of congruence monoids in Dedekind domains and of noetherian domains satisfying some natural finiteness conditions. In this chapter, we introduce C-monoids as a common generalization of all these types of auxiliary monoids and investigate their algebraic and arithmetical properties. We discuss congruence monoids in Krull domains as examples of arithmetical interest.

1

Introduction

In a recent paper [9], devoted to the arithmetic of congruence monoids in Dedekind domains, we introduced abstract congruence monoids (AC-monoids) and C0 monoids as appropriate tools for the arithmetical investigations of the more complicated arithmetically interesting objects. A similar concept (that of a Z-monoid) was used by W. Hassler [17] for the investigation of the multiplicative arithmetic of a large class of finitely generated integral domains. In this chapter, we introduce C-monoids as a common generalization of C0 -, Z- and (all interesting) AC-monoids. C-monoids have a nice algebraic structure: They are v-noetherian (Theorem 4.8) and their complete integral closures are Krull monoids with finite class group and have a nonempty conductor (Theorem 4.6). Every Krull monoid with finite class group is a C-monoid (Theorem 4.7). The arithmetically most important examples of C-monoids are congruence monoids in Krull domains. Among them are the congruence monoids in Dedekind domains investigated in [9] and the noetherian integral domains considered in [17]. 1 This

work was supported by the Austrian Science Fund FWF (Project-Nr. P16770-N12)

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By means of a suitable transfer principle, the arithmetical investigation of a C-monoid can essentially be reduced to that of a C0 -monoid (see [8, Theorem 7.2]). Apart from the arithmetical results already proved in [9], we present here a necessary and sufficient criterion for a C-monoid to have finite elasticity (Theorem 5.1), which is even new for congruence monoids of natural numbers (see [8, Theorem 7.8]). For finitely generated integral domains in general, a stronger criterion for finite elasticity was proved by F. Kainrath [18, Theorem 6.1]. This chapter is organized as follows. In Section 2 we fix our notations concerning monoids and factorization theory. In Section 3 we recall and refine the concept of class semigroups. Section 4 is devoted to the algebraic theory of C-monoids and contains the main results of this chapter. In Section 5 we present a criterion for the finiteness of the elasticity of a C-monoid, and finally in Section 6 we discuss congruence monoids in Krull domains as a main application of this new concept.

2

Preliminaries and Notations

Let N denote the set of positive integers and N0 = N ∪ {0}. For m, n ∈ Z with m ≤ n, we set [m, n] = {x ∈ Z | m ≤ x ≤ n}. For a subset X ⊂ N0 , we denote by gcd(X) ∈ N0 its greatest common divisor. In particular, gcd(X) = 0 if and only if X ⊂ {0}. For any set X, we denote by |X| ∈ N0 ∪ {∞} its cardinality. For convenience and to fix notations, we recall some notions from the theory of monoids and nonunique factorizations which we use in this chapter. Our main reference for the theory of divisibility and the theory of v-ideals is [16, Ch. 10 and 11]. For the theory of Krull monoids we refer to [16, Ch. 20, 22 and 23] and to the survey article [3]. For the basic concepts of nonunique factorizations, we refer to the survey articles in [1], in particular to [7] and [15]. Basic notions. By a semigroup we mean a nonempty set with a commutative and associative law of composition possessing a unit element. By a monoid we mean a cancellative semigroup. Unless stated otherwise, we use multiplicative notation and denote the unit element by 1. For subsets A, B of a monoid H, we set AB = {ab | a ∈ A , b ∈ B}, and for n ∈ N, we set An = A · . . . · A. Semigroup and monoid homomorphisms are assumed to respect the unit element. Subsemigroups and submonoids are assumed to contain the unit element.

For each monoid H, we fix a quotient group q(H) of H, and we denote by H the complete integral closure of H in q(H), consisting of all elements x ∈ q(H) for which there exists some c ∈ H such that cxn ∈ H for all n ∈ N. For subsets X, Y ⊂ q(H), we define (X : Y ) = {z ∈ q(H) | zY ⊂ X}. If H ⊂ D is a submonoid, we tacitly assume that q(H) ⊂ q(D). We denote by H × the group of invertible elements of H and by Hred = {aH × | a ∈ H} the associated reduced monoid of H. We call H reduced , if H × = {1} (and in this case we will identify H with Hred ). For a monoid homomorphism ϕ : H → D we denote by ϕred : Hred → Dred the induced homomorphism of reduced monoids, given by ϕred (aH × ) = ϕ(a)D× . For an integral domain R, we denote by R• = R \ {0} its multiplicative monoid. Let H be a monoid and a, b ∈ H. We call a a divisor of b and we write a | b , if b ∈ aH. We call a and b associated and write a  b , if aH × = bH × . An element u ∈ H is called an atom if u ∈ / H × , and for all a, b ∈ H, u = ab

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implies a ∈ H × or b ∈ H × . We denote by A(H) the set of all atoms of H, and we call the monoid H atomic , if every a ∈ H \ H × is a product of atoms. An element p ∈ H is called a prime , if H \ pH is a submonoid of H, and H is called factorial , if every a ∈ H \ H × is a product of primes. Every prime is an atom, and a monoid is factorial if and only if it is atomic and every atom is a prime. If H is atomic and p ∈ H is a prime, then every a ∈ q(H) has a representation a = pn bc−1 , where b, c ∈ H, p  bc and n ∈ Z. The exponent n is uniquely determined by the classes aH × and pH × in Hred , and we call n = vp (a) the p-adic value of a. The map vp : q(H) → Z is a surjective group homomorphism, called the p-adic valuation. If P is a set of primes of an atomic monoid H and x ∈ q(H), then suppP (x) = {p ∈ P | vp (x) = 0} is called the support of x (in P ). Free monoids. For a set P , we denote by F (P ) the free (abelian) monoid with basis P . F (P ) is a reduced factorial monoid and P is the set of primes of F (P ). Every nonempty subset X of F (P ) possesses a unique greatest common divisor, denoted by gcd(X) ∈ F(P ). Every a ∈ F(P ) has a unique representation in the form  pνp , where νp ∈ N0 and νp = 0 for all but finitely many p ∈ P , a= p∈P

and then νp = vp (a) for all p ∈ P . Sets of lengths and elasticity. Let H be an atomic monoid. For a ∈ H \ H × , we denote by L(a) = {r ∈ N | a = u1 · . . . · ur with u1 , . . . , ur ∈ A(H)} the set of lengths of a . The monoid H is called a BF-monoid , if L(a) is finite for all a ∈ H \ H × . If H is a BF-monoid and a ∈ H, then ρ(a) =

max L(a) ∈ Q>0 min L(a)

is called the elasticity of a and ρ(H) = sup{ρ(a) | a ∈ H \ H × } the elasticity of H. The elasticity is one of the best investigated invariants of nonunique factorizations (see [2] for a survey). Submonoids and class groups. Let D be a monoid and H ⊂ D a submonoid. For x ∈ q(D), we set [x]D/H = xq(H) ∈ q(D)/q(H); we call D/H = {[a]D/H | a ∈ D} the class monoid and q(D)/q(H) = q(D/H) the class group of D modulo H. If H ⊂ D× is a subgroup, then [a]D/H = aH ∈ q(D)/H for all a ∈ D, there are natural epimorphisms D → D/H → D/D× = Dred ⊂ q(D)/D× = q(Dred ), and Hred = H/H × is a submonoid of D/H × . In this special case, we write D/H multiplicatively. In the general situation however, we use additive notation for D/H, so that [ab]D/H = [a]D/H + [b]D/H for all a, b ∈ D, and [1]D/H is the zero element of D/H. A submonoid H ⊂ D is called • cofinal , if aD ∩ H = ∅ for all a ∈ D (equivalently, D/H = q(D/H)). • saturated , if q(H) ∩ D = H (equivalently, H = {a ∈ D | [a]D/H = [1]D/H }).

74

Halter-Koch • divisor-closed , if a ∈ H, b ∈ D and b | a implies b ∈ H (equivalently, yD ∩ H = yH for all y ∈ D).

If H ⊂ D is divisor-closed, then H ⊂ D is saturated, A(H) = A(D)∩H, and factorizations of an element of H into atoms of H amounts to the same as factorizations into atoms of D. A subset E ⊂ H is a divisor-closed submonoid if and only if H \ E is a prime s-ideal of H [16, Definition 6.1]. For any subset E ⊂ H, we denote by [[E]] the smallest divisor-closed submonoid of H containing E. In particular, [[∅]] = H × . Divisor homomorphisms, divisor theories and Krull monoids. A monoid homomorphism ϕ : H → D is called a divisor homomorphism , if ϕ(a) | ϕ(b) implies a | b for all a, b ∈ H. If H ⊂ D is a submonoid, then the injection H → D is a divisor homomorphism if and only if H is a saturated submonoid of D. An arbitrary monoid homomorphism ϕ : H → D is a divisor homomorphism if and only if ϕred : Hred → Dred is a divisor homomorphism. A divisor theory is a divisor homomorphism ϕ : H → F (P ) into a free monoid F (P ) such that for every p ∈ P there exists a finite set E ⊂ H such that p = gcd(ϕ(E)). A monoid homomorphism ϕ : H → F (P ) is a divisor theory if and only ∼ if it induces an isomorphism Hred → ϕ(H) and the injection ϕ(H) → F (P ) is a divisor theory. If ϕ : H → F (P ) and ϕ : H → F (P  ) are divisor theories, then there ∼ exists a unique isomorphism Φ : F (P ) → F (P  ) such that Φ ◦ ϕ = ϕ . A monoid H is a Krull monoid , if it possesses a divisor theory. Every saturated submonoid of a Krull monoid is a Krull monoid. If ∂ : H → F is a divisor theory, then ∂ is cofinal, and we call C(H) = F/∂(H) the (divisor) class group of H. It is (up to canonical isomorphisms) uniquely determined by H. A monoid H is factorial if and only if it is a Krull monoid with trivial divisor class group. A monoid H is a Krull monoid if and only if Hred is a Krull monoid, and then C(H) = C(Hred ). An integral domain R is a Krull domain if and only if R• is a Krull monoid, and then C(R• ) = C(R) is the divisor class group of R as defined in [5]. Products. For two monoids H1 , H2 , we denote by H1×H2 their direct product, and we view H1 and H2 as (divisor-closed) submonoids of H = H1×H2 . Then every a ∈ H has a unique factorization in the form a = a1 a2 with a1 ∈ H1 and a2 ∈ H2 . We have (H1×H2 )× = H1××H2× , q(H1×H2 ) = q(H1 )×q(H2 ) and H 1 ×H2 = H1×H2 . If H1 ⊂ D1 and H2 ⊂ D2 are submonoids, then H1×H2 ⊂ D1×D2 , and there is an ∼ isomorphism D1 /H1×D2 /H2 → D1×D2 /H1×H2 , given by a1 q(H1 ), a2 q(H2 ) → a1 a2 q(H1 ×H2 ). Also, H1 ×H2 is saturated [ cofinal ] in D1 ×D2 if and only if both H1 ⊂ D1 and H2 ⊂ D2 are saturated [ cofinal ]. A monoid H is factorial and P ⊂ H is a maximal set of representatives of pairwise nonassociated primes of H if and only if H = H × × F(P ). Every Krull monoid H splits in the form H = H × ×H0 , where H0 is a reduced Krull monoid, and there is a divisor theory H0 → F (P ). If H1 and H2 are Krull monoids with divisor theories ϕ1 : H1 → F1 and ϕ2 : H2 → F2 , then H1 × H2 is a Krull monoid, and ϕ1 × ϕ2 : H1 × H2 → F1 × F2 is a divisor theory.

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3

75

Class Semigroups

Class semigroups were introduced in [9, Definition 4.1] as a appropriate refinement of the usual concept of ordinary class groups in algebraic number theory. Later on, in Propositions 3.6 and 3.7 we shall prove that, for saturated and cofinal submonoids, the two concepts coincide. We recall the definition given in [9] in a refined version. Definition 3.1. Let D be a monoid, S ⊂ D a submonoid and A ⊂ D a subset. Two elements y, y  ∈ D are called (A, S)-equivalent, if y −1 A ∩ S = y −1 A ∩ S (equivalently: if x ∈ S, then xy ∈ A if and only if xy  ∈ A). By definition, (A, S)-equivalence is an equivalence relation on D. For each y ∈ D, we denote by [y]SA the (A, S)-equivalence class of y. For a subset T ⊂ D,  we define CT (A, S) =

[y]SA | y ∈ T , and in particular C(A, S) = CS (A, S)

∗

and C (A, S) = C(S\S × )∪{1} (A, S). We shall prove in Lemma 3.2 that (A, S)equivalence is a congruence relation on S. The quotient law on C(A, S) will be written additively, that is, [y1 ]SA + [y2 ]SA = [y1 y2 ]SA for all y1 , y2 ∈ S. Endowed with the quotient law, C(A, S) is an additive abelian semigroup with zero element [1]SA , and the canonical map S → C(A, S) , defined by y → [y]SA , is an epimorphism. Consequently, if T ⊂ S is a submonoid, then CT (A, S) is a subsemigroup of C(A, S). In particular, C ∗ (A, S) is a subsemigroup of C(A, S). We call C(A, S) the class semigroup and C ∗ (A, S) the reduced class semigroup of A in S. Lemma 3.2. Let D be a monoid, S ⊂ D a submonoid and A ⊂ D a subset. 1. (A, S)-equivalence is a congruence relation on S. 2. For every subset T ⊂ D, there is a bijective map CT (A, S) → {y −1 A ∩ S | y ∈ T } ,

given by

[y]SA → y −1 A ∩ S.

3. If y ∈ D and [y]SA ∩ A = ∅, then [y]SA ⊂ A. Proof. 1. We must prove that, for all x, y, y  ∈ S, [y]SA = [y  ]SA implies [xy]SA = [xy  ]SA . Thus let x, y, y  ∈ S be such that y −1 A ∩ S = y −1 A ∩ S. By symmetry, it suffices to prove that (xy)−1 A ∩ S ⊂ (xy  )−1 A ∩ S. If z ∈ (xy)−1 A ∩ S, then zx ∈ y −1 A ∩ xS = (y −1 A ∩ S) ∩ xS = (y −1 A ∩ S) ∩ xS = (y −1 A ∩ S) ∩ xS, and therefore z ∈ (xy  )−1 A ∩ S. 2. By the very definition of (A, S)-equivalence, for all y, y  ∈ T we have [y]SA =  S [y ]A if and only if y −1 A ∩ S = y −1 A ∩ S. 3. If y  ∈ [y]SA ∩ A and z ∈ [y]SA , then [z]SA = [y  ]SA . Hence 1 ∈ y −1 A ∩ S = −1 z A ∩ S, and thus z ∈ A. Lemma 3.3. Let H ⊂ D be a submonoid. D × ∗ 1. [u]D H = [1]H for all u ∈ H , C (H, D) = C(D\D× )∪H × (H, D), and there is a × × monomorphism Ψ : D /H → C(H, D), given by Ψ(uH × ) = [u]D H for all u ∈ D× .

76

Halter-Koch 2. C(H, D) is finite if and only if both C ∗ (H, D) and D× /H × are finite. 

3. There is an isomorphism Φ : C(H, D) → C(Hred , D/H × ), given by Φ [y]D H =

 D/H × [yH × ]Hred for all y ∈ D. It satisfies Φ C ∗ (H, D) = C ∗ (Hred , D/H × ). 4. If U ⊂ D× is a subgroup, then there is an epimorphism 

D C(HU, D), given by Ψ [y]H = [y]D HU for all y ∈ D.

Ψ : C(H, D) →

5. If D ⊂ D is a submonoid and H  = H ∩ D , then there is an epimorphism D
 CD
(H, D) → C(H  , D ), given by [y]D H → [y]H
for all y ∈ D . In particular, ∗ ×  × ∗   if C (H, D) is finite and D = D ∩ D , then C (H , D ) is also finite. Proof. 1. The map Ψ0 : D× → C(H, D), defined by Ψ0 (u) = [u]D H , is a group homomorphism, and we shall prove that Ker(Ψ0 ) = H × . Then it follows in particular D × that [u]D and C ∗ (H, D) = C(D\D× )∪H × (H, D). H = [1]H for all u ∈ H D If u ∈ D× , then u−1 H ∩ D = u−1 H. Therefore [u]D H = [1]H holds if and only if u−1 H = H, which is equivalent to u ∈ H × . 2. By 1., since C(H, D) = C ∗ (H, D) ∪ Ψ(D× /H × ). 3. If ρ : D → D/H × denotes the canonical epimorphism, then ρ(D× ) = (D/H × )× and ρ(H) = Hred . For all y, y  ∈ D, we have y −1 H ∩ D = y −1 H ∩ D if and only if ρ(y)−1 ρ(H) ∩ ρ(D) = ρ(y  )−1 ρ(H) ∩ ρ(D). Hence Φ is an isomorphism as asserted. 4. It is sufficient to prove that (H, D)-equivalence on D implies (HU, D)equivalence. We must prove that, for all y, y  ∈ D, y −1 H ∩ D = y −1 H ∩ D entails y −1 HU ∩ D ⊂ y −1 HU ∩ D. Thus suppose that y −1 H ∩ D = y −1 H ∩ D and z ∈ y −1 HU ∩ D. Then zy = cu for some u ∈ U and c ∈ H, and therefore u−1 z ∈ y −1 H ∩ D = y −1 H ∩ D, which implies z ∈ y −1 HU ∩ D.





 D D  D 5. We shall prove that, for all y, y  ∈ D , [y]D H = [y ]H implies [y]H
= [y ]H
.   −1 −1 Thus suppose that y, y ∈ D and y H ∩ D = y H ∩ D. By symmetry, it suffices to prove that y −1 H  ∩ D ⊂ y −1 H  ∩ D . If z ∈ y −1 H  ∩ D , then z ∈ y −1 H ∩ D = y −1 H ∩D, hence y  z ∈ H ∩D = H  and z ∈ y −1 H  ∩D . Therefore the assignment D
  ∗   [y]D H → [y]H
defines an epimorphism CD
(H, D) → C(H , D ), and C (H , D ) is the ∗  ×

× image of CD \D (H, D), which is a subset of C (H, D), if D \ D ⊂ D \ D× .

The following Lemma 3.4 justifies the subsequent restriction to cofinal submonoids. Lemma 3.4. Let H ⊂ D be a submonoid and D0 = {y ∈ D | yD ∩ H = ∅}. 1. D0 is the greatest monoid D such that H ⊂ D ⊂ D and H is cofinal in D. 2. D0× = D× and q(H) ∩ D = q(H) ∩ D0 . In particular, H is saturated in D if and only if H is saturated in D0 .

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∼ 0 = 3. There is an isomorphism Φ : C(H, D0 ) → CD0 (H, D) satisfying Φ [y]D H  D [y]D H for all y ∈ D0 . If y0 ∈ D \ D0 , then C(H, D) = CD0 (H, D)  [y0 ]H 

  and C ∗ (H, D) = Φ C ∗ (H, D0 )  [y0 ]D H . Proof. 1. Obvious by the definition. 2. If u ∈ D× , then uu−1 = 1 ∈ H implies u ∈ D0× . If a ∈ q(H) ∩ D, then there exists some y ∈ H such that ay ∈ H and thus a ∈ D0 . 3. It is easily checked that, for all y, y  ∈ D0 , we have y −1 H ∩ D = y −1 H ∩ D if and only if y −1 H ∩D0 = y −1 H ∩D0 . Thus Φ is an isomorphism as asserted. Since y0−1 H ∩  D = ∅ for all y0 ∈ D \ D0 , Lemma 3.2.2 implies C(H, D) = CD0 (H, D)  [y0 ]D H . Hence we obtain



  C ∗ (H, D) = C(D0 \D× )∪{1} (H, D) ∪ CD\D0 (H, D) = Φ C ∗ (H, D0 )  [y0 ]D H . 0

Lemma 3.5. Let H1 ⊂ D1 and H2 ⊂ D2 be cofinal submonoids. ∼

1. There is an isomorphism Φ : C(H1 ×H2 , D1 ×D2 ) → C(H1 , D1 ) × C(H2 , D2 ) satisfying

  D1 D2 1×D2 Φ [y1 y2 ]D = [y for all (y1 , y2 ) ∈ D1 ×D2 . ] , [y ] 1 H1 2 H2 H1×H2 2. C ∗ (H1 ×H2 , D1 ×D2 ) is finite if and only if the following two conditions are satisfied: (a) C ∗ (H1 , D1 ) and C ∗ (H1 , D1 ) are both finite. (b) Either D1 = D1× , or D2 = D2× , or the groups D1× /H1× and D2× /H2× are both finite. Proof. 1. If (y1 , y2 ) ∈ D1 × D2 , then (y1 y2 )−1 (H1 × H2 ) ∩ (D1 × D2 ) = (y1−1 H1 ∩ D1 ) × (y2−1 H2 ∩ D2 ) = ∅, since H1 ⊂ D1 and H2 ⊂ D2 are cofinal. Thus we obtain, for all (y1 , y2 ), (y1 , y2 ) ∈ D1 × D2 , (y1 y2 )−1 (H1 × H2 ) ∩ (D1 × D2 ) = (y1 y2 )−1 (H1 ×H2 ) ∩ (D1 ×D2 ) if and only if y1−1 H1 ∩ D1 = (y1 )−1 H1 ∩ D1 and y2−1 H2 ∩ D2 = (y2 )−1 H2 ∩ D2 . Therefore there is an isomorphism Φ as asserted. 

 2. Since (D1×D2 ) \ (D1××D2× ) = (D1 \ D1× )×(D2 \ D2× )  (D1 \ D1× )×D2× 

 D1× × (D2 \ D2× ) , the isomorphism Φ given in 1. maps C ∗ (H1 × H2 , D1 × D2 ) onto



 C = C ∗ (H1 , D1 )×C ∗ (H2 , D2 ) ∪ CD1 \D× (H1 , D1 )×CD× (H2 , D2 ) ∪ 1 2

 CD× (H1 , D1 )×CD2 \D× (H2 , D2 ) . 1

2

By Lemma 3.3.1 there are isomorphisms CD× (H1 , D1 ) ∼ = D1× /H1× and CD2× (H2 , D2 ) 1 ∼ = D2× /H2× . Hence C is finite if and only if (a) and (b) are satisfied.

78

Halter-Koch

Proposition 3.6. Let H ⊂ D be a submonoid. 1. C(H, D) is a group if and only if C ∗ (H, D) is a group, and in this case we have either D = D× or C(H, D) = C ∗ (H, D). 2. If C(H, D) is a group, then H ⊂ D is cofinal. 3. If C(H, D) is a torsion group, then H ⊂ D is cofinal and saturated. × Proof. 1. By definition, C(H, D) = C ∗ (H, D) ∪ C  , where C  = {[u]D H | u ∈ D } ∗ is a subgroup of C(H, D). Hence C(H, D) is a group if and only if C (H, D) is a D group. Thus suppose that D = D× , a ∈ D \ D× and u ∈ D× . Then [a]D H , [ua]H ∈ ∗ D D D ∗ C (H, D), and therefore also [u]H = [ua]H − [a]H ∈ C (H, D).

2. Let C(H, D) be a group. For every c ∈ D, there exists some c ∈ D such that  D  D D  D D  −[c]D H = [c ]H , and therefore [cc ]H = [c]H + [c ]H = [1]H , which implies cc ∈ H. 3. Let C(H, D) be a torsion group, a, b ∈ H and c ∈ D be such that a = bc. D n−1 D D Let n ∈ N be such that n[b]D ]H = [a]D H = [1]H . Then [ab H + (n − 1)[b]H = D D D n−1 ∈ H implies c ∈ H. Hence H ⊂ D is saturated. [c]H + n[b]H = [c]H , and ab Proposition 3.7. Let D be a monoid and H ⊂ D a cofinal submonoid. 1. There are epimorphisms θ : C(H, D) → D/H and θ∗ : C ∗ (H, D) → D/D× H ,

  ∗ D given by θ [y]D H = [y]D/H for all y ∈ D, and θ ([y]H = [y]D/D × H for all y ∈ (D \ D× ) ∪ {1}. 2. [1]D H ⊂ H ⊂ [1]D/H . 3. The following conditions are equivalent: (a) H is saturated in D. (b) [y]D H = [y]D/H ∩ D for all y ∈ D. (c) [1]D H = [1]D/H ∩ D. (d) The epimorphism θ : C(H, D) → D/H defined in 1. is an isomorphism.  D  Proof. 1. We assert that, for all y, y  ∈ D, [y]D H = [y ]H implies yq(H) = y q(H).  D  D Indeed, suppose that y, y ∈ D and [y]H = [y ]H . Since H is cofinal in D, there exists some z ∈ D such that yz ∈ H. Hence y  z ∈ H, and thus y  y −1 = (y  z)(yz)−1 ∈ q(H). Now the assignment [y]D H → [y]D/H = yq(H) defines a map θ : C(H, D) → D/H, and obviously θ is an epimorphism. Hence the map θ1 : C(H, D) → D/D× H, de× × D fined by θ1 ([y]D H ) = yD q(H), is also an epimorphism. If y ∈ D , then θ1 ([y]H ) = D ∗ ∗ ∗ × θ1 ([1]H ), and therefore θ = θ1 | C (H, D) : C (H, D) → D/D H is also an epimorphism.

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79

2. By Lemma 3.2.3 we obtain [1]D H ⊂ H, and by definition we have [1]D/H = q(H) ⊃ H.  D D  3. (a) ⇒ (b) If y ∈ D and y  ∈ [y]D H , then [y ]H = [y]H , hence [y ]D/H = [y]D/H  by 1., and therefore y ∈ [y]D/H ∩ D. To prove the reverse inclusion, suppose that y  ∈ [y]D/H ∩ D = yq(H) ∩ D. Then there exist u, u ∈ H such that u y  = uy. By symmetry, it is sufficient to prove that y −1 H ∩ D ⊂ y −1 H ∩ D. If x ∈ y −1 H ∩ D, then xy  u = xyu ∈ H. Hence xy ∈ q(H) ∩ D = H, and thus x ∈ y −1 H ∩ D. (b) ⇒ (c) and (b) ⇒ (d) are obvious. (c) ⇒ (a) If [1]D H = [1]D/H ∩ D, then 2. implies H = [1]D/H ∩ D = q(H) ∩ D, and therefore H ⊂ D is saturated. (d) ⇒ (c) If (c) fails, then [1]D H  [1]D/H ∩ D by 2., and thus there exists some D D . Since [u] u ∈ [1]D/H ∩ D \ [1]D H H = [1]H and [u]D/H = [1]D/H , it follows that θ is not injective. The following example is due to A. Geroldinger. Example 3.8. To illustrate the difference between D/H and C(H, D), we consider the additive monoid H = (N × N≥6 ) ∪ {(2n , 3) , (2n , 4) , (2n + 2, 5) | n ∈ N} ∪ {(0, 0} ⊂ D = N20 . Since q(H) = q(D) = Z2 , we have D/H = 0, but nevertheless C(H, D) is infinite, since any two elements of the set {(2n , 0) | n ∈ N} are not (H, D)-equivalent. Indeed, n D n n suppose that m, n ∈ N, m ≤ n and [(2m , 0)]D H = [(2 , 0)]H . Then (2 , 3) + (2 , 0) = n+1 n m m n−m n−m , 3) ∈ H implies (2 , 3)+(2 , 0) = (2 (2 +1), 3) ∈ H, hence 2 +1 = 2l (2 for some l ∈ N and thus m = n. Note that H is a finitely primary monoid of rank 2 and exponent 6 satisfying

= D, and (6, 6) + D ⊂ H (for the theory of finitely primary monoids see [6]). H Theorem 3.9. Let D be a monoid and S ⊂ H ⊂ D submonoids such that S is saturated in H and H/S is finite. If T ⊂ D is a subset such that CT (H, D) is finite, then CT (S, D) is also finite. Proof. For y, y  ∈ T , we set y ∼ y  if either y −1 H ∩ D = y −1 H ∩ D = ∅ or if there exists some z ∈ y −1 H ∩ D = y −1 H ∩ D such that [zy]H/S = [zy  ]H/S . Since H/S is finite and {y −1 H ∩ D | y ∈ T } is finite by Lemma 3.2.2, ∼ is an equivalence relation with only finitely many equivalence classes on T . We shall prove that, for all y, y  ∈ T , y ∼ y  implies y −1 S ∩ D = y −1 S ∩ D. Then {[y]D S | y ∈ T } is also finite. If y, y  ∈ T and y −1 H ∩ D = y −1 H ∩ D = ∅, then y −1 S ∩ D = y −1 S ∩ D = ∅. Thus we may assume that there exists some z ∈ y −1 H ∩ D = y −1 H ∩ D such that [zy]H/S = [zy  ]H/S . Then y −1 y  = (zy)−1 (zy  ) ∈ q(S), and since S is saturated in H, we obtain y −1 S ∩ D = y −1 (H ∩ q(S)) ∩ D = y −1 H ∩ D ∩ y −1 q(S) = y −1 H ∩ D ∩ y −1 q(S) = y −1 (H ∩ q(S)) ∩ D = y −1 S ∩ D.

80

Halter-Koch

The arithmetical importance of C ∗ (H, D) is (inter alia) revealed by the following Proposition 3.10 (see also [9, Proposition 4.5]). Proposition 3.10. Let H ⊂ D be a submonoid such that C ∗ (H, D) is finite, and     D × . V = u ∈ D×  [ua]D H = [a]H for all a ∈ D \ D 1. V ⊂ D× is a subgroup of finite index, H × ⊂ V , and V (H \ H × ) ⊂ H. 2. There exists some α ∈ N such that (D× : V ) | α and q 2α D ∩ H = q α (q α D ∩ H) for all q ∈ D \ D× . × ∗ Proof. 1. D× operates on the finite set C = {[a]D H | a ∈ D \ D } ⊂ C (H, D) by D D × × means of u [a]H = [ua]H for all u ∈ D and a ∈ D \ D . This operation defines a homomorphism ϕ : D× → Perm(C) into the (finite) permutation group of C, V = Ker(ϕ) is a subgroup of finite index, and H × ⊂ V by Lemma 3.3.1. If u ∈ V D and a ∈ H \ H × , then [ua]D H = [a]H ⊂ H and thus ua ∈ H by Lemma 3.2.3.

2. For every g ∈ C ∗ (H, D), there exist γg , λg ∈ N such that (λg + γg )g = λg g, and then (λg + γg )g = λg g for all γg , λg ∈ N such that γg | γg and λg ≥ λg . Let α ∈ N be such that (D× : V ) | α, γg | α and α ≥ λg for all g ∈ C ∗ (H, D). Then 2αg = αg for all g ∈ C ∗ (H, D). If q ∈ D \ D× and a ∈ D, then D D D D α D [q 2α a]D H = 2α[q]H + [a]H = α[q]H + [a]H = [q a]H ,

and thus q 2α a ∈ H if and only if q α a ∈ H.

4

Algebraic Theory of C-Monoids

Definition 4.1. Let H be a monoid. 1. H is called a C-monoid , if it is a submonoid of a factorial monoid F such that H ∩ F × = H × and C ∗ (H, F ) is finite. Then it follows by Proposition 3.10 that there exists some α ∈ N and a subgroup V ⊂ F × such that H × ⊂ V , (F × : V ) | α, V (H \ H × ) ⊂ H and q 2α F ∩ H = q α (q α F ∩ H) for all q ∈ F \ F × . We refer to these properties by saying that H is defined in F with exponent α and subgroup V ⊂ F × . 2. Let F = F × ×F(P ) be a factorial monoid and H ⊂ F a submonoid. (a) H is called dense in F , if vp (H) ⊂ N0 is a numerical monoid for all p ∈ P. (b) A subset E ⊂ P is called H-essential a ∈ H \ F ×.

if E = suppP (a) for some

(c) H is called simple in F , if every minimal H-essential subset of P is a singleton.

C-Monoids and Congruence Monoids in Krull Domains

81

Note that the conditions (a) and (c) do not depend on the particular choice of P . 3. H is called a C0 -monoid , if it is a C-monoid defined in a factorial monoid F possessing only finitely many pairwise nonassociated prime elements. By definition, every factorial monoid is a C-monoid. If H is a C-monoid defined in a factorial monoid F , then F is far from being uniquely determined by H. However, in Theorem 4.6 we shall prove that there is a canonical choice for F . Without giving details, we mention that the Z-monoids used in [17] and the ACmonoids defined in factorial monoids which were introduced in [9] are C-monoids, and that the C0 -monoids as defined above coincide with the C0 -monoids introduced in [9]. For Z-monoids, this follows immediately from the definitions, and for AC-monoids defined in a factorial monoid this is true by [9, Theorem 5.4]. The conformity of the concepts of C0 -monoids will be explained after the subsequent Theorem 4.3. The following technical Lemma 4.2 is basic for the further theory of C-monoids. Lemma 4.2. Let F = F × × F(P ) be a factorial monoid, H ⊂ F a submonoid, α ∈ N, and let ρα : F → F be defined by     np ρα u p pnp −αlp =u p∈P

p∈P

for all u ∈ F × and (np )p∈P ∈ N0 , where (lp )p∈P ∈ N0 if np < 2α, and α ≤ np − αlp < 2α, if np ≥ 2α. (P )

(P )

is chosen so that lp = 0,

1. The following assertions are equivalent: (a) For all p ∈ P and a ∈ pα F , we have a ∈ H if and only if pα a ∈ H. (b) If u ∈ F and (np )p∈P , (np )p∈P ∈ N0 are such that, for each p ∈ P , either np = np or [ np ≡ np mod α and min{np , np } ≥ α ], then (P )

u



pnp ∈ H

implies

u

p∈P






pnp ∈ H .

p∈P

(c) For all x, y ∈ F we have xy ∈ H if and only if ρα (x)y ∈ H. 2. Let H be a C-monoid defined in F with subgroup V ⊂ F × and exponent α ∈ N. (a) The equivalent conditions stated in 1. hold. (b) If Q ⊂ P is H-essential, u ∈ V and (lp )p∈Q ∈ NQ , then  u pαlp ∈ H . p∈Q

82

Halter-Koch (c) If H is dense in F , then every finite subset of P is contained in an H-essential subset of P . (d ) If x, y ∈ H, then suppP (x) ⊂ suppP (y) if and only if x ∈ [[y]].

Proof. 1. (a) ⇒ (b) The assertion follows by induction on n= |np − np | . p∈P

(b) ⇒ (c) The elements xy and ρα (x)y satisfy the conditions stated in (b). (c) ⇒ (a) If p ∈ P and a ∈ pα F , then ρα (pα a) = ρα (a). 2. (a) Note that 1.(a) is equivalent to p2α F ∩ H = pα (pα F ∩ H) for all p ∈ P , and this holds by Proposition 3.10.2. (b) Suppose that a ∈ H \ F × , Q = suppP (a), u ∈ V and (lp )p∈Q ∈ NQ . If  a=ε pnp , where ε ∈ F × and (np )p∈Q ∈ NQ , p∈Q

then uε−α ∈ V and a ∈ H \ H × implies uε−α aα = u



pαnp ∈ H .

p∈Q

Since αnp ≡ αlp mod α and min{αnp , αlp } ≥ α for all p ∈ Q, the assertion follows by 1.(b). (c) If H is dense in F and {p1 , . . . , pd } ⊂ P , then there exist elements a1 , . . . , ad ∈ H such that vpi (ai ) > 0 for all i ∈ [1, d], and if a = a1 · . . . · ad , then {p1 , . . . , pd } ⊂ suppP (a). (d) If x ∈ [[y]], then x | y n for some n ∈ N, and therefore suppP (x) ⊂ suppP (y). Thus suppose that x, y ∈ H and E = suppP (y) ⊃ suppP (x). Then   x=ε pnp and y = η p mp , p∈E

p∈E

where ε, η ∈ F × , np ∈ N0 and mp ∈ N for all p ∈ E. Let k ∈ N be such that kmp > np for all p ∈ E. Then (b) implies  pα(kmp −np ) ∈ H , x−1 y kα = xα−1 (ε−1 η)α p∈E

and therefore x ∈ [[y]]. The following Theorem 4.3 characterizes C0 -monoids without referring to the class semigroup, and in Proposition 4.4 we shall see that locally C-monoids behave like C0 -monoids. Theorem 4.3 (Characterization of C0 -monoids). Let F = F × ×[p1 , . . . , ps ] be a factorial monoid with pairwise nonassociated prime elements p1 , . . . , ps , and let H ⊂ F be a submonoid such that H ∩ F × = H × . Then the following assertions are equivalent:

C-Monoids and Congruence Monoids in Krull Domains

83

(a) H is a C0 -monoid defined in F . (b) There exists some α ∈ N and a subgroup V ⊂ F × such that (F × : V ) | α, V (H \ H × ) ⊂ H, and for all j ∈ [1, s] and a ∈ pα j F we have a ∈ H if and only if pα j a ∈ H. Proof. (a) ⇒ (b) By Proposition 3.10. (b) ⇒ (a) Let {u1 , . . . , ut } ⊂ F × be a set of representatives for F ×/V , and consider the finite set     B = uτ pn1 1 · . . . · pns s  τ ∈ [1, t], (n1 , . . . , ns ) ∈ [0, 2α − 1]s . Let ρα : F → F be the reduction map defined in Lemma 4.2, and define ϕ : F → B by ϕ(uc) = uτ ρα (c), whenever u ∈ uτ V and c ∈ [p1 , . . . , ps ]. We assert that F × ∗ [x]F H = [ϕ(x)]H for all x ∈ F \ F . Whence in particular C (H, F ) is finite. To prove this assertion, suppose that x = uc ∈ F \ F × , where u ∈ F × and c ∈ [p1 , . . . , ps ] \ {1}. Let τ ∈ [1, t] be such that u ∈ uτ V . Then ϕ(x) = uτ ρα (c), and we must prove that for all y ∈ F we have yx ∈ H if and only if yϕ(x) ∈ H. If y = vb ∈ F , where v ∈ F × and b ∈ [p1 , . . . , ps ], then yx = uvbc, yϕ(x) = uτ vbρα (c), and by Lemma 4.2.1 we have yϕ(x) ∈ H if and only if uτ vbc ∈ H. Since bc = 1 and (uτ v)−1 (uv) ∈ V it follows that yx ∈ H if and only if yϕ(x) ∈ H. Let F = F × ×[p1 , . . . , ps ] be as in Theorem 4.3 and H ⊂ F a submonoid such that H ∩ F × = H × . Let V ⊂ F × be a subgroup and ≡ the congruence relation modulo V on F × (observe that every congruence relation on F × is of this form). Then F ×/V = F ×/ ≡, and the following two assertions are equivalent: • V (H \ H × ) ⊂ H. • For all u, v ∈ F × , u ≡ v implies u−1 H ∩ [p1 , . . . , ps ] \ {1} = v −1 H ∩ [p1 , . . . , ps ] \ {1}. Therefore our concept of a C0 -monoid coincides with that of [9, Definition 5.1]. Proposition 4.4. Let H be a C-monoid defined in a factorial monoid F with exponent α and subgroup V ⊂ F × . Let P1 be a set of pairwise nonassociated prime elements of F , U ⊂ F × a subgroup and F1 = U ×F(P1 ) ⊂ F . Then H1 = H ∩ F1 is a C-monoid defined in F1 with exponent α and subgroup U ∩ V . In particular, (a) If P1 is finite, then H1 is a C0 -monoid. (b) If U = F × , then H1 ⊂ H is divisor-closed. Proof. F1 is factorial, F1× = U , and clearly H1× ⊂ U ∩ H1 . If ε ∈ U ∩ H1 , then ε ∈ F × ∩ H = H × and ε−1 ∈ U ∩ H ⊂ H1 , hence ε ∈ H1× . Whence H1× = F1× ∩ H1 . Since F1× = F1 ∩ F × , the reduced class semigroup C ∗ (H1 , F1 ) is finite by Lemma 3.3.5, and therefore H1 is a C-monoid defined in F1 . If q ∈ F1 \ F1× ⊂ F \ F × and a ∈ F1 , then q α a ∈ H if and only if q 2α a ∈ H, and therefore q α a ∈ H1 if and only if q 2α a ∈ H1 . Since H × ∩ H1 = F × ∩ H ∩ H1 = F1× ∩ H1 = H1× , we obtain (V ∩ U )(H1 \ H1× ) ⊂ V (H \ H × ) ⊂ H and therefore (V ∩ U )(H1 \ H1× ) ⊂ H ∩ F1 = H1 . Since finally (U : V ∩ U ) | (F × : V ) | α, it follows that H1 is a C-monoid defined in F1 with exponent α and subgroup U ∩ V . The more precise assertions (a) and (b) are obvious.

84

Halter-Koch

Proposition 4.5. Let H be a C-monoid defined in a factorial monoid F with exponent α and subgroup V ⊂ F × . 1. Hred is a C-monoid defined in F/H × with exponent α and subgroup V /H × . In particular, if H is a C0 -monoid, then so is Hred . 2. Let S ⊂ H be a saturated submonoid such that H/S is finite. Then S is a C-monoid, and if H is a C0 -monoid, then so is S. Proof. 1. F/H × is factorial, (F/H × )× = F × /H × , and (F/H × )× ∩ Hred = (F × ∩ H)/H × = {1}. By Lemma 3.3.3 we have C ∗ (H, F ) ∼ = C ∗ (Hred , F/H × ). Hence Hred × is a C-monoid in F/H . The remaining assertions are easily checked. 2. Since S ⊂ H is saturated, we have F × ∩ S = F × ∩ H ∩ S = H × ∩ S = S × and, by Theorem 3.9, C ∗ (S, F ) is finite. Theorem 4.6. Let H be a C-monoid defined in F = F × ×F(P ) with subgroup V and exponent α. 1. For p ∈ P , set dp = gcd(vp (H)) ∈ N0 . Define P0 = {pdp | p ∈ P, dp = 0} ⊂ F (P ), and F0 = {a ∈ F | vp (a) ∈ dp N0 for all p ∈ P } = F × ×F(P0 ) ⊂ F . Then H is a C-monoid defined in F0 with subgroup V and exponent α, and H is dense in F0 .

is a Krull monoid, (H : H)

= ∅, and C(H)

is a finite group whose exponent 2. H divides α. 3. If H is dense in F , then the following assertions hold. (a) H is cofinal in F .

= q(H)∩F ⊂ q(H), and if {p1 , . . . , pd } ⊂ P is an H-essential subset (b) H F F α

satisfying {[p1 ]F H , . . . , [pd ]H } = {[p]H | p ∈ P }, then (p1 · . . . · pd ) H ⊂ H.

(c) For every q ∈ F \ F × , we have q α ∈ H.

→ F (P ), defined by (d ) The map ∂ : H  ∂(a) = pvp (a) , p∈P

is a divisor theory, and there is an epimorphism C ∗ (H, F ) → C(H). (e) If H is defined in another factorial monoid F  such that H is dense in F  ,  then there exists an isomorphism Φ : Fred → Fred such that Φ(aF × ) = × aF for all a ∈ H.

C-Monoids and Congruence Monoids in Krull Domains

85

In particular, if H is dense in F , then (up to isomorphisms) Fred is uniquely determined by H.

=H

× ×H0 with a reduced Krull monoid H0 , and let H0 → 4. Suppose that H

××F(P ), F (P ) be a divisor theory. Then H is a C-monoid defined in F1 = H H is dense in F1 , and if H is a C0 -monoid, then P is finite. Proof. 1. By definition, we have H ⊂ F0 , F0× ∩ H = F × ∩ H = H × , and C ∗ (H, F0 ) is finite by Lemma 3.3.5. Hence H is a C-monoid defined in F0 . It is now obvious that H is defined in F0 with subgroup V and exponent α, and that H is dense in F0 . 2. and 3. Replacing F by F0 as defined in 1., we may assume that H is dense

is in F . Then it suffices to prove the assertions 3.(a) to 3.(d). Indeed, by 3.(d) H



a Krull monoid and C(H) is finite, by 3.(b) (H : H) = ∅, and by 3.(c) the exponent

divides α. To prove 3.(e), suppose that H is a C-monoid which is defined of C(H)

be the complete integral closure of H inside and dense both in F and F  ; let H

→H

 such that

 those inside F  . Then there is an isomorphism Φ0 : H F and H   

→ F → Fred and H

→ F → F Φ0 | H = idH . Since the maps H red are divisor theories, the assertions 3.(e) follows from the uniqueness of divisor theories. 3.(a) If a = εp1 · . . . · pn ∈ F , where n ∈ N0 , ε ∈ F × and p1 , . . . , pn ∈ P , then there exist elements a1 , . . . , an ∈ H such that vpi (ai ) > 0 for all i ∈ [1, n], and thus a | a1 · . . . · an . Hence H is cofinal in F . 3.(b) We define a homomorphism ×

π : q(F ) → F /V ×(Z/αZ)

(P )

    np by π ε p = εV, (np + αZ)p∈P , p∈P

where ε ∈ F × and (np )p∈P ∈ Z(P ) . We set R = π −1 (π(H)) ⊂ q(F ), and we shall

= R ∩ F . π(H) is a subsemigroup of an abelian torsion prove that R = q(H) and H group, and therefore it is itself a group. Hence R is a group, and H ⊂ R implies q(H) ⊂ R.

⊂ q(F ), then there exists some c ∈ H such that cz k ∈ H for all k ∈ N. If z ∈ H Hence vp (cz k ) = vp (c)+kvp (z) ≥ 0 for all p ∈ P and k ∈ N, which implies vp (z) ≥ 0 for all p ∈ P and thus z ∈ F . Therefore it remains to show that R ⊂ q(H) and

R ∩ F ⊂ H. Suppose that z ∈ R, and let x ∈ H be such that π(z) = π(x). Since H is dense in F , there exists some x0 ∈ H such that Q = suppP (x0 ) ⊃ suppP (x) ∪ suppP (z). We set   pnp and z = v p mp , x=u p∈Q

p∈Q

where u, v ∈ F × , v −1 u ∈ V , np ∈ N0 , mp ∈ Z, and np ≡ mp mod α for all p ∈ Q. By Lemma 4.2 we have y0 =

 p∈Q

pα ∈ H

and v −1 uy0l ∈ H

for all l ∈ N .

86

Halter-Koch

Let l ∈ N be such that mp + lα ≥ α for all p ∈ Q. Then   xy0 = u pnp +α ∈ H , v −1 uy0l z = u pmp +lα ∈ F , p∈Q

p∈Q

hence v −1 uy0l z ∈ H, and z = (v −1 uy0l )−1 (v −1 uy0l z) ∈ q(H). Assume now that z ∈ R ∩ F . Then mp ≥ 0 for all p ∈ P . Let {p1 , . . . , pd } ⊂ P F F be an H-essential subset satisfying {[p1 ]F H , . . . , [pd ]H } = {[p]H | p ∈ P }. Then α w0 = (p1 · . . . · pd ) ∈ H by Lemma 4.2, and we assert that w0 z k ∈ H for all k ∈ N. F Let Q = Q1  . . .  Qd be a partition such that Qi ⊂ {p ∈ Q | [p]F H = [pi ]H } for all i ∈ [1, d]. For all k ∈ N, we have w0 z k = v k

d  i=1



pα i

(u−1 v)k w0 xk = v k

pkmp ∈ F,

d  i=1

p∈Qi

pα i



pknp ∈ H ,

p∈Qi

and we consider the elements zk = v k

d 

pikmi +α ∈ F

and xk = v k

i=1

d 

i +α pkn ∈F i

i=1

where mi =

p∈Qi

mp and ni =



np .

p∈Qi

−1 k   v) w0 xk ]F We have [xk ]F H = [(u H , and thus xk ∈ H, hence zk ∈ H by Lemma 4.2,  F k F k and since [zk ]H = [w0 z ]H , it follows that w0 z ∈ H. 3.(c) If q ∈ F \ F × and a ∈ q α F ∩ H, then q α a ∈ H and therefore q α = −1 α

a (q a) ∈ q(H) ∩ F = H. 3.(d) If ϕ : F → q(F )/q(H) denotes the natural homomorphism defined by

Hence H

⊂ F is a saturated ϕ(a) = aq(H), then ϕ−1 (1) = F ∩ q(H) = H.

is a Krull monoid, and the composite map ∂ : H

→ F = submonoid. Therefore H F × ×F(P ) → F (P ) is a divisor homomorphism. It is given by 

, pvp (a) for all a ∈ H ∂(a) = p∈Q

and and we assert that it is a divisor theory. Indeed, if p ∈ P , then pα ∈ H, since vp (H) is a numerical monoid, there exists some z ∈ H such that vp (z) =

vp (p−nα z) = 1, and nα + 1 for some n ∈ N. Then p−nα z ∈ F ∩ q(H) = H, p = gcd ∂(pα ), ∂(p−nα z) .

We It remains to prove that there exists an epimorphism C ∗ (H, F ) → C(H).

∼



= have C(H) C(∂) = F (P )/∂ H = q(F (P ))/q(∂ H), and we assert that q(∂ H) = ×



F q(H) ∩ q(F (P )) ⊂ q(F (P )). Indeed, suppose that z ∈ F(P ). If z ∈ q(∂ H), −1 ×

then z = a b, where a, b ∈ F(P ) and aε, bη ∈ H for some ε, η ∈ F . Then z =

Conversely, if z ∈ F × q(H),

then z = ε0 (εa)−1 (ηb), (εη −1 )(aε)−1 (bη) ∈ F × q(H).

Then z ∈ F(P ) implies where ε0 , ε, η ∈ F × , a, b ∈ F(P ) and εa, ηb ∈ H. −1 −1

ε0 ε η = 1 and z = a b ∈ q(∂ H), since ∂(εa) = a and ∂(ηb) = b.

C-Monoids and Congruence Monoids in Krull Domains

87

By Proposition 3.7.1, there exists an epimorphism θ : C ∗ (H, F ) → F/F × H. We

and

= q(F )/F × q(H) combine it with the natural epimorphism F/F × H → F/F × H consider the isomorphisms

= F × q(F (P ))/F × q(H)

q(F )/F × q(H) ∼

∩ q(F (P )) = q(F (P ))/q(∂ H)

∼

= C(H) = q(F (P ))/F × q(H) to obtain the desired epimorphism. 4. Replacing F by F0 as defined in 1., we may assume that H is a C-monoid defined in a factorial monoid F = F × ×F(P ) such that H is dense in F , and if H

⊂ F , and the composite is a C0 -monoid, then P is finite. By 3. we have H ⊂ H

splits in the form map ∂ : H → F → F (P ) is a divisor theory. The Krull monoid H ×

=H

×H0 , where H0 is a reduced Krull monoid, ∂ | H0 is injective, and thus we H

× ×F(P ) ⊂ F , may assume that H0 → F (P ) is a divisor theory. Then H ⊂ F1 = H and by Proposition 4.4 H is a C-monoid defined in F1 . By construction, H is dense in F1 .

Theorem 4.7. 1. Let H be a Krull monoid with finite class group. Then H is a C-monoid, and H is a C0 -monoid if and only if Hred is finitely generated. 2. Let H be a C-monoid defined in a factorial monoid F such that C(H, F ) is a group. Then H is a Krull monoid. Proof. 1. Let ∂ : H → F (P ) be a divisor theory, and H = H × ×H0 , where H0 is a reduced Krull monoid. Then ∂ | H0 is injective and thus we may assume that H0 → F (P ) is a divisor theory. Then H ⊂ F = H × ×F(P ), F × = H × , and by Proposition 3.7 we have C(H, F ) ∼ = F/H ∼ = F0 /H0 = C(H). Hence C(H, F ) is finite, and H is a C-monoid. By [12, Satz 4], Hred is finitely generated if and only if P is finite. If P is finite, then H is a C0 -monoid by definition, and if H is a C0 -monoid, then P is finite by

since H is a Krull monoid). Theorem 4.6 (note that H = H, 2. By Proposition 3.6.3, H ⊂ F is saturated, and thus it is a Krull monoid.

Theorem 4.8. Every C-monoid is v-noetherian. The proof of Theorem 4.8 depends on the following Lemma 4.9, which is a refinement of Dickson’s Finiteness Theorem. Lemma 4.9. Let s ∈ N, ∅ = M ⊂ Ns0 , and let Min(M ) be the set of all minimal points of M . Then there exists a finite set M  such that Min(M ) ⊂ M  ⊂ M with the following property: If n = (n1 , . . . , ns ) ∈ M , n∗ = (n∗1 , . . . , n∗s ) ∈ Min(M ) and n∗ ≤ n, then there exists some n = (n1 , . . . , ns ) ∈ M  such that n∗ ≤ n ≤ n and {i ∈ [1, s] | n∗i < ni } = {i ∈ [1, s] | n∗i < ni }.

88

Halter-Koch

Proof of Lemma 4.9. For n∗ = (n∗1 , . . . , n∗s ) ∈ Min(M ) and I ⊂ [1, s], we denote by M (n∗ , I) the set of all n = (n1 , . . . , ns ) ∈ M such that n ≥ n∗ and I = {i ∈ [1, s] | ni > n∗i }. Let M0 (n∗ , I) be the set of all minimal points of M (n∗ , I). It is finite by Dickson’s Theorem [16, Theorem 1.1], and therefore the set   M = M0 (n∗ , I) n∗ ∈Min(M) I⊂[1,s]

is also finite and, by construction, it has the desired property. Proof of Theorem 4.8. Let H be a C-monoid defined in a factorial monoid F = F × × F(P ) with subgroup V and exponent α. By [16, Theorem 24.1] we must prove that for every subset X ⊂ H there exists a finite subset Y ⊂ X such that Y −1 ⊂ X −1 . We may assume that X = ∅, we fix an element x0 ∈ X and distinct primes p1 , . . . , ps ∈ P such that vp (x0 ) = 0 for all p ∈ P  = P \ {p1 , . . . , ps }. We define homomorphisms v : F → Ns0 and π : F → F ×/V ×(Z/αZ)s ×C ∗ (H, F ) as follows: If x = εpn1 1 · . . . · pns s a, where ε ∈ F × , n1 , . . . , ns ∈ N0 and a ∈ F(P  ), then v(x) = (n1 , . . . , ns ) ∈ Ns0 and



× s ∗ π(x) = εV, n1 + αZ, . . . , ns + αZ, [a]F H ∈ F /V ×(Z/αZ) ×C (H, F ) .

Let X = X1 ∪ . . . ∪ Xt be a partition of X such that |π(Xτ )| = 1 for all τ ∈ [1, t]. For τ ∈ [1, t], we set Mτ = v(Xτ ) ⊂ Ns0 , and we choose a finite subset Mτ ⊂ Mτ with the property of Lemma 4.9. Let Yτ ⊂ Xτ be a finite subset satisfying the following two conditions: 1. For every n ∈ Mτ there exists some x ∈ Yτ such that v(x) = n. 2. If there exist elements x1 , x2 ∈ Xτ such that v(x1 ) = v(x2 ) ∈ Mτ and x−1 1 x2 ∈ F × F (P  ) \ H, then Yτ also contains two elements with these properties. Let Y ⊂ X be a finite set such that Y1 ∪ . . . ∪ Yt ∪ {x0 } ⊂ Y . Then we have X −1 ⊂ Y −1 and vp (x) ≥ 0 for all x ∈ Y −1 and p ∈ P  . We shall prove that Y −1 ⊂ X −1 . Suppose that z ∈ Y −1 , τ ∈ [1, t] and x ∈ Xτ , say x=ε

s  i=1

pni i a

and z = η

s 

plii z0 ,

i=1

where ε, η ∈ F × , n = (n1 , . . . , ns ) = v(x), l1 , . . . , ls ∈ Z and a, z0 ∈ F(P  ). Let n∗ = (n∗1 , . . . , n∗s ) ∈ Min(Mτ ) and n = (n1 , . . . ns ) ∈ Mτ be such that ∗ n ≤ n ≤ n and {i ∈ [1, s] | n∗i < ni } = {i ∈ [1, s] | n∗i < ni }. For i ∈ [1, s], we set ni = n∗i + ki α and ni = n∗i + ki α, where ki , ki ∈ N0 and either ki = ki = 0 or ki ≥ ki ≥ 1. Let x∗ , x ∈ Yτ be elements such that v(x∗ ) = n∗ and v(x ) = n . Suppose that s s   n∗ n
x∗ = ε∗ pi i a∗ and x = ε p i i a , i=1

i=1

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where ε∗ , ε ∈ F × are such that ε∗−1 ε, ε−1 ε ∈ V , and a∗ , a ∈ F(P  ) are such ∗ F  F that [a]F H = [a ]H = [a ]H . Therefore we obtain zx = εη

s 

n∗ +li +ki α

pi i

z0 a , zx = ε η

i=1 ∗

s 

n∗ +li +ki
α

pi i

z0 a , zx∗ = ε∗ η

i=1

s 

n∗ +li

pi i

z 0 a∗ ,

i=1



zx , zx ∈ H, and we must prove that zx ∈ H. For all i ∈ [1, s], we have n∗i + li + ki α ≥ n∗i + li + ki α ≥ n∗i + li ≥ 0, and either n∗i + li + ki α = n∗i + li + ki α or n∗i + li + ki α ≥ n∗i + li + ki α ≥ α. Hence Lemma 4.2 implies y = ε η

s 

n∗ +li +ki α

pi i

z 0 a ∈ H ,

i=1  F F −1 × × and since [a]F εy]F H = [a ]H , we obtain [zx]H = [ε H . If y ∈ F , then y ∈ H \ H , and ε−1 εy ∈ V (H \ H × ) ⊂ H implies zx ∈ H. If y ∈ F × , then z0 = a = 1 and n∗i + li + ki α = 0, hence n∗i + li + ki α = n∗i + li = 0 for all i ∈ [1, s]. Thus we obtain y = ε η = zx ∈ F × ∩ H = H × and zx = εηa ∈ F × F (P  ). Assume that zx ∈ H. Then x−1 x = (zx )−1 zx = ε−1 εa ∈ F × F (P  ) \ H, and since v(x) = v(x ) ∈ Mτ , there exists some x1 ∈ Yτ such that  ×  −1 x−1 x1 ∈ F × F (P  ) \ H or x−1 implies 1 x ∈ F F (P ) \ H. In any case, z ∈ Y zx1 ∈ H. If x−1 x1 ∈ F × F (P  ) \ H, then zx1 = (zx )(x−1 x1 ) ∈ F × F (P  ) \ H, a contradiction. −1   ×   × × = If x−1 1 x ∈ F F (P )\H, then zx = (zx1 )(x1 x ) ∈ H implies zx1 ∈ H ∩F −1  × × H and therefore x1 x ∈ H , again a contradiction.

Proposition 4.10. Let H be a C-monoid defined in F = F × ×F(P ) with exponent α. For an H-essential subset E ⊂ P , we define  uE = pα ∈ H , p∈E

and for any subset Q ⊂ P , we set HQ = {x ∈ H | suppP (x) ⊂ Q} =  F (Q) ∩ H.

F××

1. {HQ | Q ⊂ P } is the set of all divisor-closed submonoids of H. In particular, every divisor-closed submonoid of a C-monoid is a C-monoid. 2. For every subset Q ⊂ P , we have HQ = [[{uE | E ⊂ Q is H-essential}]]. In particular, if Q ⊂ P is H-essential, then HQ = [[uQ ]]. 3. The following assertions are equivalent: (a) H is a C0 -monoid. (b) H has only finitely many divisor-closed submonoids. (c) H has only finitely many prime s-ideals.

90

Halter-Koch

Proof. Note that uE ∈ H by Lemma 4.2. 1. If Q ⊂ P , then HQ = (F ××F(Q))∩H is a C-monoid, and it is a divisor-closed submonoid of H by Proposition 4.4. Thus let S ⊂ H be a divisor-closed submonoid and set  suppP (x) . Q= x∈S

Then S ⊂ HQ by definition. If x ∈ HQ , then suppP (x) ⊂ Q, and thus there exist x1 , . . . , xm ∈ S such that suppP (x) ⊂ suppP (x1 ) ∪ . . . ∪ suppP (xm ) = suppP (x1 · . . . · xm ). Hence x ∈ [[x1 · . . . · xm ]] ⊂ S by Lemma 4.2.2(d). 2. If Q ⊂ P and E ⊂ Q is H-essential, then uE ∈ HQ . Hence [[{uE | E ⊂ Q is H-essential}]] ⊂ HQ . Conversely, suppose that Q ⊂ P , x ∈ HQ and E  = suppP (x). Since E  = suppP (uE
), Lemma 4.2.2(d) implies x ∈ [[uE
]] ⊂ [[{uE | E ⊂ Q is H-essential}]]. 3. The equivalence of (b) and (c) follows from the definitions. (a) ⇒ (b) If H is a C0 -monoid, then we may assume that P is finite. Then H has only finitely many divisor-closed submonoids by 1. (b) ⇒ (a) By Theorem 4.6 we may assume that H is dense in F . If Q and Q are H-essential subsets of P and Q = Q , then HQ = HQ
. Hence P has only finitely many H-essential subsets. However, by Lemma 4.2.2(c) every finite subset of P is contained in an H-essential subset of P . Therefore P is finite and H is a C0 -monoid. Theorem 4.11. Let H1 and H2 be monoids. Then H1 ×H2 is a C-monoid if and only if the following two conditions are satisfied: (a) H1 and H2 are both C-monoids. 1 × /H × and H 2 × /H × (b) Either H1 = H1× , or H2 = H2× , or the groups H 1 2 are both finite. Proof. If H1 × H2 is a C-monoid, then H1 and H2 are C-monoids by Proposition 4.10.1, since they are divisor-closed submonoids. Thus suppose that H1 and H2 are C-monoids. Then Theorem 4.6 implies that for i ∈ {1, 2} the monoid Hi is a C-monoid defined in a factorial monoid Fi = i × × F(Pi ) such that there is a divisor theory H i → F (Pi ). Then H1 × H2 ⊂ H × × ×  H1 × H2 × F(P1 ) × F(P2 ) = (H1 ×H2 ) × F(P1 ∪ P2 ) = F1 × F2 , and there is a divisor theory H 1 ×H2 = H1 × H2 → F (P1 ) × F(P2 ) = F (P1 ∪ P2 ). Again by Theorem 4.6 and by the uniqueness of divisor theories, H1 ×H2 is a C-monoid if × and only if it is a C-monoid defined in F1 ×F2 , and since (H 1 ×H2 ) ∩ (H1 ×H2 ) = × × × × × (H1 ∩ H1 )× (H2 ∩ H2 ) = H1 × H2 = (H1 × H2 ) , this is the case if and only if C ∗ (H1 × H2 , F1 × F2 ) is finite. The criterion given in Lemma 3.5 completes the proof. We close this section with some remarks concerning finitary monoids and Gmonoids. We recall the relevant definitions from [10]. A subset U of a monoid H is called an almost generating set , if U ⊂ H \ H × , and there exists some n ∈ N such that (H \ H × )n ⊂ U H. A monoid H is called

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• finitary , if it is a BF-monoid and possesses a finite almost generating set. • a G-monoid , if there exists an element a ∈ H lying in all nonempty prime s-ideals of H. A v-noetherian monoid is a G-monoid if and only if it has only finitely many prime s-ideals, and then it is also finitary [10, Proposition 4.5 and Theorem 4.7]. In [9, Theorem 5.9] it is proved that every C0 -monoid is finitary. We can do a little bit more. Proposition 4.12. Let H be a C-monoid. Then H is a G-monoid if and only if it is a C0 -monoid. Proof. By Theorem 4.8, H is v-noetherian, and therefore H is a G-monoid if and only if it has only finitely many prime s-ideals. Thus the assertion follows by Proposition 4.10.3. Example 4.13. The Hilbert monoid H = 2N ∪ {1} is a C-monoid defined in N [8, Theorem 7.4]. Hence it is v-noetherian by Theorem 4.8. Since (H \ {1})2 ⊂ 2H, it follows that {2} is an almost generating set, and therefore H is finitary. For every prime number p, the set 2pN is a prime s-ideal of H. Hence H has infinitely many prime s-ideals and thus it is not a G-monoid (see also [10, Remark 4.11.2]).

5

The Elasticity of C-Monoids

In [8, Theorem 7.2] we proved that a C-monoid H fulfills all relevant finiteness conditions of nonunique factorizations: It is locally tame, has finite catenary degree, and the Structure Theorem for Sets of Lengths holds for H. In particular, it is a BF-monoid. Monotone chains of factorizations in C-monoids are studied in [4]. The purpose of this section is to present a criterion for the elasticity of a C-monoid to be finite. Theorem 5.1. Let H be a C-monoid defined in the factorial monoid F = F ××F(P ) with exponent α.     1. Assume that a ∈ H, and set supp∗P (a) = p ∈ suppP (a)  {p} is H-essential . (a) If a ∈ A(H), then



vp (a) ≤ (2α − 1) |C ∗ (H, F )| .

p∈supp∗ P (a)

(b) We have the estimates 1 (2α − 1)|C ∗ (H, F )|



vp (a) ≤ min L(a)

p∈supp∗ P (a)

≤



vp (a)

p∈supp∗ P (a)

+ (3α − 1) | suppP (a)| .

92

Halter-Koch (c) If every minimal H-essential subset of suppP (a) is a singleton, then

max L(a) ≤

vp (a)

and

ρ(a) ≤ (2α − 1) |C ∗ (H, F )| .

p∈supp∗ P (a)

2. If H is simple in F , then ρ(H) ≤ (2α − 1) |C ∗ (H, F )|. Otherwise ρ(H) = ∞. Proof. 1.(a) Suppose that

vp (a) > (2α − 1) |C ∗ (H, F )| .

p∈supp∗ P (a)

Then there exist (not necessarily distinct) primes p1 , . . . , p2α ∈ supp∗P (a) such that F [p1 ]F H = . . . = [p2α ]H and a = p1 · . . . · p2α b for some b ∈ F . We set q = p1 · . . . · pα ,   α F F a = pα+1 · . . . · p2α b, a1 = p2α 1 b and a1 = p1 b. Then [a1 ]H = [a]H , hence α  α F a1 ∈ H, and by Lemma 4.2 we obtain p1 ∈ H and a1 ∈ H. Since [p1 ]H = [q]F H and  F   [a1 ]F = [a ] , it follows that q ∈ H and a ∈ H, and therefore a = qa is not an H H atom of H. (b) If r = min L(a) and a = u1 ·. . .·ur with u1 , . . . , ur ∈ A(H), then (a) implies

vp (a) =

p∈supp∗ P (a)

r



vp (ui ) ≤ r(2α − 1) |C ∗ (H, F )| .

i=1 p∈supp∗ P (a)

Hence the first inequality holds. To prove the second one, we consider F0 = F × ×F(P0 ), where P0 = suppP (a), and H0 = F0 ∩ H. By Proposition 4.4, H0 is a C0 -monoid defined in F0 , and H0 is a divisor-closed submonoid of H. Hence we may replace H ⊂ F by H0 ⊂ F0 , and then the inequality follows by [9, Proposition 5.6]. (c) If every minimal H-essential subset of suppP (a) is a singleton, then every atom u ∈ A(H) dividing a satisfies vp (u) ≥ 1 for at least one p ∈ P ∗ . Hence max L(a) ≤

p∈P ∗

vp (a)

and

ρ(a) =

max L(a) ≤ (2α − 1) |C ∗ (H, F )| . min L(a)

2. If H is simple in F and a ∈ H, then every minimal H-essential subset of suppP (a) is a singleton, and thus ρ(a) ≤ (2α − 1) |C ∗ (H, F )| by 1.(c). Hence ρ(H) ≤ (2α − 1) |C ∗ (H, F )|. Assume now that H is not simple in F . Then there exists some a ∈ H such that suppP (a) is a minimal H-essential subset of P and |suppP (a)| ≥ 2. Then supp∗P (a) = ∅, and 1.(b) implies min L(an ) ≤ (3α − 1) | suppP (a)| for all n ∈ N , and thus ρ(H) = ∞.

hence

lim ρ(an ) = ∞ ,

n→∞

C-Monoids and Congruence Monoids in Krull Domains

6

93

Congruence Monoids in Krull Domains

We recall the definition and the elementary properties of congruence monoids from [9]. Let R be an integral domain. A map σ = (σ1 , . . . , σm ) : R• → {±1}m is called a sign vector of R, if there exist distinct ring monomorphisms w1 , . . . , wm : R → R such that σj (a) = sign{wj (a)} for all a ∈ R• and all j ∈ [1, m]. For m = 0, the empty sequence will also be considered as a sign vector. Let {0} = f  R be an ideal and σ = (σ1 , . . . , σm ) a sign vector of R. Two elements a, b ∈ R• are called congruent modulo fσ, a ≡ b mod fσ, if a ≡ b mod f and σ(a) = σ(b). Congruence modulo fσ is a congruence relation on R• , and the semigroup of congruence classes is denoted by R/fσ. For a ∈ R• , we denote by [a]fσ ∈ R/fσ the congruence class containing a. Note that (R/fσ)× = {[a]fσ | a + f ∈ (R/f)× } is the group of invertible elements of R/fσ. For m = 0, the congruence modulo fσ is just the ordinary congruence modulo f, and we write f instead of fσ. If ∅ = Γ ⊂ R/fσ is a multiplicatively closed subset (not necessarily containing the unit element [1]fσ ), then the (multiplicative) monoid   HΓ = a ∈ R• | [a]fσ ∈ Γ ∪ {1} ⊂ R• is called the congruence monoid defined in R modulo fσ by Γ. It is called • regular modulo f , if aR + f = R for all a ∈ H. • singular modulo f , if aR + f = R for all a ∈ H \ {1}. Note that H is regular modulo f if and only if Γ ⊂ (R/fσ)× , and H is singular modulo f if and only if Γ ∩ (R/fσ)× = ∅. A submonoid H ⊂ R• is called a congruence monoid in R, if there exists an ideal f = {0} of R, a sign vector σ of R and a multiplicatively closed subset ∅ = Γ ⊂ R/fσ such that H = HΓ . Every such ideal f is called an ideal of definition for H. If f is an ideal of definition for H and R/f is noetherian, then H has a largest ideal of definition (see [9, Proposition 3.4]). The most interesting examples of congruence monoids are Hilbert monoids of natural numbers and orders in Krull domains. Regular Hilbert monoids were discussed in [13] and regular congruence monoids in Dedekind domains in [14]. For the analytical number theory of congruence monoids we refer to [11] and for the deeper arithmetical theory of congruence monoids in Dedekind domains we refer to [9]. The arithmetical theory of Hilbert monoids is also discussed in [8]. In this chapter we investigate congruence monoids in Krull domains and give a common generalization of the theory developed in [9] and the investigations of W. Hassler [17] on certain noetherian domains. Lemma 6.1. Let R be an integral domain, f = {0} an ideal of R, σ a sign vector of R, ∅ = Γ ⊂ R/fσ a multiplicatively closed subset, and H = HΓ ⊂ R• . 1. Suppose that either H is singular modulo f or that Γ ∩ (R/fσ)× is a subgroup of (R/fσ)× . Then R× ∩ H = H × .

94

Halter-Koch 2. If (R/fσ)× is a torsion group, then either H is singular modulo f or Γ ∩ (R/fσ)× is a subgroup of (R/fσ)× .

Proof. 1. Obviously, H × ⊂ R× ∩ H, and if H is singular modulo f, then R× ∩ H = {1} = H × . If Γ ∩ (R/fσ)× is a subgroup of (R/fσ)× and a ∈ R× ∩ H, then × −1 [a−1 ]fσ = [a]−1 ∈ H and therefore a ∈ H × . fσ ∈ Γ ∩ (R/fσ) . Hence a 2. Observe that every nonempty multiplicatively closed subset of a torsion group is a subgroup. If H is a congruence monoid in a Dedekind domain, f is its largest ideal of definition, and H = HΓ for some Γ ⊂ R/fσ such that Γ ∩ (R/fσ)× is a group, then H is regular modulo f if and only if H is a Krull monoid (see [9, Theorem 3.7]). In a Krull domain, we have the following weaker result. Proposition 6.2. Let R be a Krull domain, {0} = f  R an ideal and σ a sign vector of R, ∅ = Γ ⊂ R/fσ a multiplicatively closed subset, H = HΓ ⊂ R• and H ∗ = {a ∈ H | a + f ∈ (R/f)× }. Suppose that either H is singular modulo f, or that Γ ∩ (R/fσ)× is a subgroup of (R/fσ)× . 1. H ∗ is a saturated submonoid of R• . In particular, H ∗ is a Krull monoid. 2. If H is regular modulo f, then H is a Krull monoid. Proof. 1. If H is singular modulo f, then H ∗ = {1}. Thus let Γ ∩ (R/fσ)× be a subgroup of (R/fσ)× . If a, b ∈ H ∗ and b = ac for some c ∈ R• , then [a]fσ ∈ × ∗ Γ ∩ (R/fσ)× , and [c]fσ = [a]−1 fσ [b]fσ ∈ Γ ∩ (R/fσ) implies c ∈ H . 2. If H is regular modulo f, then H = H ∗ is a Krull monoid by 1.

Theorem 6.3. Let R be a Krull domain, H ⊂ R• a congruence monoid in R and f an ideal of definition for H. Let C(R) and R/f be finite, and suppose that either R is noetherian or f is divisorial. Then H is a C-monoid. Proof. Being a Krull monoid, R• splits in the form R• = R× × D, where D is a reduced Krull monoid, and there is a divisor theory D → F0 = F (P0 ). Then the composite map ∂ : R• → D → F0 is also a divisor theory. For every a ∈ F0 , ∂ −1 (aF0 ) ∪ {0} is a divisorial ideal of R (see [16, Theorem 20.5]). Let σ be a sign vector of R and ∅ = Γ ⊂ R/fσ a multiplicatively closed subset such that H = HΓ . By definition, H ⊂ R× ×D ⊂ R× ×F0 = F , F is factorial, and F × ∩ H = R× ∩ H. Since R/f is finite, Lemma 6.1 implies R× ∩ H = H × . Thus it remains to prove that C ∗ (H, F ) is finite. We consider the group G = q(F )/q(R• ). It is finite, since it is isomorphic to C(R), and since R• ⊂ F is cofinal, we have G = {zq(R• ) | z ∈ F }. Therefore it is × sufficient to prove that for each g ∈ G the set {[x]F H | x ∈ F ∩ g \ F } is finite. × Let g ∈ G be given. If z = εz0 ∈ F ∩ (−g), where ε ∈ R and z0 ∈ F0 , then z(F ∩ g) = zF ∩ q(R• ) = R× ×(z0 F0 ∩ q(D)) = R× ×(z0 F0 ∩ F0 ∩ q(D)) = R× ×(z0 F0 ∩ D) = ∂ −1 (z0 F0 ) .

C-Monoids and Congruence Monoids in Krull Domains

95

We fix an element w ∈ F ∩ −g, and we set E = {x ∈ F ∩ g | x(F ∩ −g) ⊂ f} ,

a = w(F ∩ g) ∪ {0} and b = (wE) ∪ {0} .

Then a is a divisorial ideal of R, and b ⊂ a. We shall prove the following assertions: 1) b is an ideal of R; 2) af ⊂ b; 3) a/b is finite; 4) If x, y ∈ F ∩ g \ F × and F wx ≡ wy mod bσ, then [x]F H = [y]H . × Once this is done, the finiteness of the set {[x]F H | x ∈ F ∩ g \ F } follows. Indeed, by 4) there is a surjective map         × , given by [wx]bσ → [x]F Z = [wx]bσ  x ∈ F ∩g\F × → [x]F H  x ∈ F ∩g\F H

and clearly Z ⊂ {[a]bσ | a ∈ a}. Since a/b is finite and every residue class modulo b splits into finitely many residue classes modulo bσ (by [9, Lemma 3.2]), it follows that Z is finite. 1) We prove that b, b ∈ b and c ∈ R implies b + b ∈ b and cb ∈ b. We may assume that b, b , b + b and c are all different from zero. Then b = wx, b = wx and b + b = wy, where x, x , y ∈ F ∩ g, x(F ∩ −g) ⊂ f and x (F ∩ −g) ⊂ f. If z ∈ F ∩ −g, then yz = yw(w−1 z) = (xw + x w)w−1 z = xz + x z ∈ f. Hence b + b ∈ b, and cx(F ∩ −g) ⊂ f implies cb = wcx ∈ b. 2) If 0 = a ∈ a and c ∈ f, then a = wx, where x ∈ F ∩ g, ca = w(cx), and cx(F ∩ −g) ⊂ cR ⊂ f. 3) Since fa ⊂ b, it follows that a/b is an R/f-module, and since R/f is finite, it is sufficient to prove that a/b is a finitely generated R/f-module. If R is noetherian, then a is finitely generated, and consequently a/b is also finitely generated. If f is divisorial, then all prime ideals containing f are minimal over f. Hence they are divisorial, and there are only finitely many of them, say p1 , . . . , pk (see [16, Proposition 6.6 and Theorem 24.2]). If S = R \ (p1 ∪ . . . ∪ pk ), then S −1 R is a semilocal Krull domain and {S −1 p1 , . . . , S −1 pk } = max(S −1 R). Hence S −1 R is one-dimensional, and therefore it is a principal ideal domain, and a/b = S −1 (a/b) = S −1 a/S −1 b is a finitely generated module over S −1 (R/f) = R/f (the latter argument is due to F. Kainrath). 4) Suppose that x, y ∈ F ∩ g \ F × = F ∩ g \ R× and wx ≡ wy mod bσ. Then wx − wy ∈ b = wE ∪ {0}, say wx − wy = wu for some u ∈ E ∪ {0}. We must prove that, for all z ∈ F , zx ∈ H implies zy ∈ H. If z ∈ F and zx ∈ H, then z ∈ F ∩ −g, and therefore zu ∈ f. Since zu = (zw−1 )(wu) = zx − zy and σ(zx) = σ(zw−1 )σ(wx) = σ(zw−1 )σ(wy) = σ(zy), it follows that zx ≡ zy mod fσ. Since x ∈ / R× , we have zx = 1 and hence zy ∈ H. Corollary 6.4. Let A be a noetherian domain, R its integral closure and f = AnnA (R/A) = {0}. 1. R is a Krull domain, and A• is a congruence monoid defined in R modulo f by A/f. 2. If C(R) and R/f are finite, then A• is a C-monoid.

96

Halter-Koch

Proof. By [5, Theorem 4.3], R is a Krull domain, and obviously A• is a congruence monoid defined in R modulo f by A/f (see also [11] and [9, Example 3.3.1]). Thus the assertions follow by Theorem 6.3. An even stronger version of Corollary 6.4 was proved by W. Hassler [17, Theorem 3.2], who also showed the finiteness of the unit factor group R× /A× .

Bibliography [1] D. D. Anderson (ed.), Factorization in integral domains, Marcel Dekker, 1997. [2] D. F. Anderson, Elasticity of factorizations in integral domains: a survey, Factorization in integral domains, Lecture Notes in Pure Appl. Math., Marcel Dekker 189 (1997), 1–29. [3] S.T. Chapman and A. Geroldinger, Krull domains and monoids, their sets of lengths and associated combinatorial problems, Factorization in integral domains, Lecture Notes in Pure Appl. Math., Marcel Dekker 189 (1997), 73–112. [4] A. Foroutan and A. Geroldinger, Monotone chains of factorizations in Cmonoids, these Proceedings. [5] R.M. Fossum, The divisor class group of a Krull domain, Springer, 1973. [6] A. Geroldinger, On the structure and arithmetic of finitely primary monoids, Czech. Math. J. 46 (1996), 677–695. [7] A. Geroldinger, The catenary degree and tameness of factorizations in weakly Krull domains, Factorization in integral domains, Lecture Notes in Pure Appl. Math., Marcel Dekker 189 (1997), 113–153. [8] A. Geroldinger and F. Halter-Koch, Transfer principles in the theory of nonunique factorizations, these Proceedings. [9] A. Geroldinger and F. Halter-Koch, Congruence monoids, Acta Arith. 112 (2004), 263–296. [10] A. Geroldinger, F. Halter-Koch, W. Hassler, and F. Kainrath, Finitary Monoids, Semigroup Forum 67 (2003), 1–21. [11] A. Geroldinger, F. Halter-Koch, and J. Kaczorowski, Non-unique factorizations in orders of global fields, J. Reine Angew. Math. 459 (1995), 89–118. [12] F. Halter-Koch, Halbgruppen mit Divisorentheorie, Expo. Math. 8 (1990), 27– 66. [13] F. Halter-Koch, Arithmetical Semigroups Defined by Congruences, Semigroup Forum 42 (1991), 59–62. [14] F. Halter-Koch, Ein Approximationssatz f¨ ur Halbgruppen mit Divisorentheorie, Result. Math. 19 (1991), 74–82. 97

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[15] F. Halter-Koch, Finitely generated monoids, finitely primary monoids and factorization properties of integral domains, Factorization in integral domains, Lecture Notes in Pure Appl. Math., Marcel Dekker 189 (1997), 73–112. [16] F. Halter-Koch, Ideal systems. An introduction to multiplicative ideal theory, Marcel Dekker, 1998. [17] W. Hassler, Factorization in finitely generated domains, J. Pure Appl. Algebra 186 (2004), 151–168. [18] F. Kainrath, Elasticity of finitely generated domains, Houston J. Math. 30 (2004).

Chapter 5

Monotone Chains of Factorizations in C-Monoids by

1

Andreas Foroutan and Alfred Geroldinger1

Introduction and Main Result

C-monoids were recently introduced by F. Halter-Koch as a common generalization of various types of auxiliary monoids studied in factorization theory. C-monoids are suitably defined submonoids of factorial monoids (cf. Definition 2.1), and they include Krull monoids with finite class groups and congruence monoids in Krull domains satisfying some natural finiteness conditions. In particular, let A be a noetherian domain, R its integral closure and f = AnnA (R/A) = {0}. Then the multiplicative monoid A• of A is a congruence monoid in R, and if the class group C(R) and R/f are finite, then A• is a C-monoid ([12, Theorem 3.2] and [9, Corollary 6.4]). C-monoids have nice algebraic and arithmetical properties. Every C-monoid H

is a Krull monoid with finite class is v-noetherian, its complete integral closure H

group and the conductor (H : H) is nonempty ([9, Theorem 4.6]). Moreover, H is locally tame, has finite catenary degree, satisfies the Structure Theorem for Sets of Lengths, and there is an explicit criterion for the finiteness of its elasticity ([7, Theorem 5.9], [5, Theorem 7.2] and [9, Theorem 5.9]). This chapter continues the investigations of the arithmetic of C-monoids. We discuss the main result which is formulated below (Theorem 1.1). Let H be a monoid (i.e., a commutative, cancellative semigroup with unit element) and H × its group of invertible elements. Then H is atomic, if every nonunit a ∈ H \ H × has a factorization (product decomposition) into atoms (irreducible elements) of H, and H is factorial, if it is atomic and every nonunit has a unique factorization 1 This

work was supported by the Austrian Science Fund FWF (Project-Nr. P16770-N12)

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into atoms. Suppose that H is atomic but not factorial. Then there exists some a ∈ H \ H × having two distinct factorizations z and z  , say z = u1 · . . . · ul · v1 · . . . · vm

and z  = u1 · . . . · ul · w1 · . . . · wn

where ui , vj , wk are atoms and vj H × = wk H × for all i ∈ [1, l], j ∈ [1, m] and k ∈ [1, n]. Then |z| = l + m (resp. |z  | = l + n) is called the length of z (resp. the length of z  ), and d(z, z  ) = max{m, n} is called the distance between z and z  . A simple observation shows that for every N ∈ N there exist some aN ∈ H \ H ×   and factorizations zN , zN of aN such that d(zN , zN ) ≥ N . The monoid H has finite catenary degree, if there exists some constant M ∈ N such that for every a ∈ H \H × each two factorizations z, z  of a can be concatenated by an M -chain of factorizations (i.e., there exists a finite sequence of factorizations (z = z0 , z1 , . . . , zk+1 = z  ) of a such that d(zi−1 , zi ) ≤ M for every i ∈ [1, k + 1]). It is well-known that C-monoids have finite catenary degree, but up to now there was no information about length or structure of concatenating chains. Theorem 1.1. Let H be a C-monoid. Then there exists some constant M ∈ N having the following property: for every a ∈ H and for each two factorizations z, z  of a there exist factorizations z = z0 , z1 , . . . , zk+1 = z  of a such that, for every i ∈ [1, k + 1], d(zi−1 , zi ) ≤ M

and (either |z1 | ≤ . . . ≤ |zk | or |z1 | ≥ . . . ≥ |zk | ).

Thus there exists some M ∈ N such that for every a ∈ H and each two factorizations of a there is a concatenating M -chain for which the associated sequence of lengths (|z0 |, |z1 |, . . . , |zk |, |zk+1 |) is monotone, with possible exceptions at the first and the last step. These exceptions do not occur in Krull monoids with finite class groups (see [4, Theorem 5.1]). Let A be a one-dimensional local noetherian domain A with integral closure R such that f = AnnA (R/A) = {0} and R/f is finite, whence A• is a C-monoid. W. Hassler proved an arithmetical property for A• implying Theorem 1.1, but on the other hand he could show that these exceptions actually occur. Thus there is a domain A as above for which there is no M ∈ N such that for every a ∈ A• each two factorizations of a can be concatenated by a monotone M -chain of factorizations (see [11, Theorem 4.1 and Example 6.3]). The first step in the proof of Theorem 1.1 is to reduce the problem to those C-monoids which are submonoids of finitely generated factorial monoids (see [5, Theorem 7.2]). This allows to apply Dickson’s Finiteness Theorem on minimal points of subsets S ⊂ Ns0 . Furthermore, these simpler C-monoids are finitary in the sense of [8], and we can use the machinery of tamely generated ideals as developed in [6].

2

Preliminaries

Our terminology and our notations are consistent with [7], [9] and [5]. For convenience we recall some key notions. We denote by N the set of positive integers and

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set N0 = N ∪ {0}. For integers m, n ∈ Z let [m, n] = {x ∈ Z | m ≤ x ≤ n}, and for a set X we denote by |X| ∈ N0 ∪ {∞} its cardinality. Basic Notions on Monoids. Throughout this chapter, a semigroup is a nonempty set with a commutative and associative law of composition possessing a unit element. We use multiplicative notation and denote the unit element by 1. By a monoid, we mean a cancellative semigroup. Let H be a monoid. Then H × denotes its group of invertible elements and Hred = {aH × | a ∈ H} the associated reduced monoid of H. We say that H is reduced, if H × = {1}. An element u ∈ H is called irreducible (or an atom of H), if u ∈ / H × and, for all a, b ∈ H, u = ab × × implies that a ∈ H or b ∈ H . We denote by A(H) the set of atoms of H, and we say that H is atomic if every a ∈ H \ H × is a product of atoms. An element p ∈ H is called a prime of H, if p ∈ / H × , and if a, b ∈ H, then p | ab implies that p | a or p | b. Every prime is an atom, and a monoid is factorial if and only if it is atomic and every atom is a prime. If H is atomic and p ∈ H is a prime element of H, then every a ∈ H may be written in the form a = pn b where b ∈ H and n ∈ N0 . The exponent n is uniquely determined by the classes aH × and pH × in Hred , and we call n = vp (a) the p-adic value of a. For a set P , we denote by F (P ) the free abelian monoid with basis P . Every a ∈ F(P ) has a unique representation in the form  pvp (a) where vp (a) ∈ N0 and vp (a) = 0 for all but finitely many p ∈ P . a= p∈P

For a, b ∈ F(P ), we set |a| =

p∈P

vp (a)

  and d(a, b) = max 

   b a    ,   . gcd(a, b) gcd(a, b)

Note that | · | : F (P ) → N0 is a homomorphism, and d : F (P ) × F(P ) → N0 is a metric. For every element a ∈ H we set [[a]] = {b ∈ H | b | an for some n ∈ N} ⊂ H (then [[a]] is the smallest divisor-closed submonoid containing a). A subset a ⊂ H is called an s-ideal of H, if aH = a. Finitary Monoids. Let H be a monoid with H = H × . A subset U ⊂ H \ H × is called an almost generating set of H, if there exists some n ∈ N such that (H \ H × )n ⊂ U H. We denote by M(U ) the smallest possible n ∈ N for which the above inclusion holds. If U is an almost generating set, then U [θ] = {uθ | u ∈ U } is an almost generating set for every θ ∈ N (cf. [8, Lemma 3.4.1]). The monoid H is called finitary, if it satisfies the ascending chain condition on principal ideals, and if it has some finite almost generating set U . Finitary monoids were introduced in [8] as a powerful tool in factorization theory. Main examples of finitary monoids are finitely generated monoids, strongly primary monoids, v-noetherian G-monoids and abstract congruence monoids. Let H be a finitary monoid, a ⊂ H an s-ideal, U ⊂ H \ H × a finite almost generating set and u ∈ U . We denote by Hu the set of all a ∈ H without a divisor in [[u]] \ H × , by a(U, u) the set of all a ∈ a ∩ u2 H such that [[u]] is maximal in the set {[[v]] | v ∈ U, a ∈ v 2 H}, and we set a[U, u] = Hu ∩ (u[[u]])−1 a(U, u) .

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A simple calculation gives that    a⊂ u[[u]] a[U, u] ∪ a \ U [2] H u∈U

(cf. [6, Proposition 5.5]), and we say that a is U -generated, if equality holds. Clearly, the s-ideal H is U -generated. A finite almost generating set W ⊂ H is called full, if there exists some m ∈ N such that H[W, w] ⊂ H \ W m H for all w ∈ W , and every finitary monoid has such an almost generating set. Class semigroups. Class semigroups were recently introduced in [7, Section 4] as a refinement of ordinary class groups in algebraic number theory. We repeat the definition for a situation which is sufficient for our requirements. Let D be a monoid and H ⊂ D a submonoid. Two elements y, y  ∈ D are called H-equivalent, −1 if y −1 H ∩ D = y  H ∩ D. H-equivalence is a congruence relation on D. For y ∈ D, D let [y]H denote the congruence class of y, and let ∗ D × C(H, D) = {[y]D H | y ∈ D} and C (H, D) = {[y]H | y ∈ (D \ D ) ∪ {1}}.

Then C(H, D) is a semigroup with unit element [1]D H (called the class semigroup of H in D) and C ∗ (H, D) ⊂ C(H, D) is a subsemigroup (called the reduced class semigroup of H in D). If C(H, D) is a torsion group, then H ⊂ D is saturated and cofinal, and if H ⊂ D is saturated and cofinal, then C(H, D) is isomorphic to the factor group of the quotient groups of D and H (cf. [9, Section 3]). C-monoids and C0 -monoids. C-monoids generalize abstract congruence monoids defined in factorial monoids and Z-monoids (cf. [7] and [12]). For their relevance and their properties we refer to the paper of F. Halter-Koch in this volume (see [9]). Here we only give the definition and a characterization of C0 -monoids which is important for the present investigations. Definition 2.1. A monoid H is called a C-monoid, if it is a submonoid of a factorial monoid F such that H ∩ F × = H × and C ∗ (H, F ) is finite. A C0 -monoid is a C-monoid H defined in a factorial monoid F having only finitely many pairwise nonassociated prime elements. Theorem 2.2. Let F = F × × [p1 , . . . , ps ] be a factorial monoid with pairwise nonassociated prime elements p1 , . . . , ps , and let H ⊂ F be a submonoid such that H ∩ F × = H × . Then the following assertions are equivalent: 1. H is a C0 -monoid defined in F . 2. There exists some α ∈ N and a subgroup V ⊂ F × such that (F × : V ) | α, V (H \ H × ) ⊂ H, and for all j ∈ [1, s] and a ∈ pα j F we have a ∈ H if and a ∈ H. only if pα j In particular, every C0 -monoid in the sense of Definition 2.1 is a C0 -monoid in the sense of [7, Definition 5.1], and conversely. Let H ⊂ F be a C0 -monoid with all notations as in Theorem 2.2. Then a simple inductive argument shows that H satisfies the following congruence condition (C), which will turn out to be crucial for our investigations:

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(C) If u ∈ F and (n1 , . . . , ns ), (n1 , . . . , ns ) ∈ Ns0 are such that,

 either ni = ni or ni ≡ ni mod α and min{ni , ni } ≥ α for each i ∈ [1, s], then u

s 

pni ∈ H

implies

i=1

u

s 




pni ∈ H .

i=1

Factorization Theory. Let H be a monoid. The monoid Z(H) = F (A(Hred )) is called the factorization monoid of H. Let π : Z(H) → Hred denote the unique homomorphism satisfying π(u) = u for all u ∈ A(Hred ). Then H is atomic if and only if π is surjective. For every a ∈ H the elements in Z(a) = π −1 (aH × ) are called factorizations of a. It was a crucial new idea, introduced in [4], to study arithmetical invariants (such as sets of lengths, catenary and tame degrees) with respect to a given subset of factorizations X ⊂ Z(H). We shall make substantial use of this idea and give the necessary definitions. Suppose that H is an atomic monoid, a ∈ H and X ⊂ Z(H) a subset. Note that in case X = Z(H), the forthcoming definitions coincide with the usual ones. Let ZX (a) = Z(a) ∩ X denote the set of factorizations of a which are in X and X for k ∈ N0 let ZX k (a) = {z ∈ Z (a) | |z| = k}. The set X ⊂ Z(H) is called M -dense for some nonnegative integer M ∈ N0 , if for every b ∈ H and every z ∈ Z(b) there exists some z  ∈ ZX (b) such that d(z, z  ) ≤ M . We say that X is a dense subset, if it is M -dense for some M ∈ N0 . We denote by LX (a) = {|z| | z ∈ ZX (a)} ⊂ N0 the set of lengths of a with respect to X. H is called a BF-monoid, if all sets of lengths LZ(H) (a) are finite. Finitary monoids are BF-monoids. Sets of lengths and all invariants derived from them, such as the elasticity of a monoid, are central objects of interest in factorization theory (cf. [2], [1], [3], [13]). Distances: Let L ⊂ Z. If m ∈ L and d ∈ N are such that [m, m + d] ∩ L = {m, m + d}, then m and m + d are called successive lengths of L, and d is called a distance of L. We denote by ∆(L) ⊂ N the set of all distances of L, and call  ∆(LX (a)) ⊂ N ∆X (H) = a∈H

the set of distances of H with respect to X. For subsets Z, Z  ⊂ Z(H) we set d(Z, Z  ) = min{d(z, z ) | z ∈ Z, z  ∈ Z  } ∈ N0 . Let a ∈ H. If z, z  ∈ ZX (a), then z = z  if and only if d(z, z  ) = 0; if z = z  , then 2 ≤ d(z, z  ) ≤ sup LX (a). We set X δ X (a) = sup{d({z}, ZX k (a)) | z ∈ Z (a),

|z| and k are successive lengths in LX (a)} ∈ N0 ∪ {∞} and δ X (H) = sup{δ X (a) | a ∈ H} ∈ N0 ∪ {∞}.

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Chains of Factorizations: If a ∈ H, z, z  ∈ ZX (a) and N ∈ N0 , then an N -chain of X-factorizations from z to z  is a finite sequence (z0 , z1 , . . . , zk ) in ZX (a) such that z = z0 , zk = z  and d(zi−1 , zi ) ≤ N for all i ∈ [1, k]. We denote by cX (a) the smallest N ∈ N0 ∪ {∞} such that for each two factorizations z, z  ∈ ZX (a) there is an N -chain of X-factorizations from z to z  . We set cX (H) = sup{cX (a) | a ∈ H} ∈ N0 ∪ {∞}. If cX (H) < ∞, then ∆X (H) is finite. We say that H has finite catenary degree, if cZ(H) (H) < ∞. A chain of X-factorizations (z0 , . . . , zk ) is called monotone, if either |z0 | ≤ . . . ≤ |zk | or |z0 | ≥ . . . ≥ |zk |. We denote by cX m (a) the smallest N ∈ N0 ∪ {∞} such that for each two factorizations z, z  ∈ ZX (a) there is a monotone N -chain of Xfactorizations from z to z  . We call X cX m (H) = sup{cm (a) | a ∈ H} ∈ N0 ∪ {∞}

the monotone catenary degree of H with respect to X. Tameness: For a ∈ H and u ∈ Z(H), let tX (a, u) denote the smallest integer N ∈ N0 with the following property: if z ∈ ZX (a) and ZX (a) ∩ uZ(H) = ∅, then there exists some z  ∈ ZX (a) ∩ uZ(H) such that d(z, z  ) ≤ N . For subsets H  ⊂ H and U ⊂ Z(H) we define tX (H  , U ) = sup{tX (a, u) | a ∈ H  , u ∈ U } ∈ N0 ∪ {∞}. H is called locally tame, if tZ(H) (H, u) < ∞ for all u ∈ A(Hred ). Let a ⊂ H be an s-ideal. A subset E ⊂ a is called a tame generating set of a (with bound M ∈ N), if for every a ∈ a there exists some e ∈ E satisfying e | a, sup L(e) ≤ M and tZ(H) (a, Z(e)) ≤ M (cf. [6, Definition 3.3 and Proposition 3.8]). For a ∈ H and u ∈ Z(H), let tX m (a, u) denote the smallest integer N ∈ N0 with the following property: if z ∈ ZX (a) and z ∈ ZX (a) ∩ uZ(H), then there exists some z  ∈ ZX (a) ∩ uZ(H) such that d(z, z  ) ≤ N and (either |z| ≤ |z  | ≤ |z| or |z| ≤ |z  | ≤ |z|). For subsets H  ⊂ H and U ⊂ Z(H) we define  X  tX m (H , U ) = sup{tm (a, u) | a ∈ H , u ∈ U } ∈ N0 ∪ {∞}.

Whenever an arithmetical invariant appears without an upper index specifying the set of factorizations X ⊂ Z(H), then it is supposed that X = Z(H), so L(a) = LZ(H) (a), c(H) = cZ(H) (H) and so on. All arithmetical invariants introduced above are in fact invariants of the associated reduced monoid Hred . Hence we may suppose that H is reduced whenever this is convenient.

3

Reduction of the Theorem to the Main Proposition

In this section we formulate a Main Proposition and show how it implies Theorem 1.1. The proof of the Main Proposition will be given in Sections 4 and 5.

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Proposition 3.1 (Main Proposition). Let H be a reduced C0 -monoid defined in a factorial monoid F having a finite group of units F × . Then there exist a finite set A ⊂ A(H), a finite almost generating set W of H, sets of factorizations Xw ⊂ Z(H) for every w ∈ W, a set of factorizations Y ⊂ Z(H) and a constant M ∈ N such that the following conditions are satisfied: 1. W is full and H=



w[[w]]H[W, w] ∪ (H \ W[2] H).

w∈W w 2. A is an almost generating set, AXw ⊂ Xw , A−1 Xw ∩Z(H) ⊂ Xw and ZX k (a)∩ Xw AZ(H) = ∅ for every a ∈ H, w ∈ W and every k ∈ L (a) with k ≥ M.

3. Y is M-dense. 4. Let w ∈ W, X = Xw and a ∈ w[[w]]H[W, w]. H[W, w], b ∈ Z(b) and a ∈ [[w]] such that a = b. Furthermore, for every z ∈ ZY (a) there are factorizations z = z0 , . . . , zk = z  ∈ Z(a) such d(zi−1 , zi ) = 2 for every i ∈ [1, k].

Then there exist some b ∈ wa b and ZY (a) ⊂ Z(wa ) · some z  ∈ ZX (wa ) · b and that |z0 | = . . . = |zk | and

5. tXw (H, v) < ∞ and ∆Xw (H) is finite for every v ∈ A and every w ∈ W. 6. δ Xw (H) < ∞ for every w ∈ W. Lemma 3.2. Let H be a reduced BF-monoid, X ⊂ Z(H) and A ⊂ A(H) such that AX ⊂ X and A−1 X ∩ Z(H) ⊂ X. 1. If there is some M ∈ N such that ZX (a) ∩ AZ(H) = ∅ for every a ∈ H with max LX (a) > M , then cX (H) ≤ sup{M, tX (H, A)}. 2. If there is some M ∈ N such that ZX k (a) ∩ AZ(H) = ∅ for every a ∈ H and every k ∈ LX (a) with k ≥ M , then X cX m (H) ≤ sup{M, tm (H, A)}.

Proof. 1. If tX (H, A) = ∞, then there is nothing to show. Suppose that this is not the case, and let a ∈ H. We prove that cX (a) ≤ max{M, tX (H, A)} = M ∗ . We proceed by induction on max LX (a). If max LX (a) ≤ M , then cX (a) ≤ max LX (a) ≤ M . Suppose that max LX (a) > M , and let z, z  ∈ ZX (a). By assumption there exists some u ∈ A such that ZX (a) ∩ uZ(H) = ∅, and we set b = u−1 a. Then there are y = ux, y  = ux ∈ ZX (a), with x, x ∈ Z(b), such that d(z, ux) ≤ tX (H, u) and d(z, ux ) ≤ tX (H, u). Since

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A−1 X ∩ Z(H) ⊂ X, it follows that x, x ∈ ZX (b). Since max LX (b) < max LX (a), there exists an M ∗ -chain (x, x1 , . . . , xk , x ) of X-factorizations from x to x . Since AX ⊂ X, (z, ux, ux1 , . . . , uxk , ux , z  ) is an M ∗ -chain of X-factorizations from z to z. 2. [4, Lemma 3.5]. We briefly analyze the assumption occurring in part 1. of Lemma 3.2. If A = A(H), then it holds with M = 1. Suppose that X = Z(H). Then there exists some finite set A ⊂ A(H) such that the assumption holds if and only if H is finitary. Lemma 3.3. Let H be a reduced atomic monoid and X ⊂ Z(H). 1. If u ∈ A(H) such that uX ⊂ X and u−1 X ∩ Z(H) ⊂ X, then X  X X X tX m (H, u) ≤ δ (H) t (H, u) + sup ∆ (H) + t (H, u). 2. If there exists some a ∈ H with |ZX (a)| ≥ 2, then 2 + sup ∆X (H) ≤ cX (H). In particular, if cX (H) < ∞, then ∆X (H) is finite. Proof. 1. follows from [4, Lemma 3.7], and 2. from [4, Lemma 3.3]. Proof of Theorem 1.1. A suitable transfer homomorphism (as derived in [5, Lemma 3.5 and Theorem 7.2]) shows that it suffices to prove Theorem 1.1 for C0 -monoids defined in factorial monoids having finite groups of units. By [9, Proposition 4.5], we may restrict to reduced monoids. Let H be a reduced C0 -monoid defined in a factorial monoid F having a finite group of units F × . Thus H is as in the Main Proposition, and let A, W, Xw , Y and M be as there. First we show that, for every X ∈ {Xw | w ∈ W}, we have cX m (H) < ∞. By Proposition 3.1(5. and 6.) and Lemma 3.3.1, it follows that tX m (H, u) < ∞ for all u ∈ A. Therefore, Lemma 3.2.2 implies that cX (H) < ∞. m Now we show that the assertion of Theorem 1.1 holds with the constant w M = max{M(W[2] ), M, 2, cX m (H) | w ∈ W} ∈ N.

Let a ∈ H and z, z  ∈ Z(a). If a ∈ H \ W[2] H, then d(z, z  ) ≤ max L(a) ≤ M(W[2] ). Suppose that a ∈ W[2] H whence a ∈ w[[w]]H[W, w] for some w ∈ W and set X = Xw . Since Y is M-dense, there are z, z ∈ ZY (a) such that d(z, z) ≤ M and d(z  , z ) ≤ M. By Proposition 3.1.4, there are y, y ∈ ZX (H), some b ∈ Z(H) and 2-chains of factorizations, all factorizations having the same lengths, from z to yb ∈ Z(a) and from z to y b. Since cX m (H) < ∞, there exists a monotone M -chain of factorizations ( y b, y1 b, . . . , yk b, y b) from yb to y b. Thus we obtain a monotone M -chain of factorizations from z to z , and the assertion follows.

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4

107

Construction of A, W, Xw , Y and M

We fix our notations which remain valid throughout the rest of this chapter. Let H be as in the Main Proposition, whence H ⊂ F = F × × [p1 , . . . , ps ] is a reduced C0 -monoid in the factorial monoid F , where F × is finite and p1 , . . . , ps ∈ F are pairwise nonassociated primes of F . Let α ∈ N be as in Theorem 2.2. In particular, we have H ∩ F × = H × = {1}, and H satisfies the congruence condition (C). By [7, Theorem 5.9], H is finitary, locally tame and has finite catenary degree. For x ∈ F we call supp(x) = {i ∈ [1, s] | vpi (x) > 0} the support of x. A subset E ⊂ [1, s] is called H-essential, if there exists some a ∈ H such that E = supp(a), and we denote by EH the set of all nonempty H-essential subsets of [1, s]. Construction 4.1. We define A = {u ∈ A(H) | vpi (u) < 4α for all i ∈ [1, s]}.  Since F × is finite, A is finite, and obviously every uI = i∈I pα i , where I ∈ EH , has some irreducible divisor lying in A. Since {uI | I ∈ EH } is an almost generating set of H by [7, Proposition 5.6], A is an almost generating set by [8, Lemma 3.4.2]. Lemma 4.2. There exists a map Θ : H → H satisfying the following two properties: 1. For every a ∈ H and every i ∈ [1, s] we have: if vpi (a) < 4α, then vpi (a) = vpi (Θ(a)). If vpi (a) ≥ 4α, then vpi (a) ≡ vpi (Θ(a)) mod α and vpi (Θ(a)) ∈ [2α, 3α − 1]. 2. Θ(A(H)) ⊂ A. Proof. Let a ∈ H and i ∈ [1, s]. If vpi (a) < 4α, we set ki = 0. If vpi (a) ≥ 4α, let ki ∈ N such that vpi (a) − ki α ∈ [2α, 3α − 1]. We define Θ(a) = a ·

s 

iα p−k . i

i=1

Then (C) implies that Θ(a) ∈ H. Suppose that a ∈ A(H) and assume to the contrary that Θ(a) = a1 a2 with a1 , a2 ∈ H \ H × . We construct a1 , a2 ∈ H \ H × such that a = a1 a2 , a contradiction. We set a1 = a1 ·

s  i=1

l

pi1,i

α

and a2 = a2 ·

s 

l

pi2,i

α

i=1

where l1,1 , ..., l1,s , l2,1 , ..., l2,s ∈ N0 are defined as follows. Let i ∈ [1, s]. If ki = 0, then l1,i = l2,i = 0 whence vpi (aj ) = vpi (aj ) for j ∈ [1, 2]. Suppose that ki ∈ N. If vpi (a1 ) ≥ α, then l1,i = ki and l2,i = 0. If vpi (a1 ) < α, then vpi (a2 ) = vpi (Θ(a)) − vpi (a1 ) ≥ α and we set l1,i = 0 and l2,i = ki . Then (C) implies that a1 , a2 ∈ H \ F × = H \ {1}, and by construction, we have a = a1 a2 .

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Lemma 4.3. Let u ∈ A(H), i ∈ [1, s] and n ∈ N0 . ∈ A(H). 1. If vpi (u) − nα ≥ 2α, then up−nα i 2. If vpi (u) ≥ 4α, then upnα ∈ A(H). i . Since pnα Proof. 1. Suppose that vpi (u) − nα ≥ 2α and set a = up−nα i a = u ∈ H, i it follows from (C) that a ∈ H. Assume to the contrary that a = a1 a2 with a1 , a2 ∈ H \ {1} and vpi (a1 ) ≥ vpi (a2 ). Then vpi (a1 ) ≥ α whence a1 pnα ∈ H \ {1} i and u = (a1 pnα ) · a , a contradiction. 2 i 2. Suppose that vpi (u) ≥ 4α. Again (C) implies that upnα ∈ H. Assume to i the contrary that upnα = a1 a2 with a1 , a2 ∈ H \ {1}. Without restriction we may i suppose that vpi (a1 ) ≥ 2α. We set vpi (a2 )−vpi (Θ(a2 ))−nα

a1 = a1 pi

vpi (Θ(a2 ))−vpi (a2 )

and a2 = a2 pi

.

Obviously, we have u = a1 a2 . We assert that a1 , a2 ∈ H \ {1} which implies a contradiction. Since vpi (a1 ) ≡ vpi (a1 ) mod α, and

vpi (a1 ) ≥ 2α

vpi (a1 ) = vpi (a1 ) + vpi (a2 ) − vpi (Θ(a2 )) − nα = vpi (u) + nα − vpi (Θ(a2 )) − nα ≥ 4α − 3α = α,

it follows that a1 ∈ H \ {1}. If vpi (a2 ) < 4α, then vpi (Θ(a2 )) = vpi (a2 ) whence a2 = a2 ∈ H \ {1}. If vpi (a2 ) ≥ 4α, then vpi (a2 ) = vpi (Θ(a2 )) ∈ [2α, 3α − 1] and since vpi (a2 ) ≡ vpi (a2 ) mod α, it follows that a2 ∈ H \ {1}. Let I ∈ EH be an H-essential subset. In the next step we define the subset XI ⊂ Z(H) (later, the elements of W will be indexed by EH and we shall write XI = XwI , see 4.6). Let I ⊂ I denote the set of all j ∈ I for which there exists some atom u ∈ A(H) with supp(u) ⊂ I

and vpj (u) ≥ 4α,

(∗)

and we set ΛI = {(I, j) | j ∈ I}. Let λ = (I, j) ∈ ΛI and nλ ∈ N0 . We fix some atom uλ ∈ A(H) having property (∗) and set (n ) pλ = pj and uλ λ = uλ pλnλ α . (nλ )

Then Lemma 4.3 implies that uλ

∈ A(H).

Construction 4.4. We define ΛI ΦI : NA 0 × N0

(m, n)

−→ Z(H)   (n ) u mu uλ λ −→ u∈A

and set

λ∈ΛI



ΛI . XI = Φ NA 0 × N0

Monotone Chains of Factorizations in C-Monoids

109

For X = XI , we obviously have AX ⊂ X and A−1 X∩Z(H) ⊂ X. Furthermore, for every a ∈ H and every k ∈ LX (a) with k ≥ |ΛI |+1, it follows that ZX k (a)∩AZ(H) = ∅. Thus 2. in the Main Proposition holds, provided that M ≥ |ΛI | + 1 (cf. 4.6). We continue with a construction of a set Y which is dense in the set of all factorizations. Clearly, every tame generating set of a monoid gives rise to a dense subset of factorizations. We perform this construction for the canonical tame generating sets of complete ideals. Lemma 4.5. For I ∈ EH , we define wI =



α(s+1−|I|)

pi

i∈I

and we set W = {wI | I ∈ EH }. Then there exists some θ ∈ N with the following properties: 1. W [θ] is a full almost generating set of H and  w[[w]]H[W [θ] , w] ∪ (H \ W [2θ] H). H= w∈W [θ]

2. For every I ∈ EH ,

 λ∈ΛI

uλ divides wIθ .

3. There exists some M ∈ N with the following property: for every w ∈ W [θ] and every a ∈ w[[w]]H[W [θ] , w] there exist some a ∈ [[w]] and some b ∈ H[W [θ] , w] such that a = wa b and t(a, Z(wb)) ≤ M . Proof. We show that 1., 2., and 3. hold for all sufficiently large θ ∈ N. 1. The first assertion follows from [7, Proposition 5.7.1], and the second assertion from [6, Proposition 5.5].  2. Since supp( λ∈ΛI uλ ) ⊂ I = supp(wI ), the assertion follows from [9, Lemma 4.2.2.(d)]. 3. We are going to apply [6, Theorem 5.10]. To do so, we must prove the following assertion: Assertion: There exists some M  ∈ N such that for every w ∈ W and every a ∈ H(W [θ] , wθ ) there exist a ∈ [[w]] and some b ∈ H[W [θ] , wθ ] such that a = wθ a b and t(a, Z(b)) ≤ M  . Once this is done, then the set

  E= wθ H[W [θ] , wθ ] ∪ H \ W [2θ] H w∈W

is a tame generating set of H. In fact, the proof of [6, Theorem 5.10] shows the more precise statement of the Lemma. It remains to prove the Assertion. Let I ∈ EH , w = wI and a ∈ H(W [θ] , wθ ). By [7, Lemma 5.8.4], there exist some a ∈ [[w]] and some b ∈ H[W [θ] , wθ ] such that a = wθ a b and vpi (b) ≤ 2α − 1 for all i ∈ I. Since H[W [θ] , wθ ] ⊂ [[w]]−1 [[w]]H(W [θ] , wθ ) ∩ H,

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Foroutan and Geroldinger

we may apply [7, Lemma 5.8.5 and Proposition 5.6.2] and [6, Lemma 5.9.1] to obtain that t(a, Z(b)) ≤ M  for some bound M  ∈ N depending only on H ⊂ F . Construction 4.6. Let W , θ and M be as in Lemma 4.5. We define W = W [θ]

and XwI = XI for every I ∈ EH .

For a ∈ H  \ W[2] H, we set Z (a) = Z(a). For w ∈ W, we fix a factorization w ∈ Z(w) ∩ λ∈ΛI uλ Z(H). If a = wa b ∈ w[[w]]H[W, w] where a and b are as in Lemma 4.5.3, then we fix some b ∈ Z(b) and set Z (a) = wZ(a )b. We define  Y= Z (a) ⊂ Z(H). a∈H

Then Y is M -dense by construction, and we define M = max{M, |ΛI | + 1 | I ∈ EH }. Thus 1., 2. and 3. of the Main Proposition hold true.

5

Proof of the Main Proposition

It remains to prove 4., 5. and 6. Let I ∈ EH , w = wIθ ∈ W, Λ = ΛI , X = XI and Φ = ΦI . 4. Let a ∈ w[[w]]H[W, w]. The first assertion is clear by Construction 4.6. Let z = wyb ∈ ZY (a), where a = wa b, b ∈ H[W, w], b ∈ Z(b), y ∈ Z(a ) and w ∈ Z(w) are as in 4.6. We have to find a 2-chain of factorizations, all having the same lengths, from wy to some factorization in ZX (wa ). We consider factorizations z of the form  (n ) z = uλ λ · y0 · v1 · . . . · vl (∗) λ∈Λ

where all nλ ∈ N0 , y0 ∈ F(A) and v1 , . . . , vl ∈ A([[w]]) \ A. Then for every i ∈ [1, l] z ) denote the number of all there is some j ∈ I such that vpj (vi ) ≥ 4α. Let l( z ) ≥ l, and if l( z ) = 0, then l = 0 (i, j) ∈ [1, l] × I such that vpj (vi ) ≥ 4α. Then l( and z ∈ ZX (H). Clearly, wy has form (∗), and we show that for any given z of form (∗) there exists some z of form (∗) with d( z , z ) = 2 and l( z  ) < l( z ). Then the assertion follows. Let z be as in (∗), i ∈ [1, l], j ∈ I such that vpj (vi ) ≥ 4α and λ0 = (I, j) ∈ Λ. By Lemma 4.3 we obtain that (n

)

vpj (vi )−vpj (Θ(vi ))

uλ0 = uλ0λ0 · pj and

vpj (Θ(vi ))−vpj (vi )

vi = vi · pj

∈ A(H)

∈ A(H).

Monotone Chains of Factorizations in C-Monoids (n

111

)

Then vi uλ0λ0 = vi uλ0 , 

z = uλ0

(nλ )

uλ

· y0 · vi · vi−1 · v1 · . . . · vl ∈ Z(a),

λ∈Λ\{λ0 }

z  ) < l( z ). d( z , z ) = 2 and l( 5. If tX (H, v) < ∞ for every v ∈ A, then Lemma 3.2.1 and Lemma 3.3.2 imply that ∆X (H) is finite. Thus it suffices to prove the first assertion. Recall that π : Z(H) → H denotes the factorization homomorphism and, by Construction 4.4, we have Λ Φ : NA 0 × N0

−→ Z(H)   (n ) −→ u mu uλ λ .

(m, n)

u∈A

λ∈Λ

Λ If (m, n) ∈ NA 0 ×N0 , then z = Φ(m, n) ∈ X ⊂ Z(H) has length |z| =

 u∈A

mu +|Λ|.

For v ∈ A we set Λ 2    Rv = {(m, n; m , n ) ∈ (NA 0 × N0 ) | π(Φ(m, n)) = π(Φ(m , n )) and mv > 0}.

By Dickson’s Theorem (cf. [10, Theorem 1.1]), Rv has only finitely many minimal points. We set        mu + |Λ|, mu + |Λ|  (m, n; m , n ) is a minimal point in Rv , t(v) = max u∈A

u∈A

and assert that

tX (H, v) ≤ t(v).

Let a ∈ H, z ∈ ZX (a) and ZX (a) ∩ vZ(H) = ∅. There exist (m, n), (m , n ) ∈ Λ   X  NA 0 × N0 such that z = Φ(m, n) and Φ(m , n ) ∈ Z (a) ∩ vZ(H). Then mv > 0,     ; m  ,n  ) ∈ Rv such m, n (m, n; m , n ) ∈ Rv and there exists some minimal point ( that   ) ≤ (m, n; m , n ). ; m  , n ( m, n Since a=



u mu

u∈A

=



 λ∈Λ

u

mu −g mu

u∈A

=



=

u∈A



u





g u um

u∈A mu −g mu

u∈A



(nλ )

uλ

λ∈Λ

u


m g u

u


m g u

u∈A

u

mu −g mu





λ∈Λ

u∈A

(n f λ)

uλ





(nλ −n f λ )α

pλ

λ∈Λ (n f
) uλ λ



(nλ −n f λ )α

pλ

λ∈Λ (n f
+n −n f) uλ λ λ λ

∈ F,

λ∈Λ

v + m v  > 0, it follows that and mv − m   (nf
+n −nf ) mu +g mu
umu −g uλ λ λ λ ∈ ZX (a) ∩ vZ(H), z  = u∈A

λ∈Λ

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Foroutan and Geroldinger

and clearly d(z, z  ) ≤ max



m u + |Λ|,

u∈A



  m u + |Λ| ≤ t(v).

u∈A

6. For every d ∈ ∆X (H) we set Λ 2   Rd+ = {(m, n; m , n ) ∈ (NA 0 × N0 ) | π(Φ(m, n)) = π(Φ(m , n )) and ( mu + |Λ|) = ( mu + |Λ|) + d} u∈A

u∈A

and Λ 2   Rd− = {(m, n; m , n ) ∈ (NA 0 × N0 ) | π(Φ(m, n)) = π(Φ(m , n )) and ( mu + |Λ|) = ( mu + |Λ|) − d}. u∈A

u∈A

Rd−

Rd+

and have only finitely many minimal points, Again by Dickson’s Theorem, and we set    δ(d) = max mu + |Λ|, mu + |Λ|  (m, n; m , n ) u∈A

u∈A

 is a minimal point in Rd+ or Rd− .

By 5., ∆X (H) is finite, and we assert that δ X (H) ≤ max{δ(d) | d ∈ ∆X (H)} ∈ N. Let a ∈ H, z ∈ ZX (a) and |z|, k ∈ LX (a) successive lengths. We have to verify that X d({z}, ZX k (a)) ≤ max{δ(d) | d ∈ ∆ (H)}.

Without restriction we suppose that |z| < k, and we work with Rd+ . Let (m, n), Λ     (m , n ) ∈ NA 0 × N0 such that z = Φ(m, n), z = Φ(m , n ), |z | = k and d =   X ; m  ,n  ) ∈ Rd+ such that m, n k − |z| ∈ ∆ (H). There exists some minimal point ( 

  ) ≤ (m, n; m , n ). ; m  ,n ( m, n The same calculation as in 5. shows that   (nf
+n −nf ) mu +g mu
z  = umu −g uλ λ λ λ ∈ ZX (a). u∈A

Since 

|z | − |z| =



λ∈Λ

    (mu − m u + m u ) + |Λ| − mu + |Λ| = (m u − m u ) = d, 

u∈A 

we obtain that z ∈

u∈A

u∈A

ZX k (a)

and     d(z, z ) ≤ max m u + |Λ|, m u + |Λ| ≤ δ(d). u∈A

u∈A

Acknowledgement: We would like to thank F. Halter-Koch and W. Hassler for pointing out various inaccuracies in previous versions of this manuscript.

Bibliography [1] D. D. Anderson (ed.), Factorization in integral domains, Marcel Dekker, 1997. [2] D.D. Anderson, D.F. Anderson, and M. Zafrullah, Factorizations in integral domains, J. Pure Appl. Algebra 69 (1990), 1–19. [3] S. T. Chapman and S. Glaz, Non-noetherian Commutative Ring theory, Kluwer Academic Publisher, 2000. [4] A. Foroutan, Monotone chains of factorizations, Int. J. of Commutative Rings 3 (2003). [5] A. Geroldinger and F. Halter-Koch, Transfer principles in the theory of nonunique factorizations, these Proceedings. [6] [7]

, Tamely generated ideals in finitary monoids, JP J. Algebra Number Theory Appl. 2 (2002), 205–239. , Congruence monoids, Acta Arith. 112 (2004), 263–296.

[8] A. Geroldinger, F. Halter-Koch, W. Hassler, and F. Kainrath, Finitary Monoids, Semigroup Forum 67 (2003), 1–21. [9] F. Halter-Koch, C-monoids and congruence monoids in Krull domains, these Proceedings. [10]

, Ideal systems. An introduction to multiplicative ideal theory, Marcel Dekker, 1998.

[11] W. Hassler, Properties of factorizations with successive lengths in onedimensional local domains, Int. J. of Commutative Rings. [12]

, Factorization in finitely generated domains, J. Pure Appl. Algebra 186 (2004), 151–168.

[13] F. Kainrath, Elasticity of finitely generated domains, Houston J. Math. 30 (2004).

113

Chapter 6

Transfer Principles in the Theory of Non-unique Factorizations by

Alfred Geroldinger and Franz Halter-Koch1 Abstract A successful method to investigate phenomena of nonunique factorizations occurring in an arithmetically interesting object H (an integral domain or a monoid) is to construct a simple auxiliary monoid B and a homomorphism θ : H → B which allows to transfer arithmetical properties from B to H. In this chapter we present a very general transfer principle and apply it to obtain arithmetical results for (weakly) Krull monoids, K +M -domains and congruence monoids.

1

Introduction

It is well-known that in general the ring OK of integers of an algebraic number field is not factorial, and classical philosophy in algebraic number theory states that the class group CK of OK measures the deviation from unique factorization. For a long time the only result justifying this opinion was the classical criterion that OK is factorial if and only if CK = 0. Only in 1960, L. Carlitz [8] proved that |CK | ≤ 2 if and only if OK is half-factorial, that is, any two factorizations of an element have the same length. A systematical investigation of phenomena of nonunique factorizations was initiated by W. Narkiewicz in a series of papers in the sixties and seventies (see [30, Ch. 9] for an overview). One of the most important insights of these investigations was the discovery of the connection between factorization problems in OK and combinatorial problems on zero-sum sequences in CK (see [29] and [31]). In the language we use today, this was the first application of a transfer principle: Factorization properties of OK were investigated by means of 1 This

work was supported by the Austrian Science Fund FWF (Project-Nr. P16770-N12)

114

Transfer Principles in the Theory of Non-Unique Factorizations

115

factorization properties of the much simpler monoid B(CK ) of all zero-sum sequences of CK . In the sequel, problems of nonunique factorizations, not only in rings of integers of algebraic number fields, but more generally in arbitrary integral domains, received a great deal of attention. In particular, we mention the papers [4], [5] and [2], the survey articles published in [1] and the survey articles [9] and [10]. These papers also contain an extensive bibliography. The problems and phenomena of nonunique factorizations in an integral domain R are of a purely multiplicative nature and can be formulated in an arbitrary multiplicative monoid H. Apart from its generality, this point of view turned out to be an utmost powerful tool in the theory of nonunique factorizations in integral domains. Concepts and results can be derived in a suitable auxiliary monoid of a concise structure and, by means of suitable transfer principles, the consequences for integral domains can be obtained. As mentioned above, this point of view was implicitly already used by W. Narkiewicz. It was systematically enhanced by the authors. The survey articles [11], [16] and [25] review the state of knowledge at that time. Only recently, the investigations of W. Hassler [27], A. Foroutan [12] and the authors [21] showed the need for more general and powerful transfer principles. In [21], we proved strong arithmetical finiteness results for C0 -monoids and applied them to study the arithmetic of congruence monoids in Dedekind domains. At the same time, W. Hassler noticed that C0 -monoids could also serve to investigate the multiplicative arithmetic of a large class of noetherian domains ([27]). For this purpose he introduced Z-monoids and derived a transfer principle to reduce their arithmetical structure to those of C0 -monoids. The transfer principle presented in this paper is a common generalization of all such principles published hitherto. It applies to C-monoids (and thus it contains Hassler’s transfer principle for Z-monoids), it generalizes the transfer principle for T-block monoids ([15], [16]), and a transfer principle used by F. Kainrath for certain noetherian domains ([28]). This chapter is organized as follows. In Section 2 we fix the notations concerning the algebraic and arithmetical theory of monoids and gather the basic concepts of nonunique factorizations. In Section 3 we recall the concept of a transfer homomorphism and investigate the behavior of arithmetical invariants under transfer homomorphisms. Section 4 contains the main result of the chapter, a very general transfer principle (Theorem 4.1), which will be applied to (generalized) block monoids in Section 5, to K+M -domains in Section 6 and to C-monoids and congruence monoids in Section 7. In Section 7 we shall also present an explicit criterion for a congruence monoid of natural numbers to have finite elasticity (Theorem 7.8).

2

Notations and Preliminaries on Monoids

Let N denote the set of positive integers, N0 = N ∪ {0} and P the set of all prime numbers. For g ∈ N, we denote by Ng the set of all natural numbers coprime to g, and by Pg the set of all prime numbers not dividing g. For m, n ∈ Z with m ≤ n, we set [m, n] = {x ∈ Z | m ≤ x ≤ n}. For a set X, we denote by |X| ∈ N0 ∪ {∞} its

116

Geroldinger and Halter-Koch

cardinality. For convenience and to fix notations, we recall some notions from the theory of monoids and the basic results from the theory of nonunique factorizations which we use in this chapter. Unless stated otherwise, proofs can be found in [26, Ch. 10] and in the survey articles [11], [16] and [25]. Basic notions. By a semigroup we mean a nonempty set with a commutative and associative law of composition possessing a unit element. By a monoid we mean a cancellative semigroup. Unless stated otherwise, we use multiplicative notation and denote the unit element by 1. For subsets A, B of a monoid H, we set AB = {ab | a ∈ A , b ∈ B}. Semigroup and monoid homomorphisms are assumed to respect the unit element. Subsemigroups and submonoids are assumed to contain the unit element. For each monoid H, we fix a quotient group q(H) of H, and if H ⊂ D is a submonoid, we tacitly assume that q(H) ⊂ q(D). We denote by H × the group of invertible elements of H and by Hred = {aH × | a ∈ H} the associated reduced monoid of H. We call H reduced , if H × = {1} (and then Hred = H). For a monoid homomorphism ϕ : H → D we denote by ϕred : Hred → Dred the induced homomorphism of reduced monoids, given by ϕred (aH × ) = ϕ(a)D× . For an integral domain R, we denote by R• = R \ {0} its multiplicative monoid. Let H be a monoid and a, b ∈ H. We call a a divisor of b and we write a | b, if b ∈ aH. We call a and b associated and write a  b, if aH × = bH × . An element u ∈ H is called an atom if u ∈ / H × , and for all a, b ∈ H, u = ab implies a ∈ H × or b ∈ H × . We denote by A(H) the set of all atoms of H, and we call the monoid H atomic , if every a ∈ H \ H × is a product of atoms. An element p ∈ H is called a prime , if H \ pH is a submonoid of H, and H is called factorial , if every a ∈ H \H × is a product of primes. Every prime is an atom, and a monoid is factorial if and only if it is atomic, and every atom is a prime. If H is atomic and p ∈ H is a prime, then every a ∈ q(H) has a representation a = pn bc−1 , where b, c ∈ H, p  bc and n ∈ Z. The exponent n is uniquely determined by the classes aH × and pH × in Hred , and we call n = vp (a) the p-adic value of a. The map vp : q(H) → Z is a surjective group homomorphism, called the p-adic valuation. If P is a set of primes of an atomic monoid H and x ∈ q(H), then suppP (x) = {p ∈ P | vp (x) = 0} is called the support of x (in P ). Free monoids. For a set P , we denote by F (P ) the free (abelian) monoid with basis P , and in particular F (∅) = {1}. For every map f : P → S into a monoid S there exists a unique homomorphism f : F (P ) → S extending f . F (P ) is a reduced factorial monoid and P is the set of primes of F (P ). Every subset X of F (P ) possesses a unique greatest common divisor, denoted by gcd(X) ∈ F (P ). Every a ∈ F(P ) has a unique representation in the form a=



pνp ,

where

νp ∈ N0

and νp = 0 for all but finitely many p ∈ P ,

p∈P

and then νp = vp (a) for all p ∈ P . For a, b ∈ F(P ), we set |a| =

p∈P

vp (a)

and

    a ,  d(a, b) = max  gcd(a, b) 

    b    gcd(a, b)  .

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117

We call |a| the size of a and d(a, b) the distance between a and b. Note that | · | : F (P ) → N0 is a homomorphism, and d : F (P )×F(P ) → N0 is a metric. Products. For two monoids H1 , H2 , we denote by H1×H2 their direct product, and we view H1 and H2 as submonoids of H = H1 ×H2 . Then every a ∈ H has a unique factorization in the form a = a1 a2 with a1 ∈ H1 and a2 ∈ H2 . Note that a monoid H is factorial and P is a set of representatives of pairwise nonassociated primes of H if and only if H = H × ×F(P ). and sets of lengths. Let H be a monoid. The monoid Z(H) =   Factorizations F A(Hred ) is called the factorization monoid of H. The unique homomorphism π : Z(H) → Hred satisfying π | A(Hred ) = id is called the factorization homomorphism of H. It is surjective if and only if H is atomic, and it is an isomorphism if −1 × and only if H is factorial. For a ∈ H, the elements  in Z(a) = π  (aH ) ⊂ Z(H)  are called the factorizations of a , and L(a) = |z|  z ∈ Z(a) ⊂ N0 is called the set of lengths of a. We call L(H) = {L(a) | a ∈ H} the system of sets of lengths of H. It is one of the best studied invariants in the theory of nonunique factorizations. H is called a BF-monoid (a monoid with bounded factorizations), if all sets L ∈ L(H) are finite and not empty. H is called half-factorial, if |L(a)| = 1 for all a ∈ H. Let now H be a BF-monoid. For a ∈ H \ H × , we call ρ(a) =

max L(a) ∈ Q>0 min L(a)

the elasticity of a. We call ρ(H) = sup{ρ(a) | a ∈ H \ H × } ∈ R≥0 ∪ {∞} the elasticity of H, and we say that H has accepted elasticity, if ρ(H) = ρ(a) for some a ∈ H. If H = H1 × H2 , then ρ(H) = max{ρ(H1 ), ρ(H2 )}, and H has accepted elasticity if and only if both H1 and H2 have accepted elasticity (see [24, Proposition 4]). Let H be a BF-monoid and λ : H → N0 a homomorphism satisfying λ−1 (0) = H . Then sup{λ(u) | u ∈ A(H), u not prime} ρ(H) ≤ . min{λ(u) | u ∈ A(H), u not prime} ×

For integral domains, this was proved in [3, Theorem 2.1], where λ is called a semi-length function. The proof given there remains literally valid for monoids. The Structure Theorem for Sets of Lengths. A finite set L ⊂ Z is called an almost arithmetical multiprogression with bound M ∈ N , if there exists some d ∈ [1, M ] and some subset D ⊂ [0, d] with {0, d} ⊂ D such that L = L ∪ L∗ ∪ L , where L ⊂ min L∗ + [−M, −1], L ⊂ max L∗ + [1, M ] and L∗ = [min L∗ , max L∗ ] ∩ (min L∗ + D + dZ) . We say that the Structure Theorem for Sets of Lengths holds for a monoid H, if H is a BF-monoid, and there exists some M ∈ N such that every L ∈ L(H) is an almost arithmetical multiprogression with bound M .

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The Structure Theorem for Sets of Lengths has an abstract ideal-theoretical and combinatorial basis (see [19] and [20]). In [21] we proved that it holds for AC-monoids, and as a consequence it holds for congruence monoids in Dedekind domains (see [21]) and for noetherian domains (see [27]), provided that some natural finiteness conditions are satisfied. In this chapter, we shall prove that it also holds for C-monoids (Theorem 7.2), and consequently it continues to hold for congruence monoids in Krull domains (see [23]). Next we explain two more subtle invariants of nonunique factorizations, introduced by the first author in order to investigate the system of sets of lengths (see [16]). Let H be an atomic monoid. Catenary degree: For a ∈ H, we denote by c(a) the smallest N ∈ N0 ∪ {∞} with the following property: For any two factorizations z, z  ∈ Z(a), there exist factorizations z = z0 , z1 , . . . , zk = z  in Z(a) such that d(zi−1 , zi ) ≤ N for all i ∈ [1, k] (we refer to this fact by saying that z and z  can be concatenated by an N -chain). Then c(a) ≤ sup L(a), and we call c(H) = sup{c(a) | a ∈ H} ∈ N0 ∪ {∞} the catenary degree of H. Note that c(H) = 0 if and only if H is factorial, and c(H) ≥ 2 otherwise. If c(H) = 2, then H is half-factorial. If H = H1 × H2 , then c(H) = max{c(H1 ), c(H2 )} by [17, Lemma 3.3]. Tameness: For a ∈ H and x ∈ Z(H), we denote by t(a, x) the smallest N ∈ N0 ∪ {∞} with the following property: If Z(a) ∩ xZ(H) = ∅ and z ∈ Z(a), then there exists some z  ∈ Z(a) ∩ xZ(H) such that d(z, z  ) ≤ N (note that, if c ∈ H and x ∈ Z(c), then the condition Z(a) ∩ xZ(H) = ∅ is equivalent to c | a). For subsets H  ⊂ H and X ⊂ Z(H), we define t(H  , X) = sup{t(a, x) | a ∈ H  , x ∈ X} ∈ N0 ∪ {∞} . For a ∈ H and x ∈ Z(H), we set t(a, X) = t({a}, X) and t(H  , x) = t(H  , {x}). H is called locally

tame , if t(H, u) < ∞ for all u ∈ A(Hred ), and H is called tame,

if t(H) = t H, A(Hred ) < ∞. We call t(H) the tame degree of H. Note that t(H) = 0 if and only if H is factorial, and t(H) ≥ 2 otherwise. If H = H1 × H2 , then t(H) = max{t(H1 ), t(H2 )}, and H is locally tame if and only if both H1 and H2 are locally tame. Submonoids and class groups. Let D be a monoid and H ⊂ D a submonoid. For x ∈ q(D), we set [x]D/H = xq(H) ∈ q(D)/q(H), we call D/H = {[a]D/H | a ∈ D} the class monoid and q(D)/q(H) = q(D/H) the class group of D modulo H. If H ⊂ D× is a subgroup, then [a]D/H = aH ∈ q(D)/H for all a ∈ D, there are natural epimorphisms D → D/H → D/D× = Dred ⊂ q(D)/D× = q(Dred ), and Hred = H/H × is a submonoid of D/H × . In this special case, we write D/H multiplicatively. In the general situation however, we use additive notation for D/H, so that [ab]D/H = [a]D/H + [b]D/H for all a, b ∈ D, and [1]D/H is the zero element of D/H. A submonoid H ⊂ D is called • cofinal , if aD ∩ H = ∅ for all a ∈ D (equivalently, D/H = q(D/H)).

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• saturated , if q(H) ∩ D = H (equivalently, H = {a ∈ D | [a]D/H = [1]D/H }). Block monoids and the Davenport constant. For an (additive) abelian group G and any subset G0 ⊂ G, let F (G0 ) be the free (multiplicative) monoid with basis G0 , and let σ : F (G0 ) → G be the unique homomorphism satisfying σ(g) = g for all g ∈ G0 . Then B(G0 ) = {S ∈ F(G0 ) | σ(S) = 0} is the monoid of all zero-sum sequences over G0 . It is a saturated  submonoid of F (G0 ), called   the block monoid over G0 , and D(G0 ) = |S|  S ∈ A(B(G0 )) is called the Davenport constant. It is finite whenever G0 is finite [14, Proposition 2], and it plays a prominent role in the theory of nonunique factorization (see [11] or [30, Chapter 9.1]). For every invariant ∗(H) attached to a monoid H, we write ∗(G0 )

arithmetical  instead of ∗ B(G0 ) .

If D is any atomic monoid, H ⊂ D is a saturated submonoid, G0 = {[u]D/H | u ∈ A(D)}, and if a1 , . . . , an ∈ D \ D× are such that a1 · . . . · an ∈ A(H), then n ≤ D(G0 ) (see [22, Section 3]). Class semigroups, the ω-invariant and the d-invariant. These notions were introduced and investigated in [21, Section 4], see also [23, Section 3]. We recall the definition and the basic properties. Let H ⊂ D be a submonoid. Two elements y, y  ∈ D are called H-equivalent , if y −1 H ∩ D = y −1 H ∩ D ⊂ q(D). H-equivalence is a congruence relation on D, we denote by [y]D H the congruence class of an element y ∈ D, and for a subset T ⊂ D we set    CT (H, D) = [y]D H  y ∈ T}. In particular, C(H, D) = CD (H, D) and C ∗ (H, D) = C(D\D× )∪{1} (H, D) are semigroups under the induced law of composition which we write additively, [ab]D H = D + [b] for all a, b ∈ D. [a]D H H Let H be a monoid and a, b ∈ H. Then ωH (a, b) denotes the smallest N ∈ N0 ∪ {∞} with the following property: For any n ∈ N and a1 , . . . , an ∈ H, if a = a1 · . . . · an and b | a, then there exists a subset Ω ⊂ [1, n] such that |Ω| ≤ N and    b  aν . ν∈Ω

By definition, ωH (a, b) = 0 if either b  a or b ∈ H × . If m ∈ N0 , p1 , . . . , pm are primes of H and b = p1 · . . . · pm , then ωH (a, b) ≤ m. For an additive abelian semigroup C, we denote by d(C) the smallest d ∈ N0 ∪ {∞} with the following property: For any m ∈ N and c1 , . . . , cm ∈ C there exists a subset J ⊂ [1, m] such that |J| ≤ d and m j=1

cj =



cj .

j∈J

If C is a finite additive abelian semigroup, then d(C) < ∞. If C is an abelian group, then D(C) = d(C) + 1 is the Davenport constant.

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× ×

If H ⊂ D  is a submonoid satisfying H ∩ D = H , then ωH (a, b) ≤ ωD (a, b) + ∗ d C (H, D) for all a, b ∈ H. If H ⊂ D is a cofinal submonoid, then there is a natural epimorphism C(H, D) → D/H, which is an isomorphism if and only if H ⊂ D is saturated, see [23, Proposition 3.7].

Restriction to reduced monoids: All factorization properties explained hitherto hold for a monoid H if and only if they hold for the associated reduced monoid Hred . Hence we may without restriction assume that H is reduced, and we will do this whenever it will be convenient.

3

Transfer Homomorphisms

The notion of a transfer homomorphism was first introduced in [25]. We recall the definition and investigate the behavior of the invariants of nonunique factorization under transfer homomorphisms. Definition 3.1. A monoid homomorphism θ : H → B is called a transfer homomorphism , if it has the following properties: (T 1)

B = θ(H)B × and θ−1 (B × ) = H × .

(T 2)

If u ∈ H, b, c ∈ B and θ(u) = bc, then there exist v, w ∈ H such that u = vw, θ(v)  b and θ(w)  c.

Note that θ : H → B is a transfer homomorphism if and only if θred : Hred → Bred is a transfer homomorphism. Proposition 3.2. Let θ : H → B be a transfer homomorphism and u ∈ H. 1. If n ∈ N, b1 , . . . , bn ∈ B and θ(u)  b1 ·. . .·bn , then there exist u1 , . . . , un ∈ H such that u  u1 · . . . · un and θ(uν )  bν for all ν ∈ [1, n]. 2. u is an atom of H if and only if θ(u) is an atom of B. 3. There is a unique homomorphism θ : Z(H) → Z(B) satisfying θ(uH × ) = θ(u)B × for all u ∈ A(H). It is surjective, induces the commutative diagram θ

Z(H) −−−−→ Z(B) ⏐ ⏐ ⏐ ⏐πB πH   θ

red Hred −−− −→ Bred ,

and has the following properties:

 (a) If z, z  ∈ Z(H), then |θ(z)| = |z| and d θ(z), θ(z  ) ≤ d(z, z  ). (b) θ(ZH (u)) = ZB (θ(u)) and LH (u) = LB (θ(u)).

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(c) If z ∈ Z(u) and y ∈ Z(θ(u)), then there exists some y ∈ Z(u) such that θ(y) = y, θ(gcd(z, y)) = gcd(θ(z), y) and d(z, y) = d(θ(z), y). 4. H is atomic if and only if B is atomic. 5. If H is atomic, then L(H) = L(B), ρ(H) = ρ(B), H has accepted elasticity if and only if B has accepted elasticity, and H is a BF-monoid if and only if B is a BF-monoid. Proof. We may assume that H and B are both reduced. 1. By induction on n. 2. If u ∈ A(H) and θ(u) = bc for some b, c ∈ B, then there exist v, w ∈ H such that u = vw, θ(v) = b and θ(w) = c. Hence v = 1 or w = 1 and thus b = 1 or c = 1. If θ(u) ∈ A(B) and u = vw for some v, w ∈ H, then θ(u) = θ(v)θ(w) implies θ(v) = 1 or θ(w) = 1 and thus v = 1 or w = 1. 3. Observe that Z(H) = F (A(H)), Z(B) = F (A(B)), and that the map θ | A(H) : A(H) → A(B) is surjective. Hence there is a unique homomorphism θ : Z(H) → Z(B) such that θ | A(H) = θ | A(H). It is surjective, and πB ◦ θ | A(H) = θ ◦ πH | A(H) implies πB ◦ θ = θ ◦ πH . (a) is obvious by definition, and it remains to prove the properties (b) and (c). (b) If z ∈ Z(u), then πB (θ(z)) = θ(πH (z)) = θ(u), and therefore θ(z) ∈ Z(θ(u)). Conversely, if z = u1 · . . . · un ∈ Z(θ(u)), where u1 , . . . , un ∈ A(B), then θ(u) = u1 · . . . · un (observe that the first product is performed in Z(B) and the second one in B). By 1. there exist u1 , . . . , un ∈ H such that u = u1 · . . . · un and θ(uν ) = uν for all ν ∈ [1, n]. By 2. we have u1 , . . . , un ∈ A(H), hence z = u1 · . . . · un ∈ Z(u) and z = θ(z). Now the assertion concerning the sets of lengths follows from (a). (c) If z = u1 ·. . .·un ∈ Z(u), then θ(z) = θ(u1 )·. . .·θ(un ) and, after renumbering if necessary, we may assume that y = θ(u1 ) · . . . · θ(ur )v 1 · . . . · v s , where r ∈ [0, n], s ∈ N0 , v 1 , . . . , v m ∈ A(B) and θ(u1 ) · . . . · θ(ur ) = gcd(θ(z), y). In B, we have θ(u) = θ(u1 )·. . .·θ(un ) = θ(u1 )·. . .·θ(ur )v 1 ·. . .·v s , hence θ(ur+1 ·. . .·un ) = v 1 ·. . .·vs , and therefore there exist v1 , . . . , vs ∈ H such that ur+1 · . . . · un = v1 · . . . · vs and θ(vi ) = v i for all i ∈ [1, s]. Hence v1 , . . . , vs ∈ A(H), and y = u1 · . . . · ur v1 · . . . · vs has the required properties. 4. Note that H is atomic if and only if Z(a) = ∅ for all a ∈ H. Since θ is surjective, it follows that B is atomic if and only if Z(θ(a)) = ∅ for all a ∈ H. Hence the assertion follows by 3.(b). 5. Obvious by 3.(b). Definition 3.3. Let θ : H → B be a transfer homomorphism of atomic monoids and θ : Z(H) → Z(B) the unique homomorphism satisfying θ(uH × ) = θ(u)B × for all u ∈ A(H) (see Proposition 3.2). We call θ the extension of θ to the factorization monoids. 1. For a ∈ H, we denote by c(a, θ) the smallest N ∈ N0 ∪{∞} with the following property: If z, z  ∈ Z(a) and θ(z) = θ(z  ), then there exist factorizations

122

Geroldinger and Halter-Koch z0 , . . . , zk ∈ Z(a) such that z0 = z, zk = z  , θ(zi ) = θ(z) and d(zi−1 , zi ) ≤ N for all i ∈ [1, k] (that is, z and z  can be concatenated by an N -chain in the −1 fibre Z(a) ∩ θ (θ(z))). We set c(H, θ) = sup{c(a, θ) | a ∈ H} ∈ N0 ∪ {∞} .

2. For a ∈ H and x ∈ Z(H), we denote by t(a, x, θ) the smallest N ∈ N0 ∪ {∞} with the following property: If Z(a) ∩ xZ(H) = ∅, z ∈ Z(a) and θ(z) ∈ θ(x)Z(B), then there exists some factorization z  ∈ Z(a) ∩ xZ(H) such that θ(z  ) = θ(z) and d(z, z  ) ≤ N . We set t(H, x, θ) = sup{t(a, x, θ) | a ∈ H} ∈ N0 ∪ {∞} . Theorem 3.4. Let θ : H → B be a transfer homomorphism of atomic monoids and θ : Z(H) → Z(B) its extension to the factorization monoids. 

1. If a ∈ H and x ∈ Z(H), then t(a, x) = 0 or t θ(a), θ(x) ≤ t(a, x) ≤ 

t θ(a), θ(x) + t(a, x, θ).



 In particular, if u ∈ A(Hred ), then t B, θ(u) ≤ t(H, u) ≤ t B, θ(u) + t(H, u, θ). 2. t(B) ≤ t(H), and if H is locally tame, then so is B. 3. If B is locally tame and t(H, u, θ) < ∞ for all u ∈ A(Hred ), then H is locally tame.



 4. If a ∈ H, then c θ(a) ≤ c(a) ≤ max{c θ(a) , c(a, θ)}. 5. c(B) ≤ c(H) ≤ max{c(B), c(H, θ)} Proof. 1. We may suppose that H and B are both reduced. Let a ∈ H and x ∈ Z(H). If Z(a) ∩ xZ(H) = ∅, then t(a, x) = 0. Suppose that Z(a) ∩ xZ(H) = ∅ whence Z(θ(a)) ∩ θ(x)Z(B) ⊃ θ(Z(a) ∩ xZ(H)) = ∅. a) t(θ(a), θ(x)) ≤ t(a, x) : Let z ∈ Z(θ(a)), say z = θ(z) for some z ∈ Z(a). There exists some z  ∈ Z(a) ∩ xZ(H) such that d(z, z  ) ≤ t(a, x). Then θ(z  ) ∈ Z(θ(a)) ∩ θ(x)Z(B) and d(θ(z), θ(z  )) ≤ d(z, z  ) ≤ t(a, x). b) t(a, x) ≤ t(θ(a), θ(x)) + t(a, x, θ) : Let z ∈ Z(a). There exists some y ∈ Z(θ(a)) ∩ θ(x)Z(B) such that d(θ(z), y) ≤ t(θ(a), θ(x)). By Proposition 3.2.3 (c), there is some y ∈ Z(a) such that θ(y) = y and d(z, y) = d(θ(z), y), and thus there is some z  ∈ Z(a) ∩ xZ(H) such that θ(z  ) = θ(y) and d(y, z  ) ≤ t(a, x, θ). Therefore we obtain d(z, z  ) ≤ d(z, y) + d(y, z  ) ≤ t(θ(a), θ(x)) + t(a, x, θ) . 2. and 3. are now obvious, 4. follows from the subsequent more precise Lemma 3.5, and 5. is an immediate consequence of 4.

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Lemma 3.5. Let θ : H → B be a transfer homomorphism of atomic monoids, θ : Z(H) → Z(B) its extension to the factorization monoids and a ∈ H.

 z, z  ∈ Z θ(a)

z0 , z1 , . . . , zk ∈ Z(a) be such that θ(z0 ) = z



 and θ(zk ) = z  . Then θ(z0 ), . . . , θ(zk ) ∈ Z θ(a) and d θ(zi−1 ), θ(zi ) ≤ d(zi−1 , zi ) for all i ∈ [1, k]. In particular, c(θ(a)) ≤ c(a).

1. Let

and

2. Let z, z  ∈ Z(a) and z 0 , z 1 , . . . , z k ∈ Z(θ(a)) be such that θ(z) = z 0 and θ(z  ) = z k . Then there exists some m ≥ k and there exist factorizations z = z0 , z1 , . . . , zk , zk+1 , . . . , zm = z  ∈ Z(a) such that θ(zi ) = z i ,

d(zi−1 , zi ) = d(z i−1 , z i )

for all

i ∈ [1, k]

and d(zj−1 , zj ) ≤ c(a, θ) for all   In particular, c(a) ≤ max c(θ(a)), c(a, θ) . θ(zj ) = z k ,

j ∈ [k + 1, m] .

Proof. 1. By Proposition 3.2.3 (a). 2. By Theorem 3.2.3(c) and induction on k, we obtain z = z0 , z1 , . . . , zk ∈ Z(a) such that θ(zi ) = z i and d(zi−1 , zi ) = d(z i−1 , z i ) for all i ∈ [1, k]. The existence of zk+1 , . . . , zm = z  ∈ Z(a) such that θ(zj ) = z k and d(zj−1 , zj ) ≤ c(a, θ) for all j ∈ [k + 1, m] follows by the very  definition of c(a, θ). In particular, if N = max c(θ(a)), c(a, θ)

and z 0 , z 1 , . . . z k is an N -chain



concatenating θ(z) and θ(z ) in Z(θ(a)), then z0 , z1 , . . . , zm is an N -chain concatenating z and z  in Z(a). Hence c(a) ≤ N . Proposition 3.6. Let θ : H → B and β : B → D be transfer homomorphisms of atomic monoids, and let θ : Z(H) → Z(B) and β : Z(B) → Z(D) be their extensions to the factorization monoids. Then β ◦ θ : H → D is a transfer homomorphism, and β ◦ θ : Z(H) → Z(D) is its extension to the factorization monoids. Moreover, if a ∈ H and x ∈ Z(H), then c(a, β◦θ) ≤ max{c(a, θ), c(θ(a), β)}

and

t(a, x, β◦θ) ≤ t(a, x, θ)+t(θ(a), θ(x), β) .

Proof. It is easily checked that β ◦ θ is a transfer homomorphism and that β ◦ θ is its extension to the factorization monoids. Let a ∈ H and z, z  ∈ Z(a) be such that β ◦ θ(z) = β ◦ θ(z  ). We must prove that z and z  can be concatenated by a max{c(a, θ), c(θ(a), β)}-chain in the fibre Z(a)∩(β◦θ)−1 (β◦θ(z)). Since θ(z), θ(z  ) ∈ Z(θ(a)), there exist factorizations θ(z) = −1 z 0 , z 1 , . . . , z k = θ(z  ) ∈ Z(θ(a)) ∩ β (β(θ(z)) such that d(z i−1 , z i ) ≤ c(θ(a), β) for all i ∈ [1, k]. By Lemma 3.5.2, there exists some m ≥ k and there exist factorizations z = z0 , z1 , . . . , zk , zk+1 , . . . , zm = z  ∈ Z(a) such that θ(zi ) = z i for all i ∈ [1, k], θ(zj ) = z k for all j ∈ [k + 1, m], d(zi−1 , zi ) = d(z i−1 , z i ) for all i ∈ [1, k] and d(zj−1 , zj ) ≤ c(a, θ) for all j ∈ [k + 1, m].

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Suppose now that a ∈ H, x ∈ Z(H), Z(a) ∩ xZ(H) = ∅, z ∈ Z(a) and β ◦ θ(z) ∈ β ◦ θ(x)Z(D). We must prove that there exists some z  ∈ Z(a) ∩ xZ(H) such that β ◦ θ(z  ) = β ◦ θ(z) and d(z, z  ) ≤ t(a, x, θ) + t(θ(a), θ(x),  β). Since θ(x) ∈ Z(B), θ(z) ∈ Z(θ(a)) and Z(θ(a)) ∩ θ(x)Z(B) ⊃ θ Z(a) ∩ xZ(H) = ∅, there exists some y ∈ ZB (θ(a)) ∩ θ(x)Z(B) such that β(y) = β ◦ θ(z) and d(θ(z), y) ≤ t(θ(a), θ(x), β). By Proposition 3.2.3(c), there exists some y ∈ Z(a) such that θ(y) = y and d(z, y) = d(θ(z), y). Since θ(y) ∈ θ(x)Z(B), there exists some z  ∈ Z(a) ∩ xZ(H) such that θ(z  ) = θ(y) and d(y, z  ) ≤ t(a, x, θ). Hence β ◦ θ(z  ) = β(y) = β ◦ θ(z) and d(z, z  ) ≤ d(z, y) + d(y, z  ) ≤ t(θ(a), θ(x), β) + t(a, x, θ). Our next result presents very general estimates for the invariants c(a, θ) and t(a, x, θ). The estimate for t(a, x, θ) is essentially due to W. Hassler [27, Proposition 4.7]. Proposition 3.7. Let θ : H → B be a transfer homomorphism of reduced atomic monoids. For a ∈ H and u ∈ A(H), we denote by ω ∗ (a, u, θ) the smallest N ∈ N0 ∪ {∞} with the following property: If u | a and a = u1 ·. . .·un , where n ∈ N, u1 , . . . , un ∈ A(H) and θ(u) = θ(u1 ), then there exists a subset Ω ⊂ [1, n] such that 1 ∈ Ω, |Ω| ≤ N , and    uν . u  ν∈Ω

1. If u ∈ A(H) and a ∈ uH, then t(a, u, θ) ≤ ω ∗ (a, u, θ) ≤ ω(a, u) + 1.     2. If W = sup ω ∗ (c, u, θ)  c ∈ H, u ∈ A(H) , then c(a, θ) ≤ W a ∈ H.

for all

Proof. 1. Suppose that u ∈ A(H) and a ∈ uH. a) ω ∗ (a, u, θ) ≤ ω(a, u) + 1 : If a = u1 · . . . · un , where n ∈ N, u1 , . . . , un ∈ A(H) and θ(u) = θ(u1 ), then there exists a subset Ω0 ⊂ [1, n] such that |Ω0 | ≤ ω(a, u) and    u  uν . ν∈Ω0

Then 1 ∈ Ω = Ω0 ∪{1}, and |Ω| ≤ ω(a, u)+1, which implies ω ∗ (a, u, θ) ≤ ω(a, u)+1. b) t(a, u, θ) ≤ ω ∗ (a, u, θ) : Suppose that z = u1 · . . . · un ∈ ZH (a), where u1 , . . . , un ∈ A(H) and θ(z) ∈ θ(u)Z(B). We may assume that θ(u) = θ(u1 ), and we must prove that there exists a factorization z  ∈ Z(a) ∩ uZ(H) such that θ(z  ) = θ(z) and d(z, z  ) ≤ ω ∗ (a, u, θ). We may renumber u2 , . . . , un so that u | u1 · . . . · um for some m ∈ [1, n] such that m ≤ ω ∗ (a, u, θ). If u1 · . . . · um = ub, where b ∈ H, then θ(b) = θ(u2 ) · . . . · θ(um ), and since θ is a transfer homomorphism, there exist u2 , . . . , um ∈ H such that b = u2 · . . . · um and θ(uj ) = θ(uj ) for all j ∈ [2, m]. Now we obtain z  = u·u2 ·. . .·um um+1 ·. . .·un ∈ Z(a)∩uZ(H), θ(z  ) = θ(z) and d(z, z  ) ≤ m ≤ ω ∗ (a, u, θ).

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2. Suppose that a ∈ H, z, z  ∈ Z(a) and θ(z) = θ(z  ). We may assume that z = u1 · . . . · un and z  = u1 · . . . · un , where n ∈ N, uν , uν ∈ A(H) and θ(uν ) = θ(uν ) for all ν ∈ [1, n]. We must prove that z and z  can be concatenated −1 by a W -chain in Z(a) ∩ θ (θ(z)), and we do this by induction on n. If n ≤ W , there is nothing to do. Thus suppose that n > W . After renumbering, we may assume that there exists some m ∈ [1, n] such that m ≤ W and u1 | u1 · . . . · um , say u1 · . . . · um = u1 b for some b ∈ H. Then θ(b) = θ(u2 ) · . . . · θ(um ), and thus there exist v2 , . . . , vm ∈ A(H) such that b = v2 · . . . · vm and θ(vj ) = θ(uj ) for −1 all j ∈ [2, m]. We obtain y = u1 v2 · . . . · vm um+1 · . . . · un ∈ Z(a) ∩ θ (θ(z)), −1 −1 −1 −1 −1 d(y, z  ) ≤ m ≤ W , u−1 1 y ∈ Z(u1 a), u1 z ∈ Z(u1 a) and θ(u1 y) = θ(u1 z). By induction hypothesis there exists a W -chain of factorizations y0 , y1 , . . . , yk in −1 −1 −1  Z(u−1 (θ(u−1 1 a) ∩ θ 1 z)) concatenating u1 z and u1 y. Then u1 y0 , . . . , u1 yk , z is −1  a W -chain concatenating z and z in Z(a) ∩ θ (θ(z)).

4

A Transfer Theorem

Theorem 4.1. Let D be a monoid, P a set of pairwise nonassociated primes of D and T ⊂ D a submonoid such that D = F (P ) × T . Let H ⊂ D be a submonoid such that H∩D× = H × , and U ⊂ D× a subgroup such that H × ⊂ U and U (H\H × ) ⊂ H. Let ∼ be a congruence relation on D such that, for all u, v ∈ D, u∼v

and

u∈H

implies

v∈H.

Let P ∗ = {p ∈ P | p−1 H ∩ D = H \ H × }, P0 ⊂ P ∗ a subset and p = [p]∼ the congruence class of p for p ∈ P \ P0 . Let P = { p | p ∈ P \ P0 },

 = F (P) × T /U D

: D → D  β

and

  the unique homomorphism satisfying β(p) = p for all p ∈ P \ P0 , β(p) = 1 for  = tU ∈ T /U for all t ∈ T . Finally, we set all p ∈ P0 , and β(t)   = β(H) H

and

 |H : H → H . β=β

Then we have:  −1 (H)  = HU F (P0 ), β  −1 (H  × ) = H × U F (P0 ), β −1 (H  × ) = H × and H ∩ 1. β × ×   D =H . 

2. If D = F (P \ P0 )T \ D× ∪ {1} ⊂ D, then there is an isomorphism  D)  , β ∗ : CD
(HU, D) → C ∗ (H,

given by

e D  β∗ ([y]D e HU ) = [β(y)]H

 D)  is also for all y ∈ D . In particular, if C ∗ (H, D) is finite, then C ∗ (H, finite.  is a transfer homomorphism. If H is atomic, then c(H, β) ≤ 2 3. β : H → H and t(a, uH × , β) ≤ ωH (a, u) + 1 for all a ∈ H and u ∈ A(H).

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The proof of Theorem 4.1 depends on the following technical lemma. Lemma 4.2. Let all assumptions and notations be as in Theorem 4.1. D 1. If u, v ∈ D and u ∼ v, then [u]D H = [v]H .

2. For p ∈ P , the following assertions are equivalent: (a) p ∈ P ∗ . D D × (b) pD× ∩ H = ∅ and [p]D H + [a]H = [a]H for all a ∈ D \ D .

3. If p ∈ P ∗ , q ∈ P and q ∼ p, then q ∈ P ∗ . 4. If c ∈ F(P ∗ ), then c−1 H ∩ D ⊂ H, c(H \ H × ) ⊂ H, and if c = 1, then cD× ∩ H = ∅.    ), then b ∼ b , and β(b)  5. If b, b ∈ F(P \ P0 ) and β(b) = β(b = 1 if and only if b = 1. Proof of Lemma 4.2. 1. By definition, it suffices to prove that u ∼ v implies u−1 H ∩ D ⊂ v −1 H ∩ D. If u ∼ v and z ∈ u−1 H ∩ D, then zu ∈ H and zu ∼ zv, hence zv ∈ H and z ∈ v −1 H ∩ D. 2. (a) ⇒ (b) Suppose that p ∈ P ∗ . If e ∈ pD× ∩ H, then e = pu for some u ∈ D× ∩ p−1 H = (H \ H × ) ∩ D× = ∅, a contradiction. Hence pD× ∩ H = ∅. If D D a ∈ D \ D× , then [p]D H + [a]H = [pa]H , and therefore it is sufficient to prove that a−1 H∩D = (pa)−1 H∩D. If z ∈ a−1 H∩D, then az ∈ H\D× = H\H × = p−1 H∩D, hence paz ∈ H and z ∈ (pa)−1 H ∩D. Conversely, if z ∈ (pa)−1 H ∩D, then paz ∈ H, hence az ∈ p−1 H ∩ D ⊂ H and z ∈ a−1 H ∩ D. D (b) ⇒ (a) Suppose that p ∈ P , pD× ∩ H = ∅ and [pa]D H = [a]H for all × −1 × −1 a ∈ D \ D . We must prove that p H ∩ D = H \ H . If a ∈ p H ∩ D, then D pa ∈ H and pD× ∩ H = ∅ implies a ∈ / D× . Hence [a]D H = [pa]H ⊂ H and therefore × × × D a ∈ H \ H . Conversely, if a ∈ H \ H = H \ D , then [pa]H = [a]D H ⊂ H implies pa ∈ H and a ∈ p−1 H ∩ D. D 3. Suppose that p ∈ P ∗ , q ∈ P and q ∼ p. Then [q]D H = [p]H by 1., and by 2. it × suffices to prove that qD ∩ H = ∅. Assume to the contrary that qu ∈ H for some u ∈ D× . Then pu ∼ qu implies pu ∈ H, a contradiction.

4. Suppose that c = p1 · . . . · pn , where n ∈ N0 and p1 , . . . , pn ∈ P0 , and proceed by induction on n. If n = 0, there is nothing to do. Thus suppose that n ≥ 1, b = p1 · . . . · pn−1 , p = pn , c = bp, b−1 H ∩ D ⊂ H and b(H \ H × ) ⊂ H. If a ∈ c−1 H ∩ D, then ca = pba ∈ H implies ba ∈ p−1 H ∩ D ⊂ H, and thus a ∈ b−1 H ∩ D ⊂ H. / H × . Hence If a ∈ H \ H × , then ba ∈ H, and H × = H ∩ D× implies ba ∈ × −1 ba ∈ H \ H = p H ∩ D, and therefore ca = pba ∈ H. It remains to prove that cD× ∩ H = ∅. Assume the contrary. Then there exists some u ∈ D× such that a = cu = pbu ∈ cD× ∩ H. Then u ∈ c−1 H ∩ D× ⊂ H ∩ D× = H × , and pb = u−1 a ∈ H. Hence p ∈ b−1 H ∩ D ⊂ H, a contradiction.

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 5. If b, b ∈ F(P \ P0 ) and β(b) = β(b ), then we may assume that b = p1 · . . . · pn and b = p1 ·. . .·pn , where n ∈ N0 , pi , pi ∈ P \ P0 and pi ∼ pi for all i ∈ [1, n]. Since  ∼ is a congruence relation, we obtain b ∼ b . In particular, β(b) = p1 · . . . · pn = 1 implies n = 0 and b = 1.  −1 (H)  −1 (H  ⊃ HU F (P0 ), β  ×) ⊃ Proof of Theorem 4.1. 1. We obviously have β × −1  × × × × ×     H U F (P0 ), β (H ) ⊃ H and H ∩ D = H ∩ T /U ⊃ H . Thus it remains to verify the reverse inclusions.    ). We set a = cbt and  −1 (H)  and a ∈ H be such that β(a) = β(a Let a ∈ β          ), = β(b a = c b t , where c, c ∈ F(P0 ), b, b ∈ F(P \ P0 ) and t, t ∈ T . Then β(b)  −1   −1 hence b ∼ b by Lemma 4.2.5, and t t ∈ U . Since b t ∈ c H ∩ D ⊂ H (by Lemma 4.2.4) and bt ∼ b t , we obtain bt ∈ H and a = cbt (t−1 t) ∈ HU F (P0 ).    −1 (H  × ), then β(a) × ⊂ D  × = T × /U , and therefore = β(b)(tU )∈ H If a ∈ β ×   β(b) = 1 and t ∈ T . For the same reason, we have β(b ) = 1 and t ∈ T × , hence b = b = 1 and t ∈ T × ∩ H = H × . Whence a = ct (t−1 t) ∈ F(P0 )H × U .  −1 (H  ×) = β  × ) ∩ H = F (P0 )H × U ∩ H, then a = cu, where If a ∈ β −1 (H × c ∈ F(P0 ) and u ∈ H U ⊂ D× . Hence c = 1 by Lemma 4.2.4, and a = u ∈ D× ∩ H = H × .  ∩D × = H  ∩ T × /U , then x = t U for some t ∈ T × . Let t ∈ T , If x ∈ H  = x. This b ∈ F(P \ P0 ) and c ∈ F(P0 ) such that bct ∈ H and β(bct) = β(bt)  −1 ∈ U ⊂ T × = D× , implies b = 1 and Lemma 4.2.4 shows that t = bt ∈ H. Then tt × × × ×  t ∈ D ∩ H = H and x = β(t) ∈ β(H ) ⊂ H . 

 F (P \ P0 )T \ D× = D  \D  × . Therefore it suffices to prove 2. Observe that β that, for all y, y  ∈ F(P \ P0 )T , we have  D [y]D HU = [y ]HU

e D    De . if and only if [β(y)] e = [β(y )]H e H

/ D× . Suppose that y = bt and y  = b t , where b, b ∈ F(P \P0 ), t, t ∈ T and y, y  ∈ We must prove that y −1 HU ∩ D = y −1 HU ∩ D

implies

  )−1 H  −1 H  ∩D  ⊂ β(y  ∩D  β(y)

and  −1 H   )−1 H  ∩D  = β(y  ∩D  β(y)

implies

y −1 HU ∩ D ⊂ y −1 HU ∩ D .

  −1 H  ∩ D,  Suppose first that y −1 HU ∩ D = y −1 HU ∩ D and z = β(z) ∈ β(y) −1  (H)  = HU F (P0 ) by where z = b1 t1 with b1 ∈ F(P \ P0 ) and t1 ∈ T . Then yz ∈ β 1., and therefore yz = bb1 tt1 = at0 c0 for some a ∈ H, t0 ∈ U and c0 ∈ F(P0 ). Since bb1 tt1 ∈ F(P \ P0 )T , we obtain c0 = 1 and yz ∈ HU . Hence z ∈ y −1 HU ∩ D =   )β(z)     )−1 H  ∩ D.   and β(z) ∈H ∈ β(y y −1 HU ∩D and y  z ∈ HU , which implies β(y  ∩D  = β(y   )−1 H  ∩D  and z ∈ y −1 HU ∩ D. Then  −1 H Assume now that β(y)        By 1., this implies  ∈ H. yz ∈ HU , β(y)β(z) ∈ H and therefore β(y z) = β(y  )β(z)   y z = at0 c0 , where a ∈ H, t0 ∈ U and c0 ∈ F(P0 ). Since y = b t ∈ F(P \ P0 )T , −1 ×  we obtain c0 | z (in D) and y  (c−1 / D× . By Lemma 0 z)t0 = a ∈ H \ D , since y ∈  −1 −1 4.2.4 we have c0 a = y zt0 ∈ H, and therefore z = y (c0 a)t0 ∈ y −1 HU ∩ D.

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Since CD
(HU, D) ⊂ C ∗ (HU, D), it is sufficient to prove that C ∗ (HU, D) is finite. But if C ∗ (H, D) is finite, then C ∗ (HU, D) is finite by [23, Lemma 3.3].  is surjective, and β−1 (H  × ) = H × by 1. In order 3. By definition, β : H → H to prove that β is a transfer homomorphism, we will show the following assertion:  are such that β(u) = yz, then there exist T: If u ∈ H \ H × and y, z ∈ H v, w ∈ H such that u = vw, β(v) = y and β(w) = z. Suppose that u = cp1 ·. . .·pn t ∈ H \H × , where c ∈ F(P0 ), n ∈ N0 , p1 , . . . , pn ∈ P \P0 and t ∈ T . Since cD× ∩H ∈ {H × , ∅} (by Lemma 4.2.4), we obtain p1 ·. . .·pn t ∈ / D× . We may assume that y = p1 · . . . · pm (t1 U ) and z = pm+1 · . . . · pn (t2 U ), where / D× . We set m ∈ [0, n], t1 , t2 ∈ T , t = t1 t2 t0 for some t0 ∈ U , and p1 · . . . · pm t1 ∈   −1     v = p1 · . . . · pm t1 , w = pm+1 · . . . · pn t2 ∈ β (H), and since v , w ∈ F(P \ P0 )T , 1. implies that v  = v0 t and w = wt for some v0 , w ∈ H and t , t ∈ U . By −1 / D× , hence v0 ∈ H \ H × , v0 t t t0 ∈ assumption, we have v0 = p1 · . . . · pm t1 t ∈ × ×   (H \ H )U ⊂ H \ H and v = c(v0 t t t0 ) ∈ H by Lemma 4.2.4. Now it is easily checked that u = vw, β(v) = y and β(w) = z. The estimate of t(a, uH × , β) follows by Proposition 3.7, and it remains to prove that c(H, β) ≤ 2. For this, we may assume that H is reduced.  be the extension of β to the factorization monoids, a ∈ H Let β : Z(H) → Z(H)  and z, z ∈ Z(a) such that β(z) = β(z  ). We may assume that z = u1 · . . . · un and z  = u1 · . . . · un , where n ∈ N, uν , uν ∈ A(H) and β(uν ) = β(uν ) for all −1 ν ∈ [1, n]. We must prove that there is a 2-chain of factorizations in Z(a)∩β (β(z)) concatenating z and z  , and we may assume that n ≥ 3. We consider first the case where p  a for all p ∈ P0 . We set u1 = b1 t1 and  1 ) = β(b   ) and t = t1 t for u1 = b1 t1 , where b1 , b1 ∈ F(P \ P0 ), t1 , t1 ∈ T , β(b 1 1  some t ∈ U . We proceed by induction on n + dF (P \P0 ) (b1 , b1 ) + δ, where  0, if t1 = t1 δ= 1, if t1 = t1 . If δ = 1, then u1 t = b1 t1 ∈ H, t−1 u2 ∈ H, and since β(u1 ) = β(u1 t) and β(u2 ) = β(t−1 u2 ), it follows that u1 t ∈ A(H) and t−1 u2 ∈ A(H). If z1 = (u1 t)(t−1 u2 )u3 · . . . · un , then z1 ∈ Z(a), β(z1 ) = β(z), d(z, z1 ) ≤ 2, and by the −1 induction hypothesis, there exists a 2-chain of factorizations in Z(a) ∩ β (β(z)) concatenating z1 and z  . Thus we may now assume that δ = 0, hence t1 = t1 . If dF (P \P0 ) (b1 , b1 ) = 0, then u1 = u1 , and by the induction hypothesis there exists a 2-chain of factoriza−1 −1 −1  tions y0 , y1 , . . . yk in Z(u−1 (β(u−1 1 a) ∩ β 1 z)) concatenating u1 z and u1 z . Then −1 u1 y0 , u1 y1 , . . . , u1 yk is a 2-chain of factorizations in Z(a) ∩ β (β(z)) concatenating z and z  . Now we consider the case where dF (P \P0 ) (b1 , b1 ) ≥ 1, and we set b = gcd(b1 , b1 ) ∈  1 ) = β(b   ), there exists some p ∈ P \ P0 such that p | b−1 b . F (P \ P0 ). Since β(b 1 1 −1 Then p  b b1 , and since b−1 b1 u2 · . . . · un = b−1 b1 u2 · . . . · un , there exists some  −1 b ) = β(b  −1 b1 ), ν ∈ [2, n] such that p | uν . We assume that p | u2 . Since β(b 1    −1 there exists some p ∈ P \ P0 such that p ∼ p and p | b b1 | u1 . We set v1 = pp−1 b1 t1 and v2 = p p−1 u2 . Then v1 , v2 ∈ D, v1 ∼ u1 , v2 ∼ u2 and therefore

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v1 , v2 ∈ H. Since β(v1 ) = β(u1 ) and β(v2 ) = β(u2 ), we obtain v1 , v2 ∈ A(H) and z1 = v1 v2 u3 · . . . · un ∈ Z(H). We have z1 ∈ Z(a), β(z1 ) = β(z), d(z, z1 ) ≤ 2 and dF (P \P0 ) (pp−1 b1 , b1 ) = dF (P \P0 ) (b1 , b1 ) − 1. By the induction hypothesis, there −1

exists a 2-chain of factorizations in Z(a) ∩ β (β(z)) concatenating z1 and z  . This completes the proof in the case where p  a for all p ∈ P0 . In the general case, we proceed by induction on |a|F (P0 ) . Let p ∈ P0 be such that a ∈ pD, and apply the assertions of Lemma 4.2.4. We have p−1 a ∈ p−1 H ∩ D ⊂ H, and we assume the induction hypothesis for p−1 a. For every u ∈ A(H), we have pu ∈ H, and β(u) = β(pu) implies pu ∈ A(H). Moreover, if u ∈ A(H) ∩ pD, then p−1 u ∈ H, and again β(p−1 u) = β(u) implies p−1 u ∈ A(H). We may assume that p | u1 and p | u1 . Then y = (p−1 u1 )u2 · . . . · un and y  = (p−1 u1 )u2 · . . . · un are factorizations of p−1 a satisfying β(y) = β(y  ). By the induction hypothesis, −1 there exists a 2-chain y0 , . . . , yk in Z(p−1 a) ∩ β (β(y)) concatenating y and y  . For i ∈ [0, k], we set yi = ui,1 · . . . · ui,n , where ui,ν ∈ A(H) for all ν ∈ [1, n], and in particular u0,1 = p−1 u1 , uk,1 = p−1 u1 , u0,ν = uν and uk,ν = uν for all ν ∈ [2, n]. For i ∈ [0, k], we set zi = (pui,1 )ui,2 · . . . · ui,n . Then z0 , . . . , zk is a 2-chain of −1 factorizations in Z(a) ∩ β (β(z)) concatenating z and z  .

5

(Generalized) Block Monoids

In this section we show how the transfer principles for (T -)block monoids as explained in [25] follow from Theorem 4.1. We recall the definition of a T -block monoid as introduced in [15]. Definition 5.1. Let G be an additive abelian group, G0 ⊂ G a subset, T a monoid and ι : T → G a homomorphism. Let σ : F (G0 ) → G be the unique homomorphism satisfying σ(g) = g for all g ∈ G0 . Then we call B(G0 , T, ι) = {S t ∈ F(G0 )×T | σ(S) + ι(t) = 0 } the T -block monoid over G0 defined by ι. If T = {1}, then B(G0 , T, ι) = B(G0 ), the (ordinary) block monoid of zero-sum sequences in G0 . Let B(G0 , T, ι) ⊂ F (G0 )×T be a T -block monoid as in the definition, and let ψ : F (G0 ) × T → G be defined by ψ(St) = σ(S) + ι(t). It is easily checked that B(G0 , T, ι) ⊂ F (G0 )×T is saturated, and ψ induces a monomorphism

 ψ : F (G0 )×T /B(G0 , T, ι) → G . If ψ is surjective, then B(G0 , T, ι) ⊂ F(G0 )×T is cofinal, and ψ is an isomorphism. Theorem 5.2. Let D be a reduced atomic monoid, P ⊂ D a set of primes and T ⊂ D a submonoid such that D = F (P )×T . Let H ⊂ D be a saturated submonoid, G = q(D/H),         GP = [p]D/H  p ∈ P ⊂ G and G1 = [u]D/H  u ∈ A(D) ⊂ G .

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 : D → F (GP )×T be the unique Let ι : T → G be defined by ι(t) = [t]D/H , and let β   | T = idT . homomorphism satisfying β(p) = [p]D/H for all p ∈ P and β  is a transfer homomorphism. 1. β

   −1 B(GP , T, ι) = H, and β = β  | H: H → 2. β(H) = B(GP , T, ι), β B(GP , T, ι) is a transfer homomorphism satisfying c(H, β) ≤ 2 and t(H, u, β) ≤ D(G1 ) for all u ∈ A(H). Proof. For u, v ∈ D we set u ∼ v, if [u]D/H = [v]D/H . Then ∼ is a congruence relation on D, and if u, v ∈ D, [u]D/H = [v]D/H and u ∈ H, then [u]D/H = [v]D/H = [1]D/H and therefore v ∈ H. Hence we can apply Theorem 4.1 with P0 = ∅ and U = {1}. The special case F = H in Theorem 4.1 implies 1., and for  the proof of 2. it remains to show that β(H) = B(GP , T, ι) and t(H, u, β) ≤ D(G1 ) for all u ∈ A(H). If a = p1 · . . . · pn t ∈ D, where n ∈ N0 , p1 , . . . , pn ∈ P and t ∈ T , then  β(a) = [p1 ]D/H · . . . · [p]D/H t ∈ F(GP )×T and σ([p1 ]D/H · . . . · [pn ]D/H ) + ι(t) = [p1 ]D/H + . . . + [pn ]D/H + [t]D/H = [a]D/H = 0 ∈ G if and only if a ∈ H. Hence  β(a) ∈ B(GP , T, ι) if and only if a ∈ H. To prove the estimate for t(H, u, β), we apply Proposition 3.7. Suppose that n ∈ N, u, u1 , . . . , un ∈ A(H), u | u1 · . . . · un and β(u) = β(u1 ). We must prove that we can renumber u1 , . . . , un in such a way that u | u1 · . . . · uD(G1 ) . We may suppose that u = p1 · . . . · pk t, where k ∈ N0 , p1 , . . . , pk ∈ P and t ∈ T . Then (as mentioned in Section 2) k + δ ≤ D(G1 ), where δ = 1, if t = 1 and δ = 0, if t = 1. If t = 1, we may renumber u1 , . . . , un so that u = p1 · . . . · pk | u1 · . . . · uk . If t = 1, then β(u) = β(u1 ) implies u1 = p1 · . . . · pk t, where pi ∈ P and [pi ]D/H = [pi ]D/H for all i ∈ [1, k]. Hence in this case we may renumber u2 , . . . , un so that p1 · . . . · pk | u1 · . . . · uk+1 and therefore u | u1 · . . . · uk+1 . The main application of Theorem 5.2 is the arithmetic of t-noetherian monoids, which we will sketch now using the notations and results of [26, Ch. 24]. Let H be a t-noetherian monoid whose complete integral closure admits a nonzero conductor, S the set of all strong t-maximal t-ideals, P = t-max(H) \ S, and 

S=H\

P.

P ∈S

Then Hp is a discrete valuation monoid for all p ∈ P, H=

 p∈t-max(H)

Hp =



Hp ∩ S −1 H ,

p∈P

and this intersection is of finite character. We set T = (S −1 H)red , and for p ∈ P, we denote by vp the valuation of q(H) associated with Hp . Then the map 

ϕ : H → D = F (P)×T , defined by ϕ(a) = (vp (a))p∈P , a(S −1 H)× ,

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is a divisor homomorphism, that means, ϕ(H) ⊂ D is a saturated submonoid, and ∼ ϕ induces an isomorphism ϕred : H → ϕ(H). We set G = D/H, GP = {[p]D/H | p ∈ P }, and we denote by ι : T → G the natural homomorphism. Then the map   [p]D/H t , β : H → B(GP , T, ι) , defined by β(a) = p∈P

is a transfer homomorphism for which the assertions of Theorem 5.2 hold. In general, only little is known about the arithmetic of the monoid T . If however T is well-known and GP is finite, then this transfer homomorphism can be used to obtain strong arithmetical results on H. This is the case if H is weakly Krull. Then T is a finite direct product of reduced finitely primary monoids, and strong arithmetical results were obtained in [19, Theorem 8.3]. In particular, this applies to weakly Krull Mori domains (and hence to one-dimensional Mori domains and to noetherian Cohen-Macaulay domains). If T = {1}, then Hred is a saturated submonoid of a free monoid F and thus H is a Krull monoid (see [11, Theorem 2.2]). In this case, we have the following stronger result (for simplicity, we formulate it for Hred instead of H). Theorem 5.3. Let H ⊂ F = F (P ) be a saturated submonoid, GP =  p ∈ P and D = D(GP ).



  [p]F/H 

 1. c(GP ) ≤ c(H) ≤ max c(GP ), 2} and c(H) ≤ D. 2. t(GP ) ≤ t(H) ≤ t(GP ) + D. 3. If d ∈ N, p1 , . . . , pd ∈ P and u = p1 · . . . · pd ∈ A(H), then U = [p1 ]F/H · . . . · [pd ]F/H ∈ A(G0 ), and if D ≥ 2, then   3 + (d − 1)(D − 1) d (D − 1) . t(GP , U ) ≤ t(H, u) ≤ max t(GP , U ), ≤1+ 2 2 4. We have the estimates   3 + (D − 1)2 D(D − 1) t(H) ≤ max t(GP ), , ≤1+ 2 2 and if GP = −GP ⊂ q(F/H) and H is not factorial, then t(H) ≥ D. Proof. We may assume that D ≥ 2, for otherwise H = F and the assertions are obvious. By Proposition 5.2, there is a transfer homomorphism β : H → B(GP ) satisfying c(H, β) ≤ 2 and t(H, u, β) ≤ D for all u ∈ A(H). Also, if γ ∈ N, p1 , . . . , pγ ∈ P and a = p1 · . . . · pγ ∈ H, then β(a) = [p1 ]F/H · . . . · [pγ ]F/H . 1. The first series of inequalities follows by Theorem 3.4.5. (see also [18, Proposition 4.2]). By [18, Proposition 4.3], we have c(GP ) ≤ D.

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2. If u ∈ A(H), then U = β(u) ∈ A(GP ) by Proposition 3.2.2, and Theorem 3.4.1 implies t(GP , U ) ≤ t(H, u) ≤ t(GP , U ) + D. Taking the supremum over all u ∈ A(H), the assertion follows. 3. We just proved the first inequality, and now we shall prove that   3 + (d − 1)(D − 1) d (D − 1) . a) t(H, u) ≤ max t(GP , U ), and b) t(H, u) ≤ 1+ 2 2 Then we apply b) to B(GP ) ⊂ F(GP ) instead of H ⊂ F and obtain t(GP , U ) ≤ 1 + d(D − 1)/2. From D ≥ 2 it follows that d (D − 1) 3 + (d − 1)(D − 1) ≤1+ , 2 2 which then completes the proof. a) Let a ∈ H be such that u | a and z ∈ Z(a). It is sufficient to prove that there exists some z  ∈ Z(a) ∩ uZ(H) such that   3 + (d − 1)(D − 1) d(z, z  ) ≤ t = max t(GP , U ), . 2 We proceed by induction on |z|. If z = 1, there is nothing to do. Thus suppose that z = v1 · . . . · vl , where l ∈ N and v1 , . . . , vl ∈ A(H). We may assume that for some n ∈ [1, l] we have u | v1 · . . . · vn (in F ), but u does not divide any proper subproduct of v1 · . . . · vn . If n < l, the induction hypothesis implies the existence of u2 , . . . , uk ∈ A(H) such that v1 · . . . · vn = uu2 · . . . · uk , and in Z(H) we have d(v1 · . . . · vn , uu2 · . . . · uk ) ≤ t. Then z  = uu2 · . . . · uk vn+1 · . . . · vl ∈ Z(a) ∩ uZ(H) and d(z, z  ) ≤ t. Hence we may suppose that n = l. Then l ≤ d, and if Vi = β(vi ) ∈ A(G0 ) for all i ∈ [1, l], then U | V1 · . . . · Vl . We may assume that there exists some m ∈ [1, l] such that U | V1 · . . . · Vm , but U does not divide any nonempty proper subproduct / P . Then of V1 · . . . · Vm . If u ∈ P , there is nothing to do. Thus suppose that u ∈ U = 0 and D ≥ |Vj | ≥ 2 for all j ∈ [1, m]. By the very definition of t(GP , U ), there exist U2 , . . . , Uk ∈ A(G0 ) such that V1 · . . . · Vm = U U2 · . . . · Uk , and in Z(G0 ) we have max{k, m} = d(V1 · . . . · Vm , U U2 · . . . · Uk ) ≤ t(GP , U ) ≤ t . Since β(u−1 a) = U2 · . . . · Uk Vm+1 · . . . · Vl and β : H → B(GP ) is a transfer homomorphism, there exist u2 , . . . , uk , wm+1 , . . . , wl ∈ A(H) such that β(ui ) = Ui for all i ∈ [2, k], β(wj ) = Vj for all j ∈ [m + 1, l] and u−1 a = u2 · . . . · uk wm+1 · . . . · wl . Then z  = uu2 · . . . · uk wm+1 · . . . · wl ∈ Z(a) ∩ uZ(H) and d(z, z  ) ≤ max{l, k + l − m} . If m = l, this implies d(z, z  ) ≤ t. It remains to consider the case m < l. Since u does not divide any proper subproduct of v1 · . . . · vl , there is a factorization u = u u , where u , u ∈ F \ {1}, u | v1 · . . . · vm , u | vm+1 · . . . · vl , |u | ≥ m and |u | ≥ l − m. From the trivial estimate 2(k − 1) ≤ |U2 · . . . · Uk | = |U −1 V1 · . . . · Vm | ≤ mD − |U | ≤ |u |D − d

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133

  |u |D − d + |u | max{l, k + l − m} ≤ max d, 1 + 2

we obtain

and 1+

|u |D − d 2 + |u |D − |u | + |u | + |u | = 2 2 2 + d(D − 1) − |u |(D − 2) 3 + (d − 1)(D − 1) ≤ ≤ . 2 2

b) We assume that a ∈ H, u ∈ A(H), u | a, and we estimate t(a, u). If d = 1, then t(a, u) = 0. Thus suppose that d ≥ 2 and z = u1 · . . . · ur ∈ Z(a), where r ∈ N and u1 , . . . , ur ∈ A(H). We must prove that there exists some z  ∈ Z(a) ∩ uZ(H) such that d(D − 1) . d(z, z  ) ≤ 1 + 2 We may assume that there exists some k ∈ [1, r] such that k ≤ d and u | u1 · . . . · uk (in F and thus also in H). Since p  u for all p ∈ P ∩ H, we may also assume that u1 , . . . , uk ∈ / P ∩ H. If u1 · . . . · uk = uc, where c ∈ H, and if w ∈ Z(c), then z  = uwuk+1 · . . . · ur ∈ Z(a), and d(z, z  ) ≤ max{k, |w| + 1} ≤ max{d, |w| + 1}. Since |w| ≤

|c|F |u1 |F + . . . + |uk |F − |u|F kD − d d(D − 1) = ≤ ≤ , 2 2 2 2

we obtain the asserted bound for d(z, z  ). 4. The estimate for t(H) follows immediately from 3. by taking the supremum over all u ∈ A(H). Assume now that H is not factorial, GP = −GP , and let N ∈ N be such that N < D. We shall prove that t(H) > N . By assumption, there exists some u ∈ A(H) such that u = p1 · . . . · pm , where p1 , . . . , pm ∈ P and m > N . Then there exist p1 , . . . , pm ∈ P such that [pj ]F/H = −[pj ]F/H and thus uj = pj pj ∈ A(H) for all j ∈ [1, m]. Since β is a transfer homomorphism, it follows that u = p1 · . . . · pm ∈ A(H) and Z(uu ) = {uu , u1 · . . . · um }. Since H is not factorial, we have t(H) ≥ 2, and therefore we may assume that D > N ≥ 2. Then m ≥ 3, and t(H) ≥ t(H, u) ≥ d(uu , u1 · . . . · um ) = m > N .

6

K+M-Domains

The most common examples of K+ M -domains are integral domains of the form K + XL[X] and K + XL[[X]], where K ⊂ L are integral domains. Their arithmetic has been investigated in several papers, see e. g. [7] or [2]. If K and L are fields, then the domains K +XL[X] and K +XL[[X]] are half-factorial. With our methods, we investigate more generally the case of a ring extension of the form R = K + m ⊂ L + m = D, where K ⊂ L are fields and m is a maximal ideal of D. Note that every finitely generated algebra R over an infinite perfect field with ρ(R) < ∞ is of this form (see [28, Theorem 6.2] where a transfer principle was used to investigate the elasticity of such domains).

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Proposition 6.1. Let R ⊂ D be integral domains and m ∈ max(R) such that D = RD× , D× ∩ R = R× and (R : D) = m. Then the injection j = (R• → D• ) is a transfer homomorphism satisfying c(a, j) ≤ 2

and

t(a, uR× , j) ≤ 2

for all

a ∈ R and u ∈ A(R) .

In particular, R is atomic [a BF-monoid, locally tame ] if and only if D has this property. Moreover, L(R) = L(D), ρ(R) = ρ(D), c(D) ≤ c(R) ≤ max{2, c(D)} ,

t(D) ≤ t(R) ≤ t(D) + 2 ,

and if D is factorial, then R is half-factorial. Proof. Since D = RD× and D× ∩ R = R× , j satisfies (T 1). For the proof of (T 2), let u ∈ R• and b, c ∈ D• be such that u = bc. We must prove that there exist v, w ∈ R• and ε, η ∈ D× such that u = vw, v = bε and w = cη. Since D = RD× , there exist b0 , c0 ∈ R and ε0 , η0 ∈ D× such that b = b0 ε0 and c = c0 η0 . Now we distinguish two cases. CASE 1: b0 ∈ m. Then we have ε0 η0 b0 ∈ m ⊂ R, and we set v = ε0 η0 b0 , w = c0 , ε = η0 and η = η0−1 . CASE 2: b0 ∈ / m. Since m ∈ max(R), there exists some d ∈ R such that b0 d ≡ 1 mod m, hence ε0 η0 c0 ≡ ε0 η0 b0 c0 d ≡ ud mod m and ε0 η0 c0 = ud + m ⊂ R. Now we set v = b0 , w = ε0 η0 c0 , ε = ε−1 0 and η = ε0 . We prove that jred satisfies the assumptions of Proposition 3.7 with W = 2. Thus suppose that n ∈ N≥2 , u, u1 , . . . , un ∈ A(R), a = u1 · . . . · un , u | a and u = u1 ε for some ε ∈ D× . We must prove that there exists some i ∈ [2, n] such that u | u1 ui . Let b ∈ R be such that a = ub. Then εb = u2 · . . . · un ∈ R. If b ∈ m, then ui ∈ m for some i ∈ [2, n]. Hence ε−1 ui ∈ m ⊂ R, and u1 ui = u(ε−1 ui ) implies u | u1 ui . If b ∈ / m, then there exists some c ∈ R such that bc ≡ 1 mod m and thus ε ≡ εbc ≡ u2 · . . . · un c mod m. Hence ε ∈ u2 · . . . · un c + m ⊂ R, and therefore ε ∈ R× , which implies u  u1 in R. Now Proposition 3.7 implies c(a, j) ≤ 2 and t(a, uR× , j) ≤ 2 for all a ∈ R and u ∈ A(R). The remaining assertions follow by Proposition 3.2 and Theorem 3.4.

Proposition 6.2. Let R ⊂ D be an integral domain. Let K ⊂ L ⊂ D be subfields, {0} = m ∈ max(D) such that D = L + m and R = K + m. Then R → D satisfies the assumptions of Proposition 6.1. Proof. We must show that D = RD× and D× ∩ R = R× . We obviously have L× R ⊂ D× R ⊂ D and R× ⊂ D× ∩ R. If a ∈ D, then a = u + m for some u ∈ L and m ∈ m. If u = 0, then a ∈ m ⊂ R. If u = 0, then u−1 a = 1+u−1m ∈ 1+m ⊂ R, and a = u(u−1 a) ∈ L× R. If x ∈ D× ∩ R, then x = u + m and x−1 = u + m , where u ∈ K × , u ∈ L and m, m ∈ m. Then 1 ∈ (uu + m) ∩ (K + m) implies uu ∈ K and therefore u ∈ K, x−1 ∈ R.

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7

135

C-Monoids and Congruence Monoids in N

In [21], we introduced C0 -monoids and AC-monoids as abstract tools for the arithmetical investigation of congruence monoids in Dedekind domains. In [27], W. Hassler used the concept of a Z-monoid as an abstract scheme for the arithmetic of noetherian domains satisfying some natural finiteness conditions, and in [13], a special type of C0 -monoids (those which are also Z-monoids) are used to study monotone chains of factorizations in certain congruence monoids and finitely generated domains. The concept of a C-monoid was introduced in [23] as a common generalization of all notions of auxiliary monoids just mentioned. Here we show how our general transfer principle establishes a connection between all these very similar concepts. Congruence monoids will finally serve as characteristic examples to show how our formalism works. We start by recalling the definitions. Definition 7.1. A monoid H is called a C-monoid , if it is a submonoid of a factorial monoid F such that H ∩ F × = H × and C ∗ (H, F ) is finite. Then it follows by [23, Proposition 3.10] that there exists some α ∈ N and a subgroup V ⊂ F × such that H × ⊂ V , (F × : V ) | α, V (H \ H × ) ⊂ H

and q 2α F ∩ H = q α (q α F ∩ H) for all q ∈ F \ F × .

We refer to these properties by saying that H is defined in F with exponent α and subgroup V ⊂ F × . If F = F × × F(P ) and x ∈ q(F ), we call suppP (x) = {p ∈ P | vp (x) = 0} the support of x in P . A subset E ⊂ P is called H-essential, if E = suppP (a) for some a ∈ H \ F × , and we set     κ(H, F ) = max |E|  E ⊂ P is a minimal H-essential subset . Note that κ(H, F ) only depends on F and not on the set P . By Theorem 7.2 below, H is a BF-monoid, and by [23, Theorem 5.1] we have ρ(H) < ∞ if and only if κ(H, F ) = 1. For Hilbert monoids, we shall make this criterion explicit in Theorem 7.8. A C0 -monoid is a C-monoid H defined in a factorial monoid F possessing only finitely many pairwise nonassociated prime elements. A Z-monoid is a C-monoid H defined in a factorial monoid F such that C(H, F ) is finite. We do not recall here the notion of an AC-monoid because it plays no role further on. We mention however that by [21, Theorem 5.4] every AC-monoid defined in a factorial monoid is a C-monoid, and by [23, Theorem 4.3], every C0 -monoid is an AC-monoid defined in a factorial monoid. If H is a C-monoid defined in a factorial monoid F , then C(H, F ) is finite if and only if F × /H × is finite (see [27, Proposition 5.3] and [23, Lemma 3.3]). The following theorem generalizes [27, Theorem 5.4]. Theorem 7.2. Let F = F ××F(P ) be a factorial monoid and H a C-monoid defined in F with subgroup V ⊂ F × . Let P0 ⊂ {p ∈ P | p−1 H ∩ F = H \ H × } be a subset, P = {[p]F H | p ∈ P \ P0 }, and F = F (P)×F × /V ,

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 : F → F be the unique homowhere F (P ) is the free monoid with basis P . Let β F   = 1 for all p ∈ P0 , and morphism satisfying β(p) = [p]H for all p ∈ P \ P0 , β(p) ×  β(e) = eV for all e ∈ F . Finally, we define   = β(H) H

and

 |H : H → H . β=β

 is a C0 -monoid defined in F , and F × = F × /V is finite. 1. H 2. β is a transfer homomorphism, c(H, β) ≤ 2 and t(a, uH × , β) ≤ ω(a, u) + 1 for all a ∈ H and u ∈ A(H). 3. H is locally tame, c(H) < ∞, and the Structure Theorem for Sets of Lengths holds for H.  F). 4. κ(H, F ) = κ(H, Proof. 1. and 2. (H, F )-equivalence is a congruence relation on F , and if u, v ∈ F F, [u]F H = [v]H and u ∈ H, then v ∈ H. Hence we can apply Theorem 4.1  F) are finite, and with T = F × and U = V . Then F × = F × /V , P and C ∗ (H, × ×      H ∩ F = H . In particular, H is a C0 -monoid defined in F . The estimates for c(H, β) and t(a, uH × , β) follow by Theorem 4.1.3 (note that H is a BF-monoid).  has these properties, and by Theorem 3.4, the same 3. By [21, Theorem 5.9], H is true for H. 4. It is sufficient to prove the following assertions for a finite subset E ⊂ P \ P0 :   a) If E is H-essential, then β(E) is H-essential.    b) If β(E) is H-essential and |β(E)| = |E|, then E is H-essential. Indeed, suppose that this is done, and let E ⊂ P be a minimal H-essential subset   such that |E| = κ(H, F ). Then E ∩ P0 = ∅ by Lemma 4.2.4, β(E) is H-essential     ) is a minimal H-essential by a), and there exists a subset E ⊂ E such that β(E      set, and |β(E )| = |E |. Then E is H-essential by b), hence E = E and κ(H, F ) =   )| ≤ κ(H,  F). Conversely, let E  ⊂ P be a minimal H-essential  subset such |β(E   = β(E)  = κ(H,  F ). Then there exists a subset E ⊂ P \ P0 such that E that |E|  = |E|. By b), E is H-essential, and there exists a minimal H-essential and |E|   ) is H-essential   )| = |E  | and β(E   ) ⊂ E.   subset E  ⊂ E. Now β(E by a), |β(E     and therefore E = E and κ(H,  F ) = |E| ≤ κ(H, F ). Hence β(E ) = E, a) If E ⊂ P is H-essential, then E = suppP (a) for some a ∈ H \ F × , and   β(E) = suppPe β(a) is H-essential.  1 ), . . . , β(p  r) b) Suppose that E = {p1 , . . . , pr } ⊂ P \ P0 is such that β(p    such that are distinct and β(E) is H-essential. Then there exists some y ∈ H n1 nr ×   y = β(p1 ) · . . . · β(pr ) (eV ), where n1 , . . . , nr ∈ N and e ∈ F . Hence z =  −1 (H),  and therefore, by Theorem 4.1.1, z = ytc, where y ∈ H, pn1 1 · . . . · pnr r e ∈ β t ∈ V and c ∈ F(P0 ). It follows that c = 1, y = pn1 1 · . . . · pnr r et−1 ∈ H, and thus E = suppP (y) is H-essential.

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137

As already mentioned, congruence monoids in Dedekind domains and the arithmetic of finitely generated noetherian domains are the main instances for the application of Theorem 7.2. In [23] we study more generally congruence monoids in Krull domains. Here we discuss briefly the most modest special case of congruence monoids: Hilbert monoids of natural numbers (see [21, Examples 3.3] and [6]). Definition 7.3. Let f ≥ 2 be a natural number. A subset H ⊂ N is called a congruence monoid or a Hilbert monoid defined modulo f , if there is a nonempty, multiplicatively closed subset Λ ⊂ Z/f Z such that H = {a ∈ N | a + f Z ∈ Λ} ∪ {1}. Theorem 7.4. Let f ≥ 2 be a natural number and H ⊂ N a Hilbert monoid defined modulo f . Let g ∈ N be squarefree such that H ⊂ Ng . Then g | f , H is a C-monoid defined in Ng , and there is an epimorphism θ : (Z/f Z, ·) → C ∗ (H, Ng ), given by N θ(a + f Z) = [a]Hg . Proof. If p ∈ P and p  f , then there exists some k ∈ N such that pk ≡ 1 mod f , and if 1 = a ∈ H, then also pk a ∈ H. Hence it follows that g | f . N N If a, b ∈ Ng \ {1} and a ≡ b mod f , then [a]Hg = [b]Hg by the very definition of the Hilbert monoid. Hence the canonical epimorphism Ng \ {1} → C ∗ (H, Ng ), N defined by a → [a]Hg , induces an epimorphism Z/f Z → C ∗ (H, Ng ). Proposition 7.5. Let f ≥ 2 be a natural number and Λ ⊂ Z/f Z a nonempty multiplicatively closed subset. For a ∈ Z, set a = a + f Z ∈ Z/f Z, (Λ : a) = {c ∈ Z/f Z | a c ∈ Λ}, and consider the Hilbert monoid H = {a∈ N | a ∈ Λ} ∪ {1}. Let g ∈ N be squarefree such that H ⊂ Ng , and define P0 =  H \ {1} . N

p ∈ Pg | p−1 H ∩ Ng =

N

1. For a, b ∈ Ng \ {1} we have [a]Hg = [b]Hg if and only if (Λ : a) = (Λ : b). 2. P0 = {p ∈ Pg \ H | (Λ : p) = Λ}. 3. P0 = ∅ if and only if 1 ∈ Λ. 4. If 1 ∈ Λ, then {p ∈ P | p ≡ 1 mod f } ⊂ {p ∈ Pg | (Λ : p) = Λ} ⊂ H,    N    {[p]Hg  p ∈ Pg , (Λ : p) = Λ} = 1 , and if p ∈ Pf ∩ H, then (Λ : p) = Λ. Proof. 1. Obvious by the definitions. 2. If p ∈ P0 , then p−1 H ∩ Ng = H \ {1} and therefore p ∈ / H. For the proof of (Λ : p) = Λ, suppose that a ∈ Ng \ {1}. If a ∈ (Λ : p), then ap = a p ∈ Λ, hence ap ∈ H and a ∈ p−1 H ∩ Ng = H \ {1}, which implies a ∈ Λ. Conversely, if a ∈ Λ, then a ∈ H \ {1} = p−1 H ∩ Ng , hence pa ∈ H, p a ∈ Λ, and therefore a ∈ (Λ : p). Assume now that p ∈ Pg \ H and (Λ : p) = Λ. We must prove that p−1 H ∩ Ng = H \ {1}. If a ∈ p−1 H ∩ Ng , then pa ∈ H \ {1}, hence pa = p a ∈ Λ and thus

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a ∈ (Λ : p) = Λ. Since p ∈ / H, we have a = 1 and therefore a ∈ H \ {1}. Conversely, if a ∈ H \{1}, then a ∈ Λ = (Λ : p), hence pa = p a ∈ Λ, pa ∈ H and a ∈ p−1 H ∩Ng . / Λ and p ∈ P is such that p ≡ 1 mod f , then p ∈ / H and (Λ : p) = (Λ : 3. If 1 ∈ 1) = Λ. Hence p ∈ P0 . Assume now that 1 ∈ Λ and P0 = ∅. If p ∈ P0 , then 1 ∈ Λ = (Λ : p), hence p ∈ Λ and p ∈ H, a contradiction. 4. It suffices to prove the last assertion. Thus suppose that p ∈ Pf ∩ H. Then p ∈ Λ and thus pΛ ⊂ Λ. Hence (Λ : p) ⊃ Λ. Conversely, if a ∈ N \ {1} is such that a ∈ (Λ : p), then ap = a p ∈ Λ and thus ap ∈ H. Let n ∈ N be such that pn ≡ 1 mod f . Then apn = (ap)pn−1 ∈ H, and apn ≡ a mod f . Hence a ∈ H and a ∈ Λ. We proceed with two concrete examples. We use the notations introduced in Theorem 7.2, Theorem 7.4 and Proposition 7.5. Example 7.6. f = 3, Λ = {0, 1} ⊂ Z/3Z, H = {a ∈ N | a ≡ 0 or 1 mod 3}, g = 1. We find (Λ : 0) = Z/3Z, (Λ : 1) = {0, 1} and (Λ : 2) = {0, 2}. Hence θ : (Z/3Z, ·) → C ∗ (H, N) is an isomorphism, P = { 7,  3,  5}, and  7 is a prime element of  = { 3b  5c | a ∈ N0 , (b, c) ∈ T } ∼ H 7a  = N0 ×T , where T = {(b, c) ∈ N20 | b ≥ 1 or (b = 0 , c ≡ 0 mod 2)} ⊂ (N20 , +) ,  and there is a transfer homomorphism β : H → H. If u0 = (1, 0), u1 = (1, 1) and v = (0, 2), then A(T ) = {u0 , u1 , v}, and  = c(T ) = 3, 2u0 + v = 2u1 . From these relations it is easy to derive c(H)  = t(T ) = 3. Hence we obtain t(T, u0 ) = t(T, u1 ) = t(T, v) = 3, and therefore t(H) c(H) = 3, and a detailed direct analysis shows that also t(H) = 3. Finally, we calculate the elasticity. Using Proposition 3.2.5, we obtain ρ(H) = ρ(T ). We use the semi-length function λ : T → N0 , defined by λ(b, c) = 2b + c (see Section 2) and obtain ρ(H) ≤ 3/2. Hence ρ(H) = 3/2, since 2u0 + v = 2u1 . Example 7.7. f = 9, Λ = {0, 6} ⊂ Z/9Z, H = {a ∈ N | a ≡ 0 or 6 mod 9}, g = 1. We find (Λ : 1) = (Λ : 4) = (Λ : 7) = {0, 6}, (Λ : 2) = (Λ : 5) = (Λ : 8) = {0, 3}, (Λ : 0) = Z/9Z, (Λ : 3) = {0, 2, 3, 5, 6, 8} and (Λ : 6) = {0, 1, 3, 4, 6, 7} . Hence θ(1) = θ(4) = θ(7) and θ(2) = θ(5) = θ(8), and C ∗ (H, N) = {θ(0), θ(1), θ(2), θ(3), θ(6)} is a semigroup consisting of 5 elements. Consequently, P = { 7,  2,  3}, and  7 is a prime element of  = { H 7a  2b  3c | a ∈ N0 , (b, c) ∈ T } ∼ = N0 ×T , where T = {(b, c) ∈ N20 | c ≥ 2 or (c = 1 , b ≡ 1 mod 2)} ∪ {(0, 0)} ⊂ (N20 , +) ,  and there is a transfer homomorphism β : H → H.

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139

We define w = (0, 2), w = (0, 3), and for odd k ∈ N, we set uk = (k, 1) and v k = (k, 2). Then A(T ) = {w, w  , uk , uk | k ∈ N odd }, and between these atoms the following relations hold for all odd k, l ∈ N: uk + ul = uk+l−1 + u1 , ul + v l = v k+l−1 + u1 = uk+l−1 + v 1 , v k + v l = v k+l−1 + v 1 uk + w = 2u1 + uk−2 and uk + w = 2u1 + v k−2 , 

v k + w = uk + w ,



v k + w = uk + 2w ,

if k ≥ 3 ,

3w = 2w .

From these relations we derive c(T ) = 3, and consequently, as in Example 7.6, also c(H) = 3. For odd k ∈ N, we set ak = (3k, 3k + 1), and we consider the factorizations ak = kw + u3k = (3k − 1)u1 + v 1 . They show that ρ(ak ) = (3k − 1)/(k + 1) and t(T, u3k ) ≥ k. Hence t(T ) = ∞, and consequently, as in Example 7.6, also t(H) = ∞. Moreover,    3k − 1  k ∈ N odd = 3. ρ(T ) ≥ sup k+1  Again as in Example 7.6, we use a semi-length function to obtain an upper bound for the elasticity. We define λ : T → N0 by λ(b, c) = c and obtain ρ(T ) ≤ 3. Hence ρ(T ) = 3 and it is easily seen that T does not have accepted elasticity. Therefore it follows that also ρ(H) = 3, and H does not have accepted elasticity. Theorem 7.8. Let H ⊂ N be a congruence monoid. Then the following statements are equivalent: (a) ρ(H) < ∞. (b) For every a ∈ H \ {1} there exists some p ∈ P and α ∈ N such that p | a and pα ∈ H. Proof. By Theorem 7.4, there exists a squarefree integer g ∈ N such that H is a C-monoid defined in Ng , and therefore ρ(H) < ∞ if and only if κ(H, Ng ) = 1, that is, every minimal H-essential subset of Pg is a singleton (see [23, Theorem 5.1]). But this is just condition (b).

Bibliography [1] D. D. Anderson (ed.), Factorization in integral domains, Marcel Dekker, 1997. [2] D. F. Anderson and D. N. El Abidine, Factorization in integral domains. III., J. Pure Appl. Algebra 135 (1999), no. 2, 107–127. [3] D.D. Anderson and D.F. Anderson, Elasticity of factorizations in integral domains, J. Pure Appl. Algebra 80 (1992), 217–235. [4] D.D. Anderson, D.F. Anderson, and M. Zafrullah, Factorizations in integral domains, J. Pure Appl. Algebra 69 (1990), 1–19. [5] D.D. Anderson, D.F. Anderson, and M. Zafrullah, Factorizations in integral domains II, J. Algebra 152 (1992), 78–93. [6] M. Banister, S. Chapman, J. Chiaka, and W. Myerson, On the arithmetic of arithmetical congruence monoids. [7] V. Barucci, L. Izelgue, and S. Kabbaj, Some factorization properties of A + XB[X]-domains, Commutative ring theory, Lecture Notes in Pure Appl. Math., Marcel Dekker 185 (1997), 69–78. [8] L. Carlitz, A characterization of algebraic number fields with class number two, Proc. AMS 11 (1960), 391–392. [9] S. T. Chapman and J. Coykendall, Half-factorial domains, a survey, Nonnoetherian commutative ring theory, Mathematics and its Applications, Kluwer Academic Publisher 520 (2000), 459–476. [10] S. T. Chapman, M. Freeze, and W. Smith, On generalized lengths of factorizations in Dedekind and Krull domains, Non-noetherian commutative ring theory, Mathematics and its Applications, Kluwer Academic Publisher 520 (2000), 117–137. [11] S.T. Chapman and A. Geroldinger, Krull domains and monoids, their sets of lengths and associated combinatorial problems, Factorization in integral domains, Lecture Notes in Pure Appl. Math., Marcel Dekker 189 (1997), 73–112. [12] A. Foroutan, Monotone chains of factorizations, Int. J. of Commutative Rings 2 (2003). [13] A. Foroutan and A. Geroldinger, Monotone chains of factorizations in Cmonoids, these Proceedings. 140

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¨ [14] A. Geroldinger, Uber nicht-eindeutige Zerlegungen in irreduzible Elemente, Math. Z. 197 (1988), 505–529. [15] A. Geroldinger, T-block monoids and their arithmetical applications to certain integral domains, Comm. Algebra 22 (1994), 1603–1615. [16] A. Geroldinger, The catenary degree and tameness of factorizations in weakly Krull domains, Factorization in integral domains, Lecture Notes in Pure Appl. Math., Marcel Dekker 189 (1997), 113–153. [17] A. Geroldinger, Chains of factorizations and sets of lengths, J. Algebra 188 (1997), 331–363. [18] A. Geroldinger, Chains of factorizations in weakly Krull domains, Colloq. Math. 72 (1997), 53–81. [19] A. Geroldinger, A structure theorem for sets of lengths, Colloq. Math. 78 (1998), 225–259. [20] A. Geroldinger and F. Halter-Koch, Tamely generated ideals in finitary monoids, JP J. Algebra Number Theory Appl. 2 (2002), 205–239. [21] A. Geroldinger and F. Halter-Koch, Congruence monoids, Acta Arith. 112 (2004), 263–296. [22] A. Geroldinger and G. Lettl, Factorization problems in semigroups, Semigroup Forum 40 (1990), 23–38. [23] F. Halter-Koch, C-monoids and congruence monoids in Krull domains, these Proceedings. [24] F. Halter-Koch, Elasticity of factorizations in atomic monoids and integral domains, J. Th. Nomb. Bordeaux 7 (1995), 367–385. [25] F. Halter-Koch, Finitely generated monoids, finitely primary monoids and factorization properties of integral domains, Factorization in integral domains, Lecture Notes in Pure Appl. Math., Marcel Dekker 189 (1997), 73–112. [26] F. Halter-Koch, Ideal systems. An introduction to multiplicative ideal theory, Marcel Dekker, 1998. [27] W. Hassler, Factorization in finitely generated domains, J. Pure Appl. Algebra 186 (2004), 151–168. [28] F. Kainrath, Elasticity of finitely generated domains, Houston J. Math. 30 (2004). [29] W. Narkiewicz, Finite abelian groups and factorization problems, Colloq. Math. 42 (1979), 319–330. [30] W. Narkiewicz, Elementary and analytic theory of algebraic numbers, 2nd ed., Springer, 1990. [31] W. Narkiewicz and J. Sliwa, Finite abelian groups and factorization problems II, Coll. Math. 46 (1982), 115–122.

Chapter 7

Cale Monoids, Cale Domains, and Cale Varieties by

Scott T. Chapman and Ulrich Krause

Introduction In an integral domain or monoid, two different factorizations of an element into atoms often turn out to be essentially the same when lifted to powers. In this chapter, we analyze this fascinating phenomenon, which surprisingly occurs for a great many algebraic structures. These structures range from algebraic number rings over Diophantine monoids, to varieties and their defining semigroup rings. Analytically we capture this phenomenon by the concept of a Cale representation, which is a kind of unique representation with respect to powers. Section 1 of this chapter presents the exact definitions concerning Cale concepts, as well as an array of rather different examples which illustrate the lifting of nonunique factorization to unique Cale representation. In Section 2, we consider Cale representation with respect to extraction analysis, which is one of the key tools used in the arguments throughout this chapter. Section 3 considers some useful special elements suggested by the study of the Cale property and places these elements in context with respect to the class of Krull monoids. In Section 4 we present for monoids, as well as for domains, general conditions for the Cale representation to exist. We take mainly from a joint work of the current authors with Franz Halter-Koch [4]. In Section 5 we present particular results on Cale representation in algebraic orders, Diophantine monoids and semigroup rings. Much of this work is drawn from the papers [4, 18, 21, 22]. In Section 6 we investigate affine toric varieties defined via semigroup rings for Cale monoids. The main result (Theorem 6.5) gives a description of those varieties by only a few binomials stemming from Cale representation. The authors wish to thank Alfred Geroldinger and Florian Kainrath for very useful comments on an earlier draft of this chapter. 142

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143

Cale Representation: Definition, Examples, and Counterexamples

Throughout this chapter, by a monoid M we mean an abelian semigroup with 1 (usually written multiplicatively) which is cancellative. Denote by M × its group of units. M is reduced if M × = {1}. Definition 1.1. The monoid M is inside factorial with base Q ⊂ M \M × if for each x ∈ M \M × there exists a positive integer n(x) and nonnegative integers {t(q)}q∈Q such that  (i) xn(x) = u q∈Q q t(q) where u ∈ M × and only finitely many of the t(q)’s are nonzero, and   (ii) if xn(x) = u q∈Q q t(q) = v q∈Q q s(q) where v ∈ M × and each s(q) is a nonnegative integer, then u = v and t(q) = s(q) for each q. For each x ∈ M \M × we denote by m(x) the minimal value of n(x) from part (i) of the definition and by x(q) the uniquely determined t(q). The base Q is a Cale base f (q)x(q) ∈ N0 for (or a tame base) if for each q ∈ Q there exists a f (q) ∈ N such that m(x) × all x ∈ M \M . As with n(x), in a Cale base Q each f (q) can be chosen minimally, and we denote this value as e(q). An inside factorial monoid with a Cale base is a Cale monoid and (i) is referred to as a Cale representation of x. The above notions apply to an integral domain D if its multiplicative monoid M = D• = D\{0} has the corresponding properties. In particular, a Cale domain is a domain D such that D• is inside factorial with Cale base Q. In what follows, we illustrate the above concepts by an array of quite different examples. In all these examples, the monoid considered is atomic, that is each nonunit is a product of finitely many atoms. An atom is an irreducible nonunit and we consider nonunique factorization into atoms which sometimes, but not always, can be explained by a Cale representation. Example 1.2. Algebraic number rings √ √ Let D = Z[ −5] be the principal order in Q( −5). The nonunique factorization into atoms in D √ √ 6 = 2 · 3 = (1 + −5) · (1 − −5) is commonly “explained” by the Kummer–Dedekind representation pp · qr = pq · pr into prime ideals with p = 2, 1 +

√ √ √ −5, q = 3, 1 + −5, and r = 3, 1 − −5,

which is unique up to the ordering of factors. Alternatively, by taking squares we have an explanation of this factorization by the following Cale representation √ √ √ √ 62 = 22 · (−2 + −5)(−2 − −5) = 2(−2 + −5) · 2(−2 − −5) which is unique up to ordering of factors and units.

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√ √ Whereas D is a Cale domain, the nonprincipal order R = Z[5 −1] in Q( √−1) fails to√be a Cale domain. This can be seen from the factorization 5 = (1 + 2 −1)(1 − 2 −1) into Gaussian primes for which no power belongs to R. More details on the Cale property in algebraic orders can be found in Section 5.1. Example 1.3. Diophantine monoids Consider the Diophantine monoid M = {x ∈ N30 | 2x1 + 5x2 = 3x3 } with addition (see [6]). One has the following nonunique factorization into atoms (3, 3, 7) = (1, 2, 4) + (2, 1, 3) = (3, 0, 2) + (0, 3, 5). Multiplying by 3, one obtains the following unique Cale representation 3 · (3, 3, 7) = (q1 + 2q2 ) + (2q1 + q2 ) = 3q1 + 3q2 with q1 = (3, 0, 2), q2 = (0, 3, 5). A Diophantine monoid, however, need not be a Cale monoid as the following example shows. Let M = {x ∈ N40 | x1 + x2 = x3 + x4 }. One has the following nonunique factorization into atoms (1, 1, 1, 1) = (1, 0, 1, 0) + (0, 1, 0, 1) = (1, 0, 0, 1) + (0, 1, 1, 0). It is easy to verify that (1, 0, 1, 0), (0, 1, 0, 1), (1, 0, 0, 1), and (0, 1, 1, 0) are the only 4 atoms of M . An elementary combinatorial argument can be used to show that it is impossible to combine the atoms so that a multiple of (1, 1, 1, 1) can be factored uniquely. Hence, the former nonunique factorization cannot be “explained ” by a Cale representation. More information on the Cale Property in Diophantine monoids can be found in Section 5.2. Example 1.4. Krull Monoids Examples 1.2 and 1.3 are members of the more general class of Krull Monoids. A monoid M is a Krull monoid if there exists a free Abelian monoid D and a homomorphism ∂ : M → D such that 1. x | y in M if and only if ∂(x) | ∂(y) in D, 2. every β ∈ D is the greatest common divisor of some set of elements in ∂(M ). The free basis elements of D are called the prime divisors of M and the quotient D/∂(M ) is called the divisor class group of M and denoted Cl(M ). If M is a Krull monoid and x is a nonunit of M , then there exist a unique set p1 , . . . , pk of prime divisors of M and unique positive integers n1 , . . . , nk such that ∂(x) = pn1 1 · · · pnk k .

(7.1)

Notice (7.1) implies that n1 p1 + · · · + nk pk = 0 in Cl(M ). Let M be a Krull monoid with torsion divisor class group Cl(M ) with set P = {pi | i ∈ A} of prime divisors and divisor theory ∂ : M → NP 0 . Select for each pi ∈ P with order ni in Cl(M ) a qi ∈ M with ∂(qi ) = pni i . If Q = {qi | i ∈ A}

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and x is a nonunit of M , then there exists a positive integer n such that xn factors uniquely (up to order) as a product of elements from Q. Hence, M is an inside factorial monoid. To see that M is a Cale monoid, choose pi ∈ P and suppose  mj i p ∂(x) = pm i j =i j . Let w = lcm{nj }pj |∂(x) and set dj nj = w for each pj | ∂(x).  n x(qi ) w i Then ∂(x) = (pni i )di mi j =i (pj j )dj mj and so m(x) = diwmi = m ni , completing the argument. To demonstrate how this relates to nonunique factorizations, suppose that M is a Krull monoid with Cl(M ) = Z3 . Let p1 , p1 , p2 and p2 be prime divisors of M where pi and pi lie in the divisor class ¯i (such a Krull monoid exists by [14]). If ∂(x) = p1 p2 , ∂(y) = p1 p2 , ∂(z) = p1 p2 and ∂(w) = p1 p2 , then in M v = xy = zw is a nonunique factorization. But notice that v 3 = (xy)3 = (zw)3 = abcd where ∂(a) = p31 , ∂(b) = p32 , ∂(c) = (p1 )3 and ∂(d) = (p2 )3 is the unique Cale representation of v. Example 1.5. Polynomial rings Let K be any field. In the polynomial ring D = K[X, XY 2 , XY ] one has the nonunique factorization into atoms X 2 Y 2 = X · XY 2 = XY · XY. Taking squares we meet, nevertheless, a unique Cale representation (X 2 Y 2 )2 = XX · XY 2 XY 2 = XXY 2 · XXY 2 . Similarly, in the semigroup ring R = K[X 2 , X 3 ] one has the nonunique factorization into atoms X 6 = X 2 · X 2 · X 2 = X 3 · X 3. Interestingly enough, for K = R the domain R is not Cale but for K = F2 it is Cale (see [21, Remark 3.4]). Example 1.6. Trigonometric polynomials In the domain D = R[cos x, sin x] of trigonometric polynomials, one has the following nonunique factorization into atoms √ √ cos 2x = (cos x + sin x) · (cos − sin x) = ( 2 cos x + 1) · ( 2 cos x − 1). Taking squares, however, one gets the following unique Cale representation (cos 2x)2 =

1 1 1 1 q1 q2 · q3 q4 = q1 q3 · q2 q4 2 2 2 2

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with

where

q1 (x)

=

q2 (x)

=

q3 (x)

=

q4 (x)

=

√ 2(cos x − sin x) √ − 2(cos x − sin x) √ 2(cos x + sin x) √ − 2(cos x + sin x)

+

2

+

2

+

2

+

2

1 is a unit. Actually, a Cale base is given in [23] by 2 q(x) = a cos x + b sin x + 1 +

a2 + b 2 ; a, b ∈ R, a2 + b2 ≥ 4. 4

It is of interest to note here that [23] also implies that R[cos x, sin x] is a halffactorial domain (i.e., for a given nonunit f , the length of any irreducible factorization is constant). Note moreover that the domain Q[X, Y ] with X 2 + Y 2 = 1 is a unique factorization domain, whereas Q(i)[X, Y ] with X 2 + Y 2 = 1 is not a unique factorization domain ([26, p. 16], [23]). Example 1.7. Coordinate rings (see [7, 9, 28, 30]) The quadric cone given by Y 2 = XZ in C3 is an affine toric variety with a rational double point singularity in 0. The coordinate ring of this variety is given by R = C[X, Y, Z] /

∼ C[X, XY, XY 2 ]. (Y 2 − XZ) =

By Example 1.5 the ring is not factorial but allows a Cale representation. The Neil parabola, however, given by Y 2 = X 3 in C3 is an affine toric variety with a cusp singularity in 0. The coordinate ring is given by R = C[X, Y ] /

∼ C[X 2 , X 3 ]. (Y 2 − X 3 ) =

As in Example 1.5, R is not Cale. (Replacing C by F2 , however, yields a Cale domain.) Example 1.8. Natural monoids By a natural monoid we mean a submonoid of N0 under addition (a numerical monoid ) or a submonoid of N under multiplication. For the numerical monoid M = N0 \{1, 2, 4, 7} one has the following nonunique factorization into atoms 23 = 6 · 3 + 1 · 5 = 1 · 3 + 4 · 5. Multiplying by 3 one obtains, nevertheless, a unique Cale representation 3 · 23 = 18 · 3 + 5 · 3 = 3 · 3 + 20 · 3 where {3} acts as a Cale base. For the Hilbert monoid (see [16]) M = 4N0 + 1 (considered multiplicatively) one has the following nonunique factorization into atoms 53361 = 9 · 49 · 121 = 21 · 33 · 77,

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which, in terms of “ideal numbers” ∈ / M , has the unique representation 53361 = 3 · 3 · 7 · 7 · 11 · 11 = 3 · 7 · 3 · 11 · 7 · 11. Taking squares, one has the unique Cale representation 92 · 492 · 1212 = 9 · 49 · 9 · 121 · 49 · 121. A Cale base of the Hilbert monoid is given for p prime in N by p if p ≡ 1 mod 4 and by p2 if p ≡ 3 mod 4. We note that the Hilbert monoid is but a simple example of a more complex structure known as a congruence monoid (see again [16]). Example 1.9. Semigroup rings and power series rings Generalizing Examples 1.6 and 1.8, consider the semigroup ring D[S] for an arbitrary domain D and an arbitrary additive monoid S. In general D[S] shows nonunique factorization into atoms. More precisely, by the Gilmer–Parker Theorem (see [12, 13]), D[S] has unique factorization into atoms if and only if both D and S possess unique factorization into atoms. Nevertheless, one may have unique Cale representation in case neither D nor S have unique factorization into atoms. More information on the Cale Property in semigroup rings can be found in Section 5.3. For generalized power series rings D[[S]] in the sense of Ribenboim ([24]), one can have nonunique factorizations even if both D and S possess unique factorization. A classical example was given by Samuel [26, p. 9] where D[[X]] (that is, S = N0 ) is not a unique factorization domain for a certain two–dimensional unique factorization domain D.

2

Extraction Analysis and Valuations

For a factorial domain, “extracting” the highest power of a prime p from elements of the domain defines a (p–adic exponential) valuation on the quotient field. Below we show for arbitrary inside factorial monoids that extracting maximally elements of a base from other elements obeys some nice rules. Moreover, we show for root-closed domains which are inside factorial that extracting base elements yields (exponential) valuations. First, we define the concepts of extraction analysis as introduced in [17] and review its related elementary properties (see [4], [19], and [20]). If M is a monoid then the function λ : M × M → [0, ∞] given by λ(x, y) = sup

m n

| m ∈ N0 , n ∈ N, xm | y n



is called the extraction degree on M . The monoid M is called an extraction monoid if for any x, y ∈ M \M × there exist m ∈ N0 , n ∈ N such that xm | y n and λ(x, y) = m n. An integral domain D is called an extraction domain if its multiplicative monoid D• = D\{0} is an extraction monoid. A review of elementary properties related to extraction, such as kλ(xk , xl ) = lλ(x, y) as well as others, can be found in the above references.

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Lemma 2.1. Let M be an inside factorial monoid with base Q.  x(q) q is the Cale representation of x ∈ M \M × then λ(q, x) = (i) If xm(x) = u q∈Q x(q) m(x)

for all q ∈ Q.

(ii) For every q ∈ Q, the map λ(q, ·) : M → Q≥0 is a monoid homomorphism. (iii) M is an extraction monoid. (iv) Each q ∈ Q is almost irreducible (i.e., z|q implies z m = uq n with m, n ∈ N, u ∈ M × ). Each q ∈ Q is almost primary (i.e., q|xy implies that q|xn or q|y n for some n ∈ N). Proof. See [4, Lemma 2]. Remark 2.2. In general elements of a base need neither be irreducible nor primary, but are always almost primary (see Section 3). The following lemma shows for a root-closed inside factorial domain that the extraction of base elements yields exponential valuations. Lemma 2.3. For an extraction domain D with quotient field F the following statements hold. (i) For x, y, z nonunits in D• λ(z, x + y) ≥

1 λ(z, x) · λ(z, y) ≥ min{λ(z, x), λ(z, y)} λ(z, x) + λ(z, y) 2

(ii) Suppose, D is root-closed (i.e., f ∈ F and f n ∈ D for n ∈ N implies f ∈ D). Then, for x, y, z nonunits in D• , λ(z, x + y) ≥ min{λ(z, x), λ(z, y)}. (iii) Suppose, D is root-closed and inside factorial with base Q. For q ∈ Q let vq (x) = λ(q, x) for x ∈ D• , x ∈ D× vq ( xy ) = vq (x) − vq (y) for x, y ∈ D•

and vq (x) = 0 for x ∈ D× . and vq (0) = ∞

Then vq : F → Q∪{∞} is an exponential valuation on F . That is, for f, g ∈ F we have vq (f · g) = vq (f ) + vq (g), vq (f + q) ≥ min{vq (f ), vq (g)}. If Q is a tame base, then there exists e(q) ∈ N such that e(q)vq is a discrete valuation. (iv) D is solid. That is, the root-closure D = {f ∈ F | f n ∈ D for some n ∈ N} is again an integral domain.

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m Proof. (i) Let λ(z, x) = m | xn and λ(z, y) = kl with z k | y l . For n with z nk ≤ ν ≤ nk + ml it follows that z mk | xνy nk+ml−ν . Similarly, for 0 ≤ ν ≤ nk it follows that z mk | xνy nk+ml−ν . From

(x + y)nk+ml =

nk+ml  ν=0

 nk + ml xνy nk+ml−ν ν

it follows that z mk | (x + y)nk+ml and, hence, λ(z, x + y) ≥

mk λ(z, x) · λ(z, y) 1 = ≥ min{λ(z, x), λ(z, y)}. nk + ml λ(z, x) + λ(z, y) 2

k (ii) For m, n, k, l in (i), suppose without restriction that m n ≤ l . It follows for 0 ≤ ν ≤ n that z m(n−ν)l · z knν | (xn−ν · yν)nl and, hence, z mnl | (xn−ν · yν)nl . Since D is root-closed this implies z m | xn−ν · yν for 0 ≤ ν ≤ n. Because of n    n n−ν yν we obtain that z m | (x + y)n and, hence, (x + y)n = ν x ν=0

λ(z, x + y) ≥

m = min{λ(z, x), λ(z, y)}. n

(iii) By Lemma 2.1, λ(q, x) ∈ Q≥0 and λ(q, x · y) = λ(q, x) + λ(q, y) for q ∈ Q and x, y ∈ D• \D× . Therefore, λ(q, ·) can be extended to F with νq (f ·g) = νq (f )+νq (g) for f, g ∈ F . Furthermore, for f = ff12 , g = gg12 in F and q ∈ Q we have that vq (f + g) = vq (f1 g2 + g1 f2 ) − vq (f2 g2 ) and, by step (ii), vq (f + g) ≥ min{vq (f1 g2 ) − vq (f2 g2 ), vq (g1 f2 ) − vq (f2 g2 )} ≥ min{vq (f1 ) − vq (f2 ), vq (g1 ) − vq (g2 )} = min{vq (f ), vq (g)}. (iv) Since D is multiplicatively closed, it suffices to show that D is additively closed. Let f = ff12 , g = gg12 in D and, without loss, f n , g n ∈ D for some n ∈ N. For m ∈ N0 arbitrary let m = kn + i, 0 ≤ i ≤ n − 1. It follows that f2n−1 f m = (f2n−1 f i ) · (f n )k ∈ D for all m and, similarly, g2n−1 g m ∈ D for all m. For x = f2n−1 g2n−1 ∈ D and h = f + g ∈ F it follows for arbitrary l ∈ N that x · hl = l    l n−1 f ν) · (g2n−1 g l−ν ) ∈ D. From xl−1 ∈ D and xhl ∈ D it follows from ν (f2 ν=0

(xh)l = xl−1 (xhl ) that xl−1 | (xh)l (in D) and, hence, λ(x, xh) ≥ l−1 l . Since l ∈ N is arbitrary, this implies λ(x, xh) ≥ 1 and since D is an extraction domain, there exists p ∈ N such that xp | (xh)p (in D). Thus, we arrive at (xh)p = xp y with y ∈ D and, hence, (f + g)p = hp = y ∈ D.

From the two lemmas we obtain in particular that every inside factorial integral domain is necessarily solid. In general, an integral domain need not be solid. For example, the integral domain D = R + X 2 R[X] is not solid because 1, X ∈ D but 1 + X ∈ D. Thus, this domain D cannot be inside factorial and, by Lemma 2.3 (iv), not even be an extraction domain (see also Example 1.5 in Section 2).

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Special Elements and the Cale Property in Krull Monoids

In Lemma 2.1 (iv), we defined the notion of almost primary and almost irreducible. We expand this list with the following definition. Definition 3.1. A nonunit x of a monoid M is a strong atom if for every n ∈ N the only nonunit divisors of xn are associates of powers of x. Obviously, a prime element is necessarily a strong atom and in turn must be an atom. None of these implications can be reversed. If M is a Krull monoid with torsion divisor class group, then the strong atoms of M are merely the primary atoms of M . Primary atoms need not be prime as, for example, the element (3, 0, 2) in the first case of Example 1.3. In general, a primary atom may not be a strong atom. For the numerical monoid N0 \{1}, the element 2 is clearly primary and an atom, but is not a strong atom since 23 = 32 . Thus, the notion of a strong atom lies strictly between that of an atom and a prime element. In a factorial monoid there is precisely one representation of a power of an atom, namely the trivial representation by itself. In an inside factorial monoid even the most simple Cale representation may be nontrivial. Definition 3.2. An element x ∈ M \{1} in a reduced Cale monoid M with base Q has a simple representation if xm = q n with q ∈ Q. If m, n ∈ N are minimally chosen, then x is said to be simple of type (m, n). Lemma 3.3. Let M be a reduced inside factorial monoid with base Q. 1. For x ∈ M \{1} the following properties are equivalent: (a) x is almost primary, (b) x is simple, (c) y|xn for some n ∈ N, y = 1 implies that y k = xl with k, l ≥ 1. 2. Strong atoms are almost primary and simple of type (k, 1). 3. An almost primary element has at most one strong atom as a divisor. 4. If x is a strong atom of M , then there is a unique positive integer w(x) such that xw(x) ∈ Q.  Proof. 1. We show both (a) and (c) are equivalent to (b). Let xm = q∈q q x(q) be the Cale representation of x. (a) ⇔ (b) If x is almost primary, then x|q0n for n ∈ N. Hence, q0 ∈ Q implies x(q) > 0 if and only if q = q0 by uniqueness of the Cale representation. Conversely, xm = q0n and x|yz imply y m z m = q0n w, w ∈ M . Therefore, q0 must appear in the Cale representations of y or of z. Without loss, q0r |y s and, hence, q0nr |y ns . That is, x|y ns . (b) ⇔ (c) If xm = q0t and y|xn then q0tn = xmn = y m z with z ∈ M . Therefore, the Cale representation contains only q0 (that is y r = q0s ). This implies y rt = q0st = xms . Conversely, by the Cale representation of x, q0 |xm for some q0 ∈ Q and, by (c), q0k = xl . Uniqueness implies that x(q) = 0 for q = q0 .

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2. Obviously, a strong atom possesses property 1(c) and, hence, it is almost primary. Furthermore, it is simple and by 1, xm = q n . Since x is a strong atom we must have that q = xk . 3. Let x ∈ M \{1} be almost primary with strong atoms y, z as divisors. By 1 we must have y k = xl and z r = xs with k, l, r, s ∈ N. Therefore, y ks = xls = z rl and y, z being strong atoms implies that z = y m , y = z n and hence, z = z mn . Thus, m = n = 1 and y = z. 4. Since the only nonunit divisors of xn are powers of x, the Cale property implies that Q contains at least one element of the form xn . If xn and xm ∈ Q for n = m, then xnm = (xn )m = (xm )n yields a contradiction. Remark 3.4. 1. By Lemma 3.3 part 1, primary elements in an inside factorial domain are automatically almost irreducible (see also Lemma 2.1 (iv)). Furthermore, x is almost primary if and only if some power of x is almost primary, which holds if and only if all powers of x are almost primary. 2. By Lemma 3.3 part 1, simple elements are almost primary. In particular, the elements of the Cale base Q which are simple of type (1, 1) are all almost primary (compare with Lemma 2.1 (iv).) By property 3, the elements of Q are “weak atoms” in the sense that they cannot be decomposed into different strong atoms. 3. For the notion of a strong atom see Krause [19] where these elements are called “strongly irreducible” and Stanley [27] where these elements are called “completely fundamental”. We characterize which subsets Q ⊆ M of a Krull monoid with torsion divisor class group can serve as the Cale base for M . To do this, we need to recall some definitions related to the primary elements of M . If q1 and q2 are primary elements of M , then define a relation q1 ∼ q2 if and only if q1 | q2n for some positive integer n. Obviously, ∼ is an equivalence on the set of all primary elements ℘ of M (compare to [15, Corollary 1.3]). Proposition 3.5. Let M be a Krull monoid with torsion divisor class group. Then Q ⊆ M is a base for the Cale monoid M if and only if Q consists of exactly one element chosen from each equivalence class of ℘/ ∼. Hence, Q must consist of primary elements. Proof. (⇒) We first argue that the elements of Q are primary. Our previous observaw(q ) tion combined with Lemma 3.3 part 4 imply that Q ⊇ {qi i }i∈A and Q contains no w(qi ) other powers of elements of the form qi . Suppose Q = {qi }i∈A . Then (by relabelk i ing the prime divisors if necessary) Q contains an element x = i=1 pm where each i mi > 0 and k ≥ 2. Let m = lcm{w(q1 ), w(q2 ), . . . w(qk )} and n = lcm{n1 , . . . , nk } where m = di w(qi ) and n = ci ni for each i. Then xmn = (

k 

i=1

i mn pm = i )

k  i=1

((pni i )w(qi ) )di ci mi =

k  i=1

w(qi ) di ci mi

(qi

)

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and using Lemma 3.3, we obtain a contradiction. Thus Q = {qi i }i∈A . Notice w(q ) w(q ) that if i = j then qi i ∼ qj j and that every equivalence class of ℘/ ∼ is represented in Q. (⇐) Suppose Q consists of exactly one element from each equivalence w(q ) class of ℘/ ∼. Then Q = {qi i }i∈A for some sequence {w(q1 ), w(q2 ), . . .} of posik i tive integers. Let x = i=1 pm (with relabeling of the prime divisors if necessary) i be an arbitrary element in M . Using m and n as defined in the first part of the k w(q ) proof, xmn = i=1 (qi i )di ci mi and clearly this representation is unique up to order. Using the property that an element of Q must be almost primary, we are able to characterize Krull monoids which are Cale monoids. We will require the following proposition. Proposition 3.6. Let M be a Krull monoid with divisor class group Cl(M ). Then x ∈ M is almost primary in M if and only if x is primary in M . Proof. (⇐) Follows by definition. (⇒) Let P represent the set of prime divisors of M . For p ∈ P, let | [p] | denote the order of [p] in Cl(M ). By [14, Satz 10A ii)] the primary elements of M are of the form x = pk with 1. | [p] |< ∞ and 2. | [p] | divides k. For an abelian group G and subset E ⊆ G, set E = {g1 + · · · + gs | gi ∈ E}. The proof hinges on the following result [2, Lemma 3.3]: if M is a Krull monoid with divisor class group Cl(M ) and associated set P of prime divisors, then for any p1 ∈ P, Cl(M ) = {[p] | p ∈ P and p = p1 }. Suppose x = pn1 1 pn2 2 · · · pnk k where the pi ’s are distinct prime divisors and each ni > 0. If | [p1 ] |= ∞, then k ≥ 2. We argue that such an x is not almost primary. By [2, Lemma 3.3], choose s1 , . . . , st ∈ P such that si = p1 for each i and [p1 ] = [s1 ] + · · · + [st ]. If y = sn1 1 · · · snt 1 pn2 2 · · · pnk k then y ∈ M and p1 is not a prime divisor of y. Using the same argument on the prime divisor p2 yields prime divisors r1 , . . . , rw (distinct n2 n3 p3 · · · pnk k then from p2 ) such that [p2 ] = [r1 ] + · · · + [rw ]. If z = pn1 1 r1n2 · · · rw again z ∈ M and p2 is not a prime divisor of z. Since x, y and z are in M , this n2 n3 p3 · · · pnk k ∈ M and xv = yz. Thus x | yz, implies that v = sn1 1 · · · snt 1 r1n2 · · · rw a but by construction, x divides no element of the form y b or z c . Hence, if x is divisible by a prime divisor whose image in the divisor class group is not torsion, mk 1 m2 where each then x cannot be almost primary. So suppose that x = pm 1 p2 · · · pk pi has torsion image in Cl(M ) and k ≥ 2. Using the notation of the Introduction, if w = lcm{n1 , . . . , nk } and ni di = w for each i, then x | q1m1 d1 · · · qkmk dk , but clearly xa divides no element of form (q1m1 d1 )b or (q2m2 d2 · · · qkmk dk )c . This completes the proof. Proposition 3.6 and the remarks following Definition 3.1 can be combined to show the following.

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Corollary 3.7. Let M be a Krull monoid with torsion divisor class group and suppose that x is a nonunit of M . The following statements are equivalent: 1. x is a strong atom. 2. x is an atom and primary. 3. x is an atom and almost primary. Example 3.8. While Proposition 3.6 shows that conditions 2 and 3 are equivalent in any Krull monoid, Corollary 3.7 is not true for all Krull monoids. Let M be a Krull monoid with Cl(M ) = Z such that the divisor class 1 contains the prime divisors p1 and p2 and the divisor class −1 contain prime divisors p3 and p4 . An example of such a Krull monoid is the set of solutions in nonnegative integers of the Diophantine equation x1 + x2 = x3 + x4 (see the latter part of Example 1.3). If x = p1 p3 in M , then clearly x is a strong atom since Cl(M ) is torsion-free. Setting y = p1 p4 and z = p2 p3 yields elements such that x | yz but x divides no power of y or z. Hence x is neither primary or almost primary. Theorem 3.9. For a Krull monoid M the following statements are equivalent. (1) M is a Cale monoid. (2) M is an inside factorial monoid. (3) Cl(M ) is a torsion group. Proof. (1) ⇒ (2) follows from Definition 1.1. For (2) ⇒ (3), suppose the divisor class group of M is not torsion. Suppose p1 is a prime divisor of M such that | [p1 ] |= ∞. There exists an x ∈ M with x = pn1 1 · · · pnk k where n1 > 0 and k ≥ 2. Then, for some m(x) ∈ N, xm(x) can be written uniquely as a product of elements in Q. By Proposition 3.6, Q contains no elements divisible by p1 , a contradiction. Hence Cl(M ) must be torsion. (3) ⇒ (1) follows from Example 1.4. Example 3.10. Lemma 2.1 part (iii) established that a Cale monoid is an extraction monoid. The converse of this statement is not true. It is easy to argue  n(x,pi ) that any Krull monoid M is an extractive monoid. In fact, if x = pi ∈P pi  n(y,pi ) and y = pi ∈P pi where P is the set of prime divisors of M , then λ(x, y) = n(y,pi ) min{ n(x,p | for all pi ∈ P with n(x, pi ) > 0}. By Theorem 3.9, any Krull monoid i) with class group not a torsion group is an extraction monoid which is not a Cale monoid.

4

Characterizations of Cale Monoids and Cale Domains

In Section 3, we characterized Cale monoids M which are Krull monoids. In this section, we extend this characterization and formulate it also for integral domains. We open with a characterization based on extraction within a monoid M . Some definitions are required. We note that a monoid M is said to be of finite type if increasing chains of rad(x) = {y ∈ M | x | y n for some n ∈ N} ideals are stationary.

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If y ∈ rad(x) then we say that x is a component of y. The nonunit element x is extractive if λ(x, •) is additive. Finally, x is discrete if n(x)λ(x, •) ∈ N0 for some n(x) ∈ N. Theorem 4.1. [19, Theorem 2] A monoid M is a Cale monoid if and only if (i) M is an extraction monoid of finite type (ii) each nonunit of M has an extractive and discrete component. If M is a Cale monoid with base Q, then for each q ∈ Q define fq : F → N0 by fq (x) = e(q)x(q) m(x) (see Definition 1.1). Definition 4.2. Let M be a monoid. The root closure M of M in G is defined as M = {w ∈ G | wk ∈ M for some k ≥ 1}. Further, if M is a Cale monoid with base Q, then set  = { x ∈ G | fq ( x ) ≥ 0 for all q ∈ Q}. M y y . Lemma 4.3. If M is a Cale monoid with base Q, then M = M  ⊆ M . Let x ∈ M  (i.e., fq (x)−fq (y) ≥ 0 for all q ∈ Q) Proof. We first argue that M y and hence n(q)(λ(q, x) − λ(q, y)) ≥ 0 for all q ∈ Q. Thus, λ(q, x) ≥ λ(q, y) for all x(q) y(q) ≥ m(y) for all q ∈ Q (i.e., m(y)x(q) ≥ m(x)y(q) q ∈ Q and by Lemma 2.1 (i), m(x) for all q ∈ Q). Thus    xm(x)m(y) = q x(q)m(y) = q m(x)y(q) q x(q)m(y)−m(x)y(q) = y m(x)m(y) · z q∈Q

q∈Q

q∈Q

 ⊆ M . For the where z ∈ M . Hence (x/y)m(x)m(y) ∈ M (i.e., (x/y) ∈ M ) and M x k reverse containment, let w = y ∈ G with w ∈ M for some k ≥ 1. By noting k

k

for every nonunit x ∈ M that (xm(x) )km(x ) = ((xk )m(x ) )m(x) , we obtain that xk (q) k = kx(q) m(x) for every q ∈ Q. This implies that kfq (w) = fq (w ) ≥ 0 for all m(xk ) . q ∈ Q. Thus fq (w) ≥ 0 for all q ∈ Q and x ∈ M y

Theorem 4.4. For a monoid M with quotient group G the following statements are equivalent. 1) M is a Cale monoid with base Q. 2) The root closure M of M is a Krull monoid with torsion divisor class group.  is contained in G. If w ∈ G, Proof. 1) ⇒ 2) Obviously the quotient group of M x , w is in the quotient group of then w = y with x and y in M . Since M ⊆ M   ) = G. We have that M . By Lemma 4.3, M = M and hence quot(M ) = quot(M  = {w ∈ G | fq (w) ≥ 0 for all q}. Finally, {q ∈ Q | fq (w) = 0} is finite for w ∈ G. M To see this, notice that by Lemma 2.1 (i) λ(q, x) =

x(q) m(x)

= 0 only for finitely many

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q, and hence n(q)(λ(q, x) − λ(q, y)) = 0 only for finitely many q ∈ Q. Thus, the set F = {fq | q ∈ Q} forms a defining family for M and M is a Krull monoid. To complete the argument, let F  = {fq ∈ F | fq is an essential state of M }, and for each fq ∈ F  , let fˆq be the normalization of fq . If I represents the essential (I) normalized states of M and ϕ : M → N0 defined by ϕfˆq (x) = fˆq (x), then Cl(M ) ∼ = (I)   ˆ ˆ
N0 /ϕ(M ). Let q be an element of Q such that fq ∈ I. If q ∈ Q, q = q and fq ∈ I, (I) then fˆq
(q) = 0 and fˆq (q) = z > 0. Let eq be the element of N0 which is 1 in fˆq coordinate and zero elsewhere. Then z · eq = ϕ(q) which means that in the divisor class group, z[eq ] = 0. Thus [eq ] is a torsion element of Cl(M ). Since these elements generate the divisor class group, Cl(M ) is a torsion group. 2) ⇒ 1) Let M be a monoid whose root closure M is a Krull monoid with torsion divisor class group. By our remarks in the Introduction, M is a Cale monoid. Let Q be the Cale base for M given in the Introduction. By [11, Lemma 5.4], {1} = M × =  × x(q ) M ∩ M . Let x ∈ M ⊆ M . Then xm(x) = u · qi ∈Q qi i , where u is a unit in M x(q )

and each qi i ∈ M . For each i ∈ A, let ni be a positive integer such that qini ∈ M and n0 be a positive integer such that 1 = un0 ∈ M . If n = lcm({n  i }i∈A , n0 ), then set n = ni ti (for i ∈ A) and n = n0 t0 . Thus, xm(x)n = (un0 )t0 qi ∈Q (qini )x(qi )ti =  ni x(qi )ti and clearly this representation is unique with respect to Q = qi ∈Q (qi ) ni {qi | i ∈ A}. Thus Q forms a base for M and hence M is inside factorial. Setting f (qini ) = f (qi )ni also yields that Q is tame. Hence, M is a Cale monoid. Corollary 4.5. Let M be a Cale monoid. If M is a monoid such that M ⊆ M ⊆ M , then M is Cale monoid. Proof. Since M is itself root closed ([10, p. 680]), M = M . By Theorem 4.4, M is a Krull monoid with torsion divisor class group. The converse of Theorem 4.4 completes the argument. Example 4.6. We can use Theorem 4.4 to construct an example of a Cale monoid H with base Q where Q does not consist entirely of primary elements. Let M be a Krull monoid with Cl(M ) = Z3 and countably many prime divisors in each nonzero divisor class (for instance, one could choose a reduced algebraic ring of integers with class number 3 modulo the monoid generated by its prime elements). By Theorem 4.4, M is a Cale monoid. Let a, b and c be distinct prime divisors of M taken from the divisor class ¯ 1. Consider the following atoms in M: x = a3 , y = b3 , z = c3 , 2 2 v = a b, u = ab , e = ac2 , f = b2 c. Let I represent the atoms of M . Let H be the monoid generated by I − {u}. Clearly H is contained in M . Notice that if w is in M and u does not divide w, then w is also in H. Further, u3 is in H (u3 = a3 (b3 )2 = xy 2 ). Hence, if w is in M , then either w is in H or w3 is in H. Thus M contains the root closure of H. Since M itself is root closed, the root closure of H equals M . Further, M and H have isomorphic quotient groups and hence H is a Cale monoid. Let {pi }i∈A be the complete set of prime divisors of M . By the proof of 2) implies 1) in Theorem 4.4, the set {(pi )3 }i∈A is a Cale base for H. We argue that x = a3 is almost primary, but not primary in H. To see this, we have that x divides

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v 2 z (indeed, (a2 b)2 c3 = (a3 )(ac2 )(b2 c) and v 2 z = xef ), but x does not divide v 2 (v 2 = a4 b2 = a3 (ab2 ) and u = ab2 is not in H). Clearly x does not divide any power of z. Thus x is not primary. By Lemma 2.1 part (iv), x is almost primary. Theorem 4.4 allows for the construction of some interesting integral domains which satisfy the Cale property. Example 4.7. Here is perhaps the simplest construction of a half-factorial domain not involving an algebraic number ring. Let K be any field and A ⊆ K. By [3, Theorem 5.3], R = A + XK[X] is half-factorial if and only if A is a subfield of K. Now, if K = F4 and A = F2 , then R = F2 + XF4 [X]. Notice that the quotient field of R is isomorphic to F4 (X) and that the root closure of R is R = F4 [X]. Since R is a factorial domain, R is a Cale domain by Theorem 4.4. This example easily generalizes to the case where K = Fpn and A is any subfield of K. Example 4.8. Let F be any finite field and set R = F [[X 2 , X 3 ]] and A = F [X 2 , X 3 ]. The integral closure and root closure of both R and A are unique factorization domains (F [[X]] and F [X] respectively). By Theorem 4.4, both R and A are Cale domains. √ Example 4.9. Let R = Z[ −3]. Then R is a nonintegrally closed half-factorial √ 1+ −3 domain whose integral closure and root closure (A = Z[ 2 ]) is a unique factorization domain [32, p. 285]. By Theorem 4.4, √ R is a Cale domain. But, not in √ R that all algebraic √ orders are Cale monoids. Let R = Z[ 17]. One can argue λ(2, 1 + √17) = 1 (see [19, pp. 149–150]) but there is no n ∈ N with 2n | (1 + 17)n . Thus Z[ 17] is not an extraction monoid and is hence not a Cale monoid. From Theorem 4.4, the following characterization of Krull monoids is obtained. Corollary 4.10. A monoid M is a Krull monoid with torsion divisor class group if and only if (i) M is a Cale monoid. (ii) M is root closed. Proof. (⇒) If M is Krull with torsion divisor class group, then (i) holds by Theorem 3.9. Since all Krull monoids are root closed, (ii) holds. (⇐) Suppose M is a monoid with properties (i) and (ii). By Theorem 4.4, M is a Krull monoid with torsion divisor class group. Example 4.11. An example for Corollary 4.10 is given by discrete valuation domains, which can be characterized as those domains whose multiplicative monoids satisfy (i) and (ii) with a base Q consisting of exactly one element. Condition (i) cannot be replaced by requiring M to be inside factorial only. Let D be a valuation domain with value group Q. Since D is integrally closed, it must also be root closed. Hence, D• is inside factorial with (ii), but D• is not a Krull monoid (because D is not atomic, see [8, Proposition 2.3]). We close this section by examining the situation for integral domains. The basis of a Cale domain is essentially determined by its minimal prime ideal structure.

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Theorem 4.12. [4, Theorem 2] For a Cale domain D a subset Q ⊂ D is a Cale basis if and only if q → rad(q) is a bijection of Q onto the minimal prime ideals of D• . In particular, a Cale basis is essentially uniquely determined. ˜ represent the integral closure of D. The If D is an integral domain, then let D following characterizes when D is Cale. Theorem 4.13. [4, Theorem 7] An integral domain D is a Cale domain if and only if ˜ is a Krull domain, (i) D ˜ is a torsion group, and (ii) Cl(D) ˜ is a root extension (i.e., for all x ∈ D ˜ there is an n ∈ N such that (iii) D ⊂ D n x ∈ D). Theorem 4.13 also characterizes inside factorial integral domains if “rational generalized” is inserted prior to the Krull statement in (i). Under the Noetherian hypothesis, the result above leads to the following important observation. Corollary 4.14. Let D be a noetherian domain. Then D is a Cale domain if and only if D is inside factorial.

5

Cale Representation in Particular Integral Domains and Monoids

5.1

Cale Representation in Algebraic Orders

In this section, we review some implications of the Cale property when considered for algebraic orders. By the results of the previous section, if K is an algebraic number field and D = OK the principal algebraic order in K, then D is a Cale domain. Proposition 3.5 implies that the primary atoms (up to equivalence) of D form a Cale basis. Moreover, any Cale basis of D consists of primary elements. Theorem 5.1. [18, Theorem] Let D be the principal order in an algebraic number field K. Then Cl(D) ∼ = Zpk where p is a prime integer if and only if there exists a k Cale base Q of D such that for any atom x of D, xp factors into at most pk factors of Q. Theorem 5.1 has several nice interpretations for principal orders of small class number. Corollary 5.2. Let D be the principal algebraic order in an algebraic number field K with class number h. 1. The following statements are equivalent. (a) h = 1. (b) Every atom of D is primary. (c) D is factorial.

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2. The following statements are equivalent. (a) h = 2. (b) The square of every atom of D is the product of two primary atoms of D. (c) D is a half-factorial domain. ˜ we have equivaFor arbitrary algebraic orders R ⊂ OK with integral closure R, lence of the Cale and inside factorial properties. Theorem 5.3. [4, Proposition 9] Let R ⊂ OK be an order in an algebraic number field K. The following statements are equivalent. (a) R is inside factorial. ˜ is a root extension. (b) R ⊂ R ˜ → spec(R) is bijective. (c) The map spec(R) (d) R is a Cale domain. (e) R is an extraction domain. (f) The elasticity of R is finite. Theorem 5.3 has been partly extended in a recent work of Martine Picavet– L’Hermitte (for the concepts of an almost B´ezout domain and an almost factorial domain see [1] and [31], respectively). Theorem 5.4. [22, Theorem 2.1] Let R be a 1–dimensional noetherian domain with torsion class group. The following conditions are equivalent. (a) R is inside factorial. ˜ is a root extension. (b) R ⊂ R (c) R is an almost B´ezout domain. (d) R is an almost factorial domain. ˜ is a finitely generated R–module and R is residually finite, then If in addition R (a)–(d) above are equivalent to ˜ → spec(R) is bijective. (e) The map spec(R) √ Example 5.5. Let R = Z[f ω] be a quadratic order of the field Q( d). As usual, √ √ ω = 12 (1 + d) if d ≡ 1 (mod 4) and ω = d if d ≡ 2, 3 (mod 4). Theorem 5.3 implies that R is a Cale domain if and only if ( D p ) = 1 for all primes p|f . For √ example, if d = −1 and f = 5, then R = Z[5 −1] is not a Cale domain because 5 is a product of Gaussian primes.

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159

Cale Representation in Diophantine Monoids

We focus in this section on the class of monoids introduced in Example 1.3. Let be a matrix with A ∈ Zm×n be a matrix with integer coefficients, B ∈ Nm×n 0 nonnegative integer coefficients, and c ∈ Rm×1 be a column vector with nonnegative + real entries. By a Diophantine monoid, we mean any additive monoid MA = {x ∈ Nn0 | Ax = 0} defined by equations or MB,c = {x ∈ Nn0 | Bx
c} ∪ {0} defined by inequalities. From [21, Theorem 4.1 & Theorem 4.4] we obtain the following propositions. Proposition 5.6. Let A ∈ Zm×n be a matrix for which MA = 0. The following statements are equivalent. 1. MA is a Cale monoid. 2. Cl(MA ) is a torsion group. 3. MA is isomorphic to a monoid MA
where each row of A has just one negative entry. be a matrix and c ∈ Rm×1 be a column vector Proposition 5.7. Let B ∈ Nm×n + 0 such that MB,c = 0. The following statements are equivalent. 1. MB,c is a Cale monoid. 2. MB,c = Nn0 . 3. If ci represents the ith entry of c, then ci > 0 implies that the ith row of B is positive. Example 5.8. Much of Proposition 5.6 can be demonstrated in the case where MA is a Diophantine monoid defined by one equation. In other words, A = [ a 1 , a2 , · · · a n ] and hence MA = {x ∈ Nn0 | a1 x1 + · · · + an−1 xn−1 + an xn = 0}. In the case where exactly one of the ai ’s is negative, then MA is a Cale monoid with base Q = {q1 , . . . , qn−1 } which assuming without loss that an < 0 and ai ≥ 0 for 1 ≤ i ≤ n − 1, is given by qi =

an ai ei + en . gcd{ai , an } gcd{ai , an }

Notice that Q consists of all primary atoms of MA . On the other hand, if A consists of at least two positive and at least two negative elements, then Cl(MA ) ∼ = Z [5, Theorem 2.3] and Proposition 5.6 implies that MA is not a Cale monoid.

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Cale Representation in Semigroup Rings

As with the beautiful Theorem of Gilmer and Parker [12] concerning unique factorization in semigroup rings, the Cale property has a nice characterization when S is reduced and torsion-free. An integral domain D is called solid if the root-closure of its multiplicative monoid D• is additively closed or, equivalently, if D = D• ∪ {0} is a subdomain of the quotient field of D (see Lemma 2.3 (iv)). Theorem 5.9. [21, Theorem 3.2] If D is an integral domain and S an additive reduced torsion-free monoid, then D[S] is a Cale domain if and only if 1. D is a Cale domain, 2. S is a Cale monoid, and 3. D[S] is solid. Corollary 5.10. [21, Corollary 3.3] Let D and S be as above with D integrally closed and S root-closed. Then D[S] is a Cale domain if and only if D is a Cale domain and S is a Cale monoid. In the case where D is factorial and S is a Diophantine monoid, then we can say the following. Theorem 5.11. [21, Theorem 4.1] Suppose that D is a factorial domain. 1. For a Diophantine monoid S given by equations, D[S] is a Cale domain if and only if S isomorphic to MA where each row of A has just one positive entry. 2. For a Diophantine monoid S given by inequalities where D[S] is solid, D[S] is a Cale domain if and only if ci > 0 implies that the ith row of B is positive. Example 5.12. Caution should be taken when considering Theorem 5.9. The hypothesis that D[S] be solid cannot be omitted. For example, let S = N0 \{1}. S is a numerical monoid and hence Cale. If D = R then D[S] is not solid, and therefore not a Cale domain. On the other hand, if D = F2 , then D[S] is solid, and hence a Cale domain (compare to Example 1.5).

6

Toric Ideals and Cale Varieties

In Section 1, Example 1.7, we mentioned two affine toric varieties, the quadric cone and the Neil parabola. Whereas the coordinate ring of the former is a Cale domain, this is not true for the latter. Nevertheless, in both cases the coordinate ring is generated by a Cale monoid S and both varieties are determined as the zero sets of the respective toric ideals belonging to S. Those toric varieties we call Cale varieties (see Definition 6.1 below). In the following we shall investigate this new type of toric variety and show, in particular, that they can be described by special polynomial equations given by a Cale base of S. Let D be an arbitrary integral domain and let S be any additive monoid that is finitely generated by atoms g1 , . . . , gl . The R´edei map Φ : Nl0 → S defined by Φ(a) =

l i=1

ai g i

(7.2)

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is a surjective homomorphism of additive monoids (this map appears in the proof of R´edei’s theorem that every finitely generated monoid is finitely presented [25]). The mapping Φ induces a surjective homomorphism of rings ra X a ) = ra X Φ(a) (7.3) Φ∗ : D[Nl0 ] → D[S] by Φ∗ ( where X a for a = (a1 , . . . , al ) denotes the monomial X1a1 X2a2 · · · Xlal . Therefore, D[S] ∼ = D[Nl0 ]/IS with IS = ker Φ∗ . The ideal IS is called the toric ideal of S for g1 , . . . , gl . Definition 6.1. The zero set of the toric ideal IS , V (IS ) = {x ∈ Dl | f (x) = 0 for all f ∈ IS }, is called the affine toric variety defined by S for g1 , . . . , gl . D[S] is called the coordinate ring with respect to S. For a Cale monoid S the affine toric variety V (IS ) is called a Cale variety. Remark 6.2. 1. Usually, the toric ideal is considered for D a field and S ⊂ Zd+ d or S ⊂ Z (see [28, Section 5.1], [29, Chapter 4]). The toric ideal is also called the presentation ideal [30, Chapter 7] and the kernel ideal [13, Section 7]. The latter is considered in a general framework and it is shown that IS is generated by the binomials X a − X b with a, b ∈ Nl0 , Φ(a) = Φ(b) ([13, Theorem 7.2]). Obviously, IS is already generated under the constraint on a, b ∈ Nl0 that min{ai , bi } = 0 for all i. If S is cancellative and torsion-free, then D[S] is an integral domain and the toric ideal is a prime ideal ([13, Theorem 8.1]). 2. The above definition of an affine toric variety follows [29, Chapter 4] where, however, D is assumed to be a field and S is assumed to be a monoid in Zd . A toric variety in the sense of [29, Chapter 4] (or even more in the sense of our Definition 6.1) need not be normal (see Theorem 6.8 below). When defining an affine toric variety in this generality one has to be careful. All notions in Definition 6.1 refer to a fixed S which, in general, can neither be recognized from V (IS ) nor from D[S]. For what follows, let S be an additive Cale monoid which is torsion-free with S × = {0}. For S finitely generated we may choose a Cale base consisting of atoms. Let g1 , . . . , gl be a fixed generating set of atoms and let Q = {g1 , . . . , gk } for k ≤ l be a Cale base of S. The Cale representation then yields mj g j =

k

cij gj for all k + 1 ≤ j ≤ l

(7.4)

i=1

where mj ∈ N is minimally chosen and the cij ∈ N0 are uniquely determined. Definition 6.3. Let αj = mj ej ∈ Nl0 , βj =

k i=1

cij ei ∈ Nl0 for k + 1 ≤ j ≤ l

(7.5)

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where eh is the hth unit vector in Nl0 . A binomial ±(X αj − X βj ) for k + 1 ≤ j ≤ l is called a base binomial for S in D[Nl0 ]. Let IQ ⊆ D[Nl0 ] be the ideal generated by all base binomials for S with Cale base Q = {g1 , . . . , gk }. Obviously, Φ(αj ) = mj gj and Φ(βj ) =

k 

cij gi by definition of Φ. The Cale

i=1

representation in S is, therefore, equivalent to Φ(αj ) = Φ(βj ) for all k + 1 ≤ j ≤ l. It follows that Φ∗ (X αj −X βj ) = X Φ(αj ) −X Φ(βj ) = 0 and, hence, all base binomials for S belong to the toric ideal IS . We shall show that the base binomials “almost” generate IS in the sense that IQ and IS coincide if and only if IQ is a prime ideal. Although the base binomials may not generate the toric ideal, we will nevertheless show that the Cale variety of S is given by the common zero set of all base binomials for S. For this we need the following binomial lemma which may be looked at as a generalization of the “basic formula” in [9, equation(2.1)]. Lemma 6.4 (Binomial Lemma). Let R be a commutative ring and ui , vi ∈ R, ki ∈ N0 for 1 ≤ i ≤ n. Then there exist wi ∈ R, 1 ≤ i ≤ n such that n 

uki i i=1

−

n  i=1

viki

=

n

(ui − vi )wi .

(7.6)

i=1

Proof. By induction over n. For n = 1, the assertion uk11 − v1k1 = (u1 − v1 )w1 k1  is true for w1 = uk11 −j v1j−1 ∈ R. Suppose, the assertion is true for n and set U=

n  i=1

uki i ,

V =

j=1 n  i=1

viki . For u, v ∈ R and k ∈ N0 we have that U uk − V v k = U (uk − v k ) + v k (U − V ).

By assumption U − V =

n 

(ui − vi )wi and, from n = 1, uk − v k = (u − v)w with

i=1

w ∈ R. Therefore

U uk − V v k =

n

(ui − vi )wi v k + (u − v)wU.

i=1

This proves the lemma. Now we are ready to prove the main result of this section, which states in particular that a Cale variety can be obtained as the zero set of base binomials. Theorem 6.5. Let D be an integral domain and S = g1 , . . . , gl  be a finitely generated Cale monoid which is reduced. Suppose further that Q is a fixed Cale Base of S. (i) There exists exactly one prime ideal P such that IQ ⊆ P ⊆ IS , namely P = IS . In particular, IQ = IS if and only if IQ is a prime ideal. (ii) The Cale variety for S equals the joint zero set of all base binomials for S.

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Proof. (1) In a first step, we show for a binomial X a − X b ∈ IS that we have ma +

l

mj (aj βj + bj αj ) = mb +

j=k+1

l

mj (aj αj + bj βj ),

(7.7)

j=k+1

a b where m = lcm{mk+1 , . . . , ml } and mj = mm−1 j . Since X − X ∈ IS implies l l   Φ(a) = Φ(b), by definition of Φ, it follows that aj g j = bj gj . Using (7.4) we j=1

j=1

obtain m

k

l

aj g j +

j=1

aj mj

k

cij gi = m

i=1

j=k+1

k

bj gj +

j=1

l

bj mj

j=k+1

k

cij gi .

i=1

Since Q = {g1 , . . . , gk } is a Cale base, this implies mai +

l

mj aj cij

l

= mbi +

j=k+1

mj bj cij for all 1 ≤ i ≤ k.

(7.8)

j=k+1

This shows that (7.7) holds for components with indices 1 ≤ i ≤ k. For a component with k + 1 ≤ i ≤ l equation (7.7) means that mai + mi (bi mi ) = mbi + mi (ai mi ), which holds trivially. (2) Next we show that for X a − X b ∈ IS it follows that (X ma − X mb )B (X ma − X mb )A

= =

X mb (A − B) ∈ IQ X ma (A − B) ∈ IQ

(7.9)

where A = X c and B = X d with c=

l

mj (aj αj

+ bj βj ) and d =

j=k+1

l

mj (aj βj + bj αj ).

j=k+1

From equation (7.7) we get X ma B = X mb A. Therefore, it suffices to show that A− B ∈ IQ . But this follows from the Binomial Lemma applied to the commutative ring R = D[Nl0 ]. Let n = 2(l − k) and set for 1 ≤ i ≤ l − k ui = X αk+i , vi = X βk+i , ul−k+i = vi , and vl−k+i = ui . Setting ki = ak+i mk+i and kl−k+i = bk+i mk+i for 1 ≤ i ≤ l − k, from Lemma 6.4 we obtain that A−B =

l

(X

αj

− X )fj +

j=k+1

with fj , gj ∈ D[Nl0 ]. Thus, A − B ∈ IQ .

βj

l j=k+1

(X βj − X αj )gj

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(3) To prove assertion (i) let P be a prime ideal in D[Nl0 ] such that IQ ⊆ P ⊆ IS . For a binomial X a − X b ∈ IS we show that it belongs to P , which proves (i) since IS is a prime ideal generated by these binomials. By step (2), (X ma − X mb )B ∈ IQ ⊆ P and, hence, X ma − X mb ∈ P or B ∈ P . The latter is impossible, because otherwise B ∈ IS (that is Φ∗ (B) = X Φ(d) would be the null polynomial). Therefore, X ma − X mb ∈ P . Now, if D has characteristic p = 0, it may happen that p divides m. Let m = pr q with r ≥ 0 and p  q. This notation covers also the case p = 0 by setting pr = 1. For a = pr a and b = pr b, we have that q X ma − X mb = (X a − X b ) X a(q−i) X b(i−1) . i=1

From Φ(a) = Φ(b) it follows that Φ(a) = Φ(b) and, hence, q q a(q−i) b(i−1) X )= X Φ(a)(q−i)+Φ(b)(i−1) = qX Φ(a)(q−1) Φ ( X ∗

i=1

i=1

which cannot be the null polynomial. Therefore,

q 

X a(q−i) X b(i−1) cannot be in P

i=1

and we must have that X a − X b ∈ P . In D[Nl0 ] it holds that r

r

r

X a − X b = (X a )p − (X b )p = (X a − X b )p

(which is valid also for p = 2), which implies that X a − X b ∈ P . (4) Concerning assertion (ii), from (i) we have that V (IS ) ⊆ V (IQ ). To show the reverse inclusion, let x ∈ V (IQ ). Since IS is generated by binomials X a − X b with Φ(a) = Φ(b), it suffices to show that xa = xb . From equation (7.9) we have that (xma − xmb ) · xc = 0 and (xma − xmb ) · xd = 0. If xc = 0 or xd = 0 then xma − xmb = 0 and, hence, xa = xb . We shall show that xc = 0 = xd implies that xa = 0 = xb . For this it is sufficient to show that xc = 0 implies xa = 0. By definition of c and d in step (2) together with (7.8) we have that ma + d = mb + c l l   and ci = mj bj cij , di = mj aj cij for all 1 ≤ i ≤ k as well as cj = maj j=k+1

j=k+1

and dj = mbj for k + 1 ≤ j ≤ l. Suppose xc = 0, that is xi = 0 and ci > 0 for at least one 1 ≤ i ≤ l. If k + 1 ≤ i then ci = mai and xai i = 0, which implies xa = 0. If i ≤ k then by the above ai > 0 or di > 0. If ai > 0 then again xa = 0. If di > 0 then there exists k + 1 ≤ j ≤ l such that aj > 0 and cij > 0. Because of x ∈ V (IQ ) m c c c m we have that xj j − x11j · · · xkkj = 0. From xi ij = 0 we get xj j = 0 and, hence, aj xj = 0. Thus, xa = 0. This proves (ii). The homogeneity of varieties (or ideals) is reflected by the property of halffactoriality of monoids (which is of its own interest and has been intensely studied by many authors). An affine variety V ⊆ Dn is homogeneous if for λ ∈ D\{0} and x = (x1 , . . . , xn ) ∈ V then λx = (λx1 , . . . , λxn ) ∈ V . A monoid is half-factorial if all representations of a nonunit by finitely many atoms have the same number of atoms. For a finitely generated half-factorial monoid S, this yields for the Red´ei map in equation (7.2) that Φ(a) = Φ(b) implies

l i=1

ai =

l i=1

bi .

(7.10)

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The following result shows that for Cale varieties homogeneity can be checked by the Cale base of the monoid. Corollary 6.6. Let D be an integral domain, let S be a finitely generated monoid (with fixed generators), and let V = V (IS ). Suppose that char(D) = 0 and that V contains a point x with xi = 0 for all 1 ≤ i ≤ n. Then (i) V is homogeneous if and only if S is half-factorial. (ii) For S a Cale monoid, V is homogeneous if and only if in equation (7.4) one k  cij for all k + 1 ≤ j ≤ l. has mj = i=1

Proof. (i) Since IS is generated by the binomials X a − X b , where Φ(a) = Φ(b), V (IS ) is homogeneous if and only if xa − xb = 0 implies that (λx)a − (λx)b for λ ∈ D\{0}. Because there is at least one x ∈ V with xa = 0, we have λa1 +···+al = λb1 +···+bl . Because D is an integral domain with char(D) = 0, the l l   ai = bi . latter is equivalent to i=1

i=1

(ii) Suppose now, S is a Cale monoid. Let Q = {g1 , . . . , gk } be a Cale base according to equation (7.4). Obviously, if S is half-factorial, equation (7.4) imk  cij for all k + 1 ≤ j ≤ l. Now, assume these equations hold. plies that mj = i=1

Then, of course, V (IQ ) is homogeneous because the base binomials generate IQ . From Theorem 6.5, part (ii), we obtain that V (IS ) = V (IQ ). Thus V (IS ) is also homogeneous. Remark 6.7. For char(D) = 0 neither (i) nor (ii) in Corollary 6.6 need to hold. Consider D = F2 and S the numerical monoid generated by 2 and 3 (see also Example 6.11 below). Then IS is generated by X13 − X22 and V (IS ) is homogeneous. The monoid S is a Cale monoid but not half-factorial. As remarked already, a Cale variety need not be normal. Thereby, V (IS ) is normal if D[S] is integrally closed. The following result derives criteria for normal Cale varieties from material in the previous section (recall by general assumption that S is cancellative, reduced, and torsion-free). Theorem 6.8. Let D = K be a field and let S be a finitely generated monoid (with generators fixed). The following statements are equivalent (a) The variety V (IS ) is normal and S is Cale. (b) S is Cale and root-closed. (c) S is isomorphic to a Diophantine monoid with a matrix having in each row just one positive entry or just one negative entry. (d) D[S] is a Cale domain and S is root-closed. Proof. (a) ⇒ (b): By assumption S is Cale and the coordinate ring K[Nl0 ]/IS ∼ = ¯ S¯ the rootK[S] is integrally closed. The integral closure of K[S] equals K[S], closure of S. Therefore, S¯ = S.

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(b) ⇒ (c): By [6, Corollary 3.3] a finitely generated monoid that is root-closed must be isomorphic to a Diophantine monoid. By [21, Theorem 4.1(i)] the latter must be isomorphic to a Diophantine monoid with a matrix as in (c). (c) ⇒ (d): By [21, Theorem 4.1(i)] K[S] is a Cale domain. As a Diophantine monoid S is root-closed. (d) ⇒ (a): By [21, Corollary 3.3], K[S] Cale domain and S root-closed implies ¯ = K[S]. that S is Cale. Furthermore, the integral closure of K[S] equals K[S] Therefore, D[S] is integrally closed. Remark 6.9. For affine toric varieties that are normal see [29, Proposition 13.5]. To this the characterization may be added that V (IS ) is normal if and only if S is a Diophantine monoid. The general results obtained we shall illustrate by a couple of examples and counterexamples. Example 6.10. Consider for n ≥ 3 the additive monoid S = {x ∈ Nn0 | x1 + · · · + xn−1 = bxn }

(7.11)

for b ∈ N fixed. Obviously, S is cancellative, torsion-free, and S × = {0}. Moreover, S is a Diophantine monoid, hence finitely generated, given by the matrix A = [1 ··· 1

− b ].

By [21, Theorem 4.1 (i)] S is a Cale monoid and, taking D to be a field, by Theorem 6.8 (c) the variety V for S is a normal Cale variety. Actually, for this example a Cale base can be easily computed, Q = {qi }1≤i≤n−1 with qi = bei + en for 1 ≤ i ≤ n − 1. Namely, x = (x1 , . . . , xn ) ∈ S has as Cale representation with respect to Q in the sense of Definition 1.1 m(x)x =

n−1

x(qi )qi with m(x) = b, x(qi ) = xi .

(7.12)

i=1

According to Theorem 6.5 (i), the Cale variety V equals the joint zero set of all base binomials. To determine the latter we have to compute the Cale representation (7.12) for all atoms g not in the Cale base. This may be difficult insofar as it is difficult in general to screen all atoms. Below we calculate the atoms in special cases. Nevertheless, from equation (7.12) we can conclude that for any atom g = (g1 , . . . , gn−1 , gn ) ∈ S which is not in Q the base binomial must be of the form g

n−1 Xgb − X1g1 · · · Xn−1 .

(7.13)

For example, since g = ei + (b − 1)ej + en , 1 ≤ i, j ≤ n − 1, is an atom not in Q for i = j, particular base binomials are given by Xgb − Xi Xjb−1 . For b = 2 these exhaust all base binomials and the variety for S is homogeneous and an intersection of quadric cones with just one quadric cone for n = 3 (Example 1.7). For b ≥ 3, however, other types of base binomials occur (see b = 3 below).

Cale Monoids, Cale Domains, and Cale Varieties

167

By Theorem 6.5 (ii) the Cale variety of S is the joint zero set of all binomials given by equation (7.13) — though the toric ideal is not generated by those binomials. Actually, the number of generators may be much bigger than the number of base binomials. The Cale variety for the Diophantine monoid given by equation (7.11) turns out to be the bth Veronese embedding of projective space P n−2 . The latter [29, Chapter 14] is the toric variety V (IT ) belonging to the monoid T generated by the set A of all with component sum equal g ∈ Nn−1 0   to b. Consider the mapping ϕ : T → S defined for x = cg g by ϕ(x) = (x, cg ). ϕ is a well-defined monoid homomorphism g∈A

g∈A

that is injective and has the property n−1

n−1



i=1

i=1

g∈A

ϕi (x) =

xi =

cg

n−1



i=1

g∈A

gi = b

cg = bϕ(x)n .

This shows that ϕ : T → S is a monoid isomorphism. In particular, consider for b = 3, n = 3 the twisted cubic curve which is defined by A = {(3, 0), (2, 1), (1, 2), (0, 3)} and has a toric ideal generated by binomials Y1 Y3 − Y22 , Y1 Y4 − Y2 Y3 , and Y2 Y4 − Y32 ([29, Example 1.2 (a)]). By the above, the monoid T generated by A is isomorphic to the monoid S = {x ∈ N30 | x1 + x2 = 3x3 }. The Cale base Q = {q1 , q2 } consists of q1 = 3e1 + e3 = (3, 0, 1) and q2 = 3e2 + e3 = (0, 3, 1). There are two more atoms g3 = (1, 2, 1) and g4 = (2, 1, 1) for which the Cale representation (7.12) is 3g3 = q1 + 2q2 and 3g4 = 2q1 + q2 . Therefore, the base binomials not in Q are according to equation (7.13) given by X33 − X1 X22 and X43 − X12 X2 , setting X3 = Xg3 , X4 = Xg4 . Thus, by Corollary 6.6 (ii) the variety for S is homogeneous. It can easily be seen by identifying X1 = Y1 , X3 = Y3 , X2 = Y4 , X4 = Y2 that the ideal IQ generated by binomials in X’s as above is contained in the ideal IT generated by the binomials in Y ’s and that IQ and IT do not coincide. Nevertheless, from Theorem 6.5 we have that V (IQ ) = V (IT ). According to [29, Theorem 3.3] the toric variety V (IT ) “is cut out by its circuits” which in the case of T are the generators of IT , with the exception of Y1 Y4 − Y2 Y3 , together with the two base binomials above taken in terms of the Y ’s. By Theorem 6.5 the latter are sufficient to cut out variety V (IT ). By the way, the monoid S = {x ∈ N30 | x1 + x2 = 3x3 } is isomorphic also to the monoid T = {x ∈ N30 | 2x1 + 5x2 = 3x3 } by ϕ : S → T, ϕ(x) = (x1 , x2 , x2 + 2x3 ). The monoid T is discussed in [21] where it is argued that V (IT ) is the joint zero set of X33 − X1 X22

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and X32 − X2 X4 ([21, last line]). This is implied by Theorem 6.5 (ii), because in D for x33 = x1 x22 the equation x23 = x2 x4 holds if and only if x34 = x21 x2 . It is, however, not the case that the toric ideal IT is generated by X33 − X1 X22 and X32 − X2 X4 . In general, the number of generators for the toric ideal IS is much higher than the number of base binomials which describe the variety V (IS ) in the sense of Theorem 6.5 (ii). Consider for example the monoid S given by equation (7.11) for b = 2. As seen already, the base binomials are Xg2 − Xi Xj for atom g = ei + (b − 1)ej + en , 1 ≤ i, j ≤ n − 1 and i = j. The Cale base Q consists of k = n − 1 atoms.  and the total number of atoms is l = The number of base binomials is n−1 2 n−1 n−1 n k + 2 = (n − 1) + 2 = 2 . For the case n = 4 and for 1 ≤ i ≤ 3 let Xi correspond to qi in the Cale base and let Xij correspond to g = ei +ej +e4 . The base 2 2 2 − X1 X2 , X13 − X1 X3 , X23 − X2 X3 and the remaining binomials are given by X12 binomials to generate IS are X1 X23 − X12 X13 , X12 X3 − X13 X23 , X13 X2 − X12 X23 . See also [29, p. 142] where this minimal system of generators has been computed for the corresponding Veronese embedding. Example 6.11. We present simple examples of Cale and non-Cale affine toric varieties. In the latter case, we exhibit Cale varieties which are not normal or have a coordinate ring which may or may not be a Cale domain (depending on the field). Consider a numerical semigroup S, that is the additive monoid S = g1 N0 + · · · + gl N0 with 1 < g1 < · · · < gl pairwise relative prime and l ≥ 2. We may assume the gi to be atoms in S. Because of gcd{g1 , . . . , gl } = 1 it follows that for every n ∈ N0 there exists k ∈ N such that kn ∈ S. Therefore, S¯ = N0 and from Theorem 4.4 we obtain that S is a Cale monoid. Actually, Q = {q} with q = g1 can serve as a Cale base and the Cale representation of x ∈ S with respect to Q is given by m(x)x = x(q)q with m(x) = q and x(q) = x. Thus, the variety V for S is a Cale variety. Since the gj with 2 ≤ j ≤ l are exactly the atoms not in the Cale base, the base binomials are given by g

Xjq − X1 j for 2 ≤ j ≤ l.

(7.14)

The zero set of such a base binomial is a generalized Neil parabola. By Theorem 6.5 (ii) the variety V for S is the intersection of the generalized Neil parabolas given by equation (7.14). Since q = g1 = gj for 2 ≤ j ≤ l it follows that V is never homogeneous (see also Corollary 6.6 (ii)). To illustrate Theorem 6.8, suppose now that D = K is a field. Because of S¯ = N0 but S  N0 , S is not root-closed. By Theorem 6.8, therefore, V is not normal. Concerning Theorem 6.8 (d) the question arises if the coordinate ring of V could still be a Cale domain. This depends on the field K. For simplicity consider the case of the common Neil parabola, that is l = 2, g1 = 2, g2 = 3 and the base binomial X22 − X13 . For K = C and K = R, K[S] is not solid and, hence, cannot be a Cale domain by Lemmas 2.1 and 2.3. For K = F2 , however, K[S] is solid and, hence, a Cale domain (see Section 5.3 and [21, Remark 3.4]). Geometrically, for K = C or K = R the singularity of V in 0 is cusp. For K = F2 , however, x22 − x31 = (x2 + x1 )(x2 − x1 ) and, hence, V has a double point singularity in 0 (as has the quadric cone the coordinate of which is a Cale domain). In the “standard toric varieties”, D is an algebraically closed field and the variety for D is normal. For those varieties, to be Cale means that the coordinate ring is

Cale Monoids, Cale Domains, and Cale Varieties

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a Cale domain. The above discussion shows that this may fail for varieties not being normal. Finally, a simple example of a standard variety which is not a Cale variety is given by C[S] for S the Diophantine monoid given by just one equation, S = {x ∈ N40 : x1 + x2 = x3 + x4 }. Obviously, S is root-closed and by Theorem 6.8 the variety of S cannot be a Cale variety and its coordinate ring cannot be a Cale domain.

Bibliography [1] D.D. Anderson and M. Zafrullah, Almost B´ezout domains, J. Algebra 142 (1991), 285–309. [2] D.D. Anderson, S. T. Chapman, F. Halter-Koch and M. Zafrullah, Criteria for unique factorization in integral domains, J. Pure Appl. Algebra 127 (1998), 205–218. [3] D.D. Anderson, D.F. Anderson and M. Zafrullah, Rings between D[X] and K[X], Houston J. Math. 17(1991), 109–129. [4] S.T. Chapman, F. Halter-Koch and U. Krause, Inside factorial monoids and integral domains, J. Algebra 252 (2002), 350–375. [5] S.T. Chapman, U. Krause and E. Oeljeklaus, Monoids determined by a homogenous linear Diophantine equation and the half-factorial property, J. Pure Appl. Algebra 151 (2000), 107–133. [6] S.T. Chapman, U. Krause and E. Oeljeklaus, On diophantine monoids and their class groups, Pac. J. Math. 207 (2002), 125–147. [7] D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms, New York, Springer Verlag, 1992. [8] J. Coykendall, D. Dobbs and B. Mullins, On integral domains with no atoms, Comm. Algebra 27 (1999), 5813–5831. [9] D. Eisenbud and B. Sturmfels, Binomial ideals, Duke J. Math 84 (1996), 1–45. [10] A. Geroldinger, On the structure and arithmetic of finitely primary monoids, Czech. Math. J. 46 (1996), 677–695. [11] A. Geroldinger and F. Halter-Koch, Arithmetical theory of monoid homomorphisms, Semigroup Forum 48 (1994), 333–362. [12] R. Gilmer and T. Parker, Divisibility properties in semigroup rings, Mich. Math. J. 21 (1974), 65–86. [13] R. Gilmer, Commutative Semigroup Rings, The University of Chicago Press, Chicago, 1984. [14] F. Halter-Koch, Halbgruppen mit Divisorentheorie, Expo. Math 8 (1990), 27– 66. 170

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[15] F. Halter-Koch, Divisor theories with primary elements and weakly Krull domains, Bol. Un. Mat. Ital. B 9 (1995), 417–441. [16] F. Halter-Koch, Ideal Systems, Marcel Dekker, New York, 1998. [17] U. Krause, Eindeutige Faktorisierung ohne ideale Elemente, Abh Braunschweigische Wiss Ges 33 (1982), 169–177. [18] U. Krause, A characterization of algebraic number fields with cyclic class group of prime power order, Math. Zeitschrift 186 (1984), 143–148. [19] U. Krause, Semigroups that are factorial from inside or from outside, Lattices, Semigroups and Universal Algebra, Plenum Press, New York, 1990, 147–161. [20] U. Krause, A general principle of marked extraction, Advances in Mathematical Systems Theory: A Volume in Honor of Diederich Hinrichsen, Birkh¨ auser, Boston, 2001, 169–183. [21] U. Krause, Semigroup rings that are inside factorial and their Cale representation, Rings, Modules, Algebras, and Abelian Groups, Lecture Notes in Pure and Appl. Math., Marcel Dekker 236 (2004), 353–363. [22] M. Picavet–L’Hermitte, Cale bases in algebraic order, Annales Math´ematiques Blaise Pascal 10 (2003), 117–131. [23] G. Picavet and M. Picavet–L’Hermitte, Trigonometric polynomial rings, Commutative Ring Theory and Applications, Lecture Notes in Pure and Appl. Math., Marcel Dekker 231 (2003), 419–433. [24] P. Ribenboim, Generalized power series rings, Lattices, Semigroups and Universal Algebra, New York, Plenum Press, 1990, 271–277. [25] J.C. Rosales and P. A. Garc´ıa–S´ anchez, Finitely Generated Commutative Monoids, New York, Nova Science Publishers, 1999. [26] P. Samuel, On unique factorization domains, Ill. Jour. Math. 5 (1961), 1–17. [27] R. Stanley, Combinatorics and Commutative Algebra, Birkhauser, Boston, 1983. [28] B. Sturmfels, Equations defining toric varieties, Proc. Sympos. Pure Math. 62 (1997), 437–449. [29] B. Sturmfels, Gr¨ obner Bases and Convex Polytopes, American Mathematical Society, Providence, 1996. [30] R. H. Villarreal, Monomial Algebras, New York, Marcel Dekker Inc, 2001. [31] M. Zafrullah, A general theory of almost factoriality, Manuscripta Math. 51 (1985), 29–62. [32] A. Zaks, Half-factorial domains, Israel J. of Math. 37(1980), 281–302.

Chapter 8

Weakly Krull Inside Factorial Domains Daniel D. Anderson, Muhammed Zafrullah, and Gyu Whan Chang by

1

Introduction

Chapman, Halter-Koch, and Krause [8] introduced the notion of an inside factorial monoid and integral domain. Throughout we will confine ourselves to the integral domain case, but the interested reader may easily supply definitions and proofs for the monoid case. An integral domain D is inside factorial if there exists a divisor homomorphism ϕ: F → D∗ = D − {0} where F is a factorial monoid and for each x ∈ D∗ , there exists an n ≥ 1 with xn ∈ ϕ(F ). They showed [8, Proposition 4] that an inside factorial domain may be defined in terms of a Cale basis and it will be this characterization that we use for the definition of an inside factorial domain. A subset Q ⊆ D∗ is a Cale basis for D if Q = {uqα1 · · · qαn | u ∈ U (D), qαi ∈ Q} is a factorial monoid with primes Q and for each d ∈ D∗ there exists an n ≥ 1 with dn ∈ Q. Here U (D) is the group of units of D. A domain D is inside factorial if and only if D has a Cale basis. They showed [8, Theorem 4] that D is inside factorial if and only if D, the integral closure of D, is a generalized Krull domain with torsion t-class group Clt (D), for each P ∈ X (1) (D), the valuation domain DP has value group order-isomorphic to a subgroup of (Q, +) (we say that D is rational), and D ⊆ D is a root extension (i.e., for each x ∈ D, there exists an n ≥ 1 with xn ∈ D).
Recall that an integral domain D is weakly Krull [4] if D = P ∈X (1) (D) DP where the intersection is locally finite. While an integrally closed inside factorial domain is weakly Krull, an inside factorial domain need not be weakly Krull (see Example 3.2 below). The purpose of this chapter is threefold. First we show (Theorem 3.1) that an inside factorial domain has a unique minimal root extension which is weakly Krull (and necessarily inside factorial). Second, we give (Theorem 3.3) a number of 172

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characterizations of weakly Krull inside factorial domains. For example, we show that for an inside factorial domain the following conditions are equivalent: (1) D is weakly Krull, (2) D is an almost GCD domain, (3) t-dim D = 1, and (4) for each element q of a Cale basis for D, (q) is primary. Third, Theorems 3.1 and 3.3 are then used to give a new proof that if D is inside factorial, then D is a rational generalized Krull domain with torsion t-class group and D ⊆ D is a root extension.

2

Preliminaries

In this section we review some results on Cale bases, AGCD domains, and root extensions. Let D be an inside factorial domain with Cale basis Q. Now by definition the elements map Q → X (1) (D) given by  of Q are primes in the monoid Q and the  q → (q) is a bijection [8, Theorem 2]. (In fact, if Q ⊆ D∗ with bijection as above, then Q is a Cale basis.) However, the elements of Q need not be irreducible nor primary as elements of D. Also, recall that an integral domain D is an almost GCD domain (AGCD domain) [10] if for x, y ∈ D∗ , there exists an n ≥ 1 with xn D ∩ y n D, or equivalently (xn , y n )v , principal. An AGCD domain D has torsion t-class group [5, Theorem 3.4], D ⊆ D is a root extension [10, Theorem 3.1], and D is a PVMD with torsion t-class group [10, Theorem 3.4 and Corollary 3.8]. Now a PVMD with torsion tclass group is an AGCD domain [10, Theorem 3.9], but if D is an integral domain with D ⊆ D a root extension and D a PVMD with torsion t-class group, or equivalently, D is an AGCD domain, D need not be an AGCD domain. The following example is [3, Theorem 3.1] (also see Example 3.2 of this chapter). Let K  L be a purely inseparable field extension and let D = K + (X)L[X] where X is a set of indeterminates with |X| > 1. Then D  D = L [X] is a root extension and L[X] is a UFD (and hence an AGCD domain), but D is not an AGCD domain. Note that if |X| = 1, D is an AGCD domain. The following results on root extensions will be frequently used. (1) [5, Theorem 2.1], [8, Proposition 5] Let R ⊆ S be a root extension of commutative rings. The map Spec(S) → Spec(R) given by Q → Q √ ∩ R is an order isomorphism and homeomorphism with inverse given by P → P = {s ∈ S | sn ∈ P for some n ≥ 1}. Moreover, RP ⊆ SQ is a root extension. (2) [8, Proposition 5] Let D ⊆ S be a root extension of integral domains. Then D is inside factorial if and only if S is inside factorial. Of course, in the case where D is inside factorial and S ⊆ K, the quotient field of D, D ⊆ S is a root extension if and only if D ⊆ S is integral. We have remarked that an inside factorial domain need not be weakly Krull. The previously mentioned example D = K +(X)L[X] where K  L is purely inseparable and |X| > 1 is such an example, see Example 3.2 below. In [6, Proposition 2.4] we showed that the following conditions are equivalent for an inside factorial domain  D with Cale basis Q: (1) for each q ∈ Q, (q) is a maximal t-ideal, (2) D is weakly Krull, (3) D is an AGCD domain, (4) each q ∈ Q is a primary element, and (5) distinct elements of Q are v-coprime. See Theorem 3.3 below where these and several more equivalences are given.

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Weakly Krull Inside Factorial Domains

We begin this section by showing that an inside factorial domain has a unique minimal root extension which is weakly Krull. We then give an example of an inside factorial domain that is not weakly Krull and give a number of characterizations of weakly Krull inside factorial domains.
Theorem 3.1. Let D be an inside factorial domain and let D# = P ∈X (1) (D) DP . (1 ) D ⊆ D# is a root extension and hence D# is inside factorial. (2 ) D# is weakly Krull. (3 ) Let S be an integral domain containing D such that S is weakly Krull and D ⊆ S is a root extension. Then D# ⊆ S. (4 ) Suppose that S is an integral domain with D# ⊆ S a root extension. Then S is a weakly Krull inside factorial domain. Proof. Here D is an inside factorialdomain with quotient field K. Let Q = {qα } be a Cale basis for D and let Pα = (qα ). So Q → X (1) (D) given by qα → Pα is a bijection. (1) Let 0 = x ∈ D# . Write x = a/b where a, b ∈ D∗ . Since suitable powers of a and b lie in Q, we can choose n ≥ 1 with xn = qαn11 · · · qαnss where ∈ U (D) and each ni ∈ Z∗ . Suppose that some ni < 0. With a change of notation, if necessary, we can assume that n1 , · · · , ni < 0 and ni+1 , · · · , ns > 0. Now xn ∈ DPαi , so qαn11 · · · qαnss = r/t where r ∈ D and t ∈ D − Pαi . Then ni+1 1 i tqαi+1 · · · qαnss = rqα−n · · · qα−n ∈ Pαi . But no factor on the left hand side lies in 1 i Pαi , a contradiction. Hence each ni > 0, so xn ∈ D. Since D ⊆ D# is a root extension, D# is inside factorial. (2) For N ∈ X (1) (D# ), let S = D# − N and P = N ∩ D. Since D ⊆ D# is a root # extension, P ∈ X (1) (D). Now D# ⊆ DP gives DN ⊆ (DP )S = DP where the # , so equality follows since ht PP = 1 and PP ∩ S = ∅. But certainly DP ⊆ DN

# # . Thus D# ⊆ N ∈X (1) (D# ) DN = P ∈X (1) (D) DP = D# and hence DP = DN
# . Let 0= x ∈ K. Then x is a unit in almost all the D# = N ∈X (1) (D# ) DN  DPα ’s. This follows since 0 = a ∈ D is in Pα = (qα ) if and only if an ∈ Q has qα as one of its factors and thus a is in only finitely many Pα . Since each # # DN is equal to a unique DPα , x is a unit in almost all the DN ’s. Hence D# is weakly Krull.
(3) Suppose that D ⊆ S is a root extension and that S = N ∈X (1) (S) SN . For N ∈
X (1) (S), P = N ∩ D ∈ X (1) (D); so DP ⊆ SN . Then D# = P ∈X (1) (D) DP ⊆
N ∈X (1) (S) SN = S. extension. (4) Suppose that D# ⊆ S is a root extension. Then D ⊆ S is also a root  # # Hence Q is a Cale basis for D and S. Since D is weakly Krull, qα D# ∈ √ We claim that qα S is qα SX (1) (D# ) gives that qα D# is qα D# -primary. √ primary. Suppose that xy ∈ qα S and x∈ / qα S. Choose n ≥ 1 with xn , y n ∈ # n n # n / qα D# , so for some m ≥ 1, (y n )m ∈ D and x y ∈ qα D . Then x ∈ # qα D ⊆ qα S. Thus S has a Cale basis Q = {qα } with each qα S primary. By the remarks preceding Theorem 3.1, S is a weakly Krull inside factorial domain.

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Of course, for D inside factorial, D = D# if and only if D is weakly Krull. Note that by [8, Proposition 5(f)] D is tamely inside factorial if and only if D# is. We next give an example of an inside factorial domain that is not weakly Krull. Example 3.2. Let D = K + (X)L[X] where K ⊆ L is a purely inseparable field extension and X is a nonempty set of indeterminates over L. Then D ⊆ D = L[X] is a root extension since K ⊆ L is purely inseparable and D being factorial is inside factorial. Thus D is inside factorial. Alternatively, if we take Q to be a complete set of nonassociate prime elements of K[X], then Q is a Cale basis for D and hence D is inside factorial. Suppose that |X| = 1. Then dim D = 1, so D = D# and hence D is weakly factorial. Suppose that |X| > 1. We claim that D# = L |X|. Let P be a height-one prime ideal of D; so P = f L[X] ∩ D where f ∈ L[X] is irreducible. Choose g ∈ D − P with g(0)=0. ( If |X| = 1 and f = X, we can’t choose such a g. But suppose |X| > 1. If f ∈ X, take any g ∈ X − {f } while if f ∈ / X, take g = X ∈ X.) Now for a ∈ L, a = ag/g ∈ DP . Hence L ⊆ D# , so D# = L[X]. We next give a number of conditions equivalent to D being a weakly Krull inside factorial domain. But first we need some more definitions. An integral domain D is almost weakly factorial [4] if some power of each nonunit of D is a product of primary elements. By [4, Theorem 3.4], D is almost weakly factorial if and only if D is weakly Krull and Clt (D) is torsion. An integral domain D is a generalized weakly factorial domain [7] if each nonzero prime ideal of D contains a nonzero primary element. Thus an almost weakly factorial domain is a generalized weakly factorial domain. Two elements x and y of an integral domain D are power associates if n m there exist n, m ≥ 1 and a unit ∈ U (D)  with x = y . Given a nonzero prime ideal P of D, b is a base for P if P = (b). Thus if (b) is P -primary, b is a base for P . Finally, t-dim D = 1 if every prime t-ideal of D is a maximal t-ideal. Theorem 3.3. For an integral domain D the following conditions are equivalent. (1 ) D is inside factorial and weakly Krull. (2 ) D is inside factorial and AGCD. (3 ) D is inside factorial and t-dim D = 1. √ (4 ) D is inside factorial with a Cale basis Q such that qD is a maximal t-ideal for each q ∈ Q. √ (5 ) D is inside factorial and for each Cale basis Q for D, qD is a maximal t-ideal for each q ∈ Q. (6 ) D is inside factorial with a Cale basis Q such that each q ∈ Q is primary. (7 ) D is inside factorial and for each Cale basis Q for D and each q ∈ Q, q is primary. (8 ) D is inside factorial with a Cale basis Q such that distinct elements of Q are v-coprime. (9 ) D is inside factorial and for each Cale basis Q for D distinct elements of Q are v-coprime. (10 ) D is almost weakly factorial and two nonzero primary elements of D are either v-coprime or power associates. (11 ) D is a weakly Krull AGCD domain and two nonzero primary elements of D are either v-coprime or power associates.

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(12 ) D is a generalized weakly factorial domain such that for each height-one prime ideal P of D, every pair of base elements of P are power associates and for each nonzero x ∈ P , there is an n ≥ 1 with xn = db where b is a base for P and d∈ / P. Proof. The equivalence of (1), (2), and (4)–(9) √ is given by [6, Proposition 2.4]. (3)⇐⇒(4) For D inside factorial, X (1) (D) = { qD | q ∈ Q} for any Cale basis Q for D. Since a height-one prime ideal is a t-ideal, the equivalence follows. (6)=⇒(10) Let x be a nonzero nonunit of D. Then some power xn ∈ Q. So xn is a product of primary elements and hence D is almost factorial. Suppose that q1 and q2  weakly are nonzero primary elements of D. If (q ) =  (q2 ), then (q1 ,  q2 )v = D (for say 1   by (q1 ) = (q2 ) = 2 ) are maximal t-ideals). So suppose  (6)=⇒(3) (q1 ) and (q (q) where q ∈ Q. Then q1n1 = 1 q m1 and q2n2 = 2 q m2 for some n1 , n2 , m1 , m2 ≥ 1 and units 1 , 2 . But then q1 and q2 are power associates. (10)=⇒(6) Since D is almost weakly factorial, D is weakly Krull and each height-one prime Pα contains a Pα -primary element qα . Let Q = {qα }. We claim that Q is a Cale basis for D. If 1 qαn11 · · · qαnss = 2 qαm11 · · · qαmss where 1 , 2 are units and ni , mi ≥ 0, then qαnii DPαi = 1 qαn11 · · · qαnss DPαi = 2 qαm11 · · · qαmss DPαi = qαmii DPαi ; so ni = mi . Hence Q is a factorial monoid. Let x be a nonzero nonunit of D. Then some power xn is a product of primary elements. But each primary element in this factorization is power associate to some qα . Thus raising xn to an appropriate power gives that some power of x is in Q. (11)=⇒(10) It suffices to show that an AGCD weakly Krull domain is almost weakly factorial. But we have already remarked that an AGCD domain has torsion t-class group and that a domain is almost weakly factorial if and only if it is a weakly Krull domain with torsion t-class group. (6)=⇒(11) Since (6)=⇒(1), D is weakly Krull and since (6)=⇒(2), D is an AGCD domain. And by (6)=⇒(10) two nonzero primary elements are either v-coprime or power associates. (10)=⇒(12) Certainly an almost weakly factorial domain is a generalized weakly factorial domain. Now an almost weakly factorial domain is weakly Krull and in  (x) = P  , P  a height-one prime, is a weakly Krull domain an element x with

primary (for (x) = x P ∈X (1) (D) DP = P ∈X (1) (D) xDP = xDP
∩ D). Thus two bases for a height-one prime ideal P are P -primary and hence by hypothesis are power associates. Finally, let 0 = x ∈ P . Since D is almost weakly factorial, some xn is a product of primary elements, say xn = q1 · · · qi qi+1 · · · qm where q1 , · · · , qi ∈ P (with necessarily i ≥ 1) and / P . Now (q1 · · · qi ) is still P -primary  qi+1 , · · · , qm ∈ (for D is weakly Krull and (q1 · · · qi ) = P ). Take b = q1 · · · qi and d = qi+1 · · · qn ; so xn = db where d ∈ / P and b is a base for P . (12)=⇒(1) A generalized weakly factorial domain is weakly Krull [7, Corollary 2.3]. For each P ∈ X (1) (D) choose a base element xP for P. We claim that Q = {xP } is a Cale basis for D; and hence D is inside factorial. As in the proof of (10)=⇒(6), Q is a factorial monoid. Let x be a nonzero nonunit of D. Let P1 , · · · , Pm be the height-one prime ideals of D containing x. By hypothesis, there is an n ≥ 1 so that xn = db where b is a base for P1 and d ∈ / P1 . Now b and xP1 are power associates, say br = uxsP1 where r, s ≥ 1 / P1 . Now P2 , · · · , Pm are and u is a unit. Then xnr = dr uxsP1 = x1 xsP1 where x1 ∈ the height-one prime ideals containing x1 . By induction, we have xt1 = βxsP22 · · · xsPmm where β is a unit and t, s2 , · · · , sm ≥ 1. Hence xtnr ∈ Q.

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4

177

Miscellaneous Results

We have already remarked that an integral domain D is inside factorial if and only if D is a rational generalized Krull domain with torsion t-class group and D ⊆ D is a root extension [8]. We begin this section by giving an alternative proof of the implication (=⇒) based on Theorems 3.1 and 3.3. We then discuss when certain extensions of an inside factorial domain are again inside factorial. Theorem 4.1. [8, Theorem 4] An integral domain D is inside factorial if and only if D is a rational generalized Krull domain with torsion t-class group and D ⊆ D is a root extension. Proof. (=⇒) Suppose that D is inside factorial. By Theorem 3.1, D ⊆ D# is a root extension and D# is a weakly Krull inside factorial domain. By Theorem 3.3 D# is an AGCD domain. Thus we can apply results from [10] concerning AGCD domains. By [10, Theorem 3.1], D# ⊆ D# = D is a root extension. Hence D ⊆ D is a root extension. By [10, Theorem 3.4 and Corollary 3.8] D is a PVMD with torsion t-class group. But by Theorem 3.1 again, D is weakly Krull. Hence D is a generalized Krull domain. It remains to show that for each P ∈ X (1) (D), the value group of D P is isomorphic to a subgroup of (Q, +). Let Q be a Cale basis for D with P = (p) where p ∈ Q. For 0 = x ∈ K, there is a natural number n with xn = pn0 q1n1 · · · qsns where n0 , · · · , ns ∈ Z, ∈ U (D) and q1 , · · · , qs ∈ Q. But q1 , · · · , qs are units in DP , so we can write xn =  pn0 where  is a unit in DP . If v: K ∗ → (R, +) is the valuation for DP , then nv(x) = v(xn ) = v(  pn0 ) = n0 v(p). Hence v(x) = nn0 v(p). Thus im v is order-isomorphic to (Q, +). (⇐=) [8, Theorem 4] We end by considering extensions of inside factorial domains. Let D be inside factorial. Then each overring E of D contained in D is again inside factorial (for D ⊆ E is a root extension). Moreover, by Theorem 3.1 if D is weakly Krull, so is E. Next suppose that S is a multiplicatively closed subset of D. Then D ⊆ D a root extension gives that DS ⊆ DS = DS is a root extension, DS is a rational generalized Krull domain, and DS has torsion t-class group since D does [2, Theorem 4.4]. Thus DS is inside factorial. Alternatively, observe that if Q is a Cale basis for D, then (1) {q ∈ Q | qD
S = DS } is a Cale basis for DS . Moreover, if ∅ = Λ ⊆ X (D), then R = P ∈Λ DP is a weakly Krull inside factorial domain. For if we set S =  

√ / Λ , then (D# )S = = P ∈Λ DP = R. q ∈ Q | qD ∈ P ∈X (1) (D) DP S

We next show that D[X] is inside factorial if and only if D is inside factorial and D[X] ⊆ D[X] is a root extension. Also see [9, Theorem 3.2]. Now D[X] is inside factorial if and only if D[X] ⊆ D[X] is a root extension and D[X] is a rational generalized Krull domain with torsion t-class group. But D[X] is a rational generalized Krull domain with torsion t-class group if and only if D is (for D is a generalized Krull domain if and only if D[X] is, the natural map Clt (D) → Clt (D[X]) is an isomorphism, and for N ∈ X (1) (D[X]), D[X]N is a DVR if N ∩D = 0 and if N ∩ D = 0, then D[X]N = DN ∩D (X) has the same value group as DN ∩D ) and D[X] ⊆ D[X] a root extension forces D ⊆ D to be a root extension. However, D ⊆ D√ a root extension need not imply the D[X] ⊆ D[X] is a root extension. Let D = Z[ 5]. Then D  D is a root extension and D is an AGCD (= weakly Krull) inside factorial domain, but D[X]  D[X] is not a root extension [1, Example 3.6].

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Hence D[X] is not inside factorial. According to [1, Theorem 3.4], for an integral domain D, D[X] is an AGCD domain if and only if D is an AGCD domain and D[X] ⊆ D[X] is a root extension. Hence if D[X] is inside factorial, D[X] is AGCD (= weakly Krull) if and only if D is.

Bibliography [1] D. D. Anderson, T. Dumitrescu, and M. Zafrullah, Almost splitting sets and AGCD domains, Comm. Algebra 32 (2004), 147–158. [2] D. D. Anderson, E. G. Houston, and M. Zafrullah, t-linked extensions, the tclass group, and Nagata’s Theorem, J. Pure Appl. Algebra 86 (1993), 109–124. [3] D. D. Anderson, K. R. Knopp, and R. L. Lewin, Almost B´ezout domains, II, J. Algebra 167 (1994), 547–556. [4] D. D. Anderson, J. L. Mott, and M. Zafrullah, Finite character representations for integral domains, Boll. Un. Mat. Ital. B (7 ) 6 (1992), 613–630. [5] D. D. Anderson and M. Zafrullah, Almost B´ezout domains, J. Algebra 142 (1991), 285–309. [6] D. D. Anderson and M. Zafrullah, A note on almost GCD monoids, Semigroup Forum 24 (2004), 811–828. [7] D. F. Anderson, G. W. Chang, and J. Park, Generalized weakly factorial domains, Houston J. Math 29 (2003), 1–13. [8] S. Chapman, F. Halter-Koch, and U. Krause, Inside factorial monoids and integral domains, J. Algebra 252 (2002), 350–375. [9] U. Krause, Semigroup rings that are inside factorial and their Cale representation, Rings, Modules, Algebras, and Abelian Groups (A. Facchini, E. Houston, and L. Salce, eds.), Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 236 (2004), 353–363. [10] M. Zafrullah, A general theory of almost factoriality, Manuscripta Math. 51 (1985), 29–62.

179

Chapter 9

The m-Complement of a Multiplicative Set by

David F. Anderson and Gyu Whan Chang Abstract Let D be an integral domain. For any ∅ = S ⊆ D \ {0}, we define the mcomplement for S (in D) to be ND (S) = {0 = x ∈ D|xD ∩ sD = xsD for all s ∈ S}. In this chapter, we investigate the relationship between S and ND (S).

1

Introduction

Let D be an integral domain with quotient field qf (D), nonzero elements D∗ = D\{0}, and group of units U (D). For any ∅ = S ⊆ D∗ , we define the m-complement for S (in D) to be ND (S) = {x ∈ D∗ |xD ∩ sD = xsD for all s ∈ S}. Then ND (S) is a saturated multiplicative subset of D. When the context is clear, we write N (S) for ND (S). As in [1], a saturated multiplicative subset S of D is called a splitting set if SN (S) = D∗ . It is well-known, and easily verified, that if S is a splitting set, then N (S) is also a splitting set, S ∩ N (S) = U (D), N (N (S)) = S, and D = DS ∩ DN (S) . As an example, U (D) is a splitting set with N (U (D)) = D∗ , and thus D∗ is a splitting set with N (D∗ ) = U (D). Also, if S is a splitting set, then Cl(D) = Cl(DS ) ⊕ Cl(DN (S) ) [1, Corollary 3.8], where Cl(D) is the t-class group of D (see [6]). In [7], we investigated when Cl(D) = Cl(DS ) ⊕ Cl(DN (S) ) in the case that SN (S) = D \ P for some prime ideal P of D. For other properties and applications of splitting sets, see [1], [2], [3], [5], and [9]. The m-complement of S has also been studied in [3], where they used the notation S ⊥ for N (S). In this chapter, we investigate the relationship between S and N (S). We give some basic properties for N (S) and compute NR (S) for certain ring extensions R of D. Several of these results recover as special cases earlier results when S is a splitting set. As usual, Iv = (I −1 )−1 and It = ∪{(a1 , . . . , an )v |0 = (a1 , . . . , an ) ⊆ I} for a nonzero fractional ideal I of D, and I is a t-(resp., v-)ideal of D if It = I (resp., Iv = I). Note that for x, s ∈ D∗ , xD ∩ sD = xsD if and only if (x, s)v = D; so ND (S) = {x ∈ D∗ |(x, s)v = D for all s ∈ S}. We will often use the fact that 180

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(x, s)v = D if and only if (x, s) is contained in no prime t-ideal of D. Any undefined notation is standard, see [10].

2

Basic Properties of the m-Complement

In this section, we give some basic properties of the m-complement and several examples. Our first lemma is implicit in [1] and shows that N (S) is a saturated multiplicative subset of D (cf.[3, Proposition 2.4] and [5, page 2210]). Moreover, N (S) = N (S) = N (S), where S = {ux1 · · · xn |u ∈ U (D) and xi ∈ S} is the multiplicative subset of D generated by S and the saturation of a multiplicative subset T of D is T = {x ∈ D|xy ∈ T for some y ∈ D}. We may thus assume that S is a (saturated) multiplicative subset of D. The proofs of these results are routine, and hence are omitted. Lemma 2.1. Let D be an integral domain. Then the following statements are equivalent for all x, y, z ∈ D∗ . 1. xD ∩ zD = xzD and yD ∩ zD = yzD. 2. xyD ∩ zD = xyzD. Proposition 2.2. Let D be an integral domain and ∅ = S ⊆ D∗ . Then N (S) is a saturated multiplicative subset of D. Moreover, N (S) = N (S) = N (S). Let Σ be the set of all saturated multiplicative subsets of D and P(D∗ ) the power set of D∗ . By Proposition 2.2, we have a mapping N : P(D∗ ) \ {∅} → Σ which restricts to a mapping Σ → Σ. We next list several properties of the N operator. These properties all follow directly from definitions and Lemma 2.1, and they will be used without further reference ((1), (2), (3), and (4) have been observed in [3, Proposition 2.4], and part (8) is a special case of [5, Proposition 1.1]). Proposition 2.3. Let D be an integral domain, S, S1 , and S2 nonempty subsets of D∗ , and {Sα } a family of nonempty subsets of D∗ . 1. If S1 ⊆ S2 , then N (S2 ) ⊆ N (S1 ). 2. S ∩ N (S) ⊆ U (D). (Equality holds if U (D) ⊆ S.) 3. S ⊆ N (N (S)). 4. N (S) = N (N (N (S))). 5. N (∪Sα ) = N (∪Sα ) = ∩N (Sα ). 6. N (S1 S2 ) = N (S1 ) ∩ N (S2 ). 7. If S1 ∩ S2 = ∅, then N (S1 )N (S2 ) ⊆ N (S1 ∩ S2 ). 8. D = DS ∩ DN (S) . We next give several examples to show that the inclusions in Proposition 2.3 may be proper.

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Example 2.4. (a) Let p be a prime element of an integral domain D and S = p = {upn |u ∈ U (D) and n ≥ 0}. Then N (S) = D \ pD and SN (S) = D \ Q, where Q = ∩pn D. More generally, for p1 , . . . , pm prime elements of D and S = p1 , . . . , pm  = {upk11 · · · pkmm |u ∈ U (D) and each ki ≥ 0}, N (S) = D \ (∪pi D), and SN (S) = D \ (∪Qi ), where Qi = ∩(pni D). In particular, S is a splitting set if and only if each Qi = {0}, i.e., each pi D has height-one (cf. [2, Proposition 1.6]). For an infinite set {pα |α ∈ A} of nonassociate prime elements of D, let S = {pα }. Then N (S) = D \ (∪pα D) and SN (S) = D \ [(∪Qα ∪ (∪JB )], where Qα = ∩pnα D and JB = ∩{pβ D|β ∈ B} for each countably infinite B ⊆ A. Thus S is a splitting set if and only if each Qα and each JB is {0} (cf. [2, Proposition 1.6]). (b) Let R = Z + XQ[X]. Then S = Z∗ is a saturated multiplicative subset generated by prime elements for both Z and R with NZ (S) = U (Z) and NR (S) = {a + Xf |a ∈ U (Z) and f ∈ Q[X]} (cf. Proposition 3.3(2)). However, X ∈ NR (NR (S)); so the inclusion in Proposition 2.3(3) may be proper even when S is a saturated multiplicative subset generated by prime elements (for another example, see [3, Example 2.5]). (c) The inclusion in Proposition 2.3(7) may be proper. For example, let S1 = {1, 2}, S2 = {1, 4}, and D = Z. Then N (S1 ∩ S2 ) = N ({1}) = Z∗ , while N (S1 )N (S2 ) = N (S1 ) = Z \ 2Z by part (a) above. For another example, let S be a nonsplitting multiplicative subset of an integral domain D such that S = N (N (S)). Then S ∩ N (S) = U (D), and hence N (N (S))N (S) = SN (S)  D∗ = N (S ∩ N (S)). However, if S1 and S2 are saturated multiplicative subsets of a UFD, then N (S1 )N (S2 ) = N (S1 ∩ S2 ). We next give a necessary and sufficient condition to have S = N (N (S)) for each saturated multiplicative subset S of D. First we recall a few definitions. A nonzero nonunit x ∈ D is said to be primary if xD is a primary ideal. As in [8], D is called a generalized weakly factorial domain (GWFD) if every nonzero prime ideal of D contains a primary element. It is known that if D is not a field, then D is a GWFD if and only if t-dim(D) = 1 and each height-one prime ideal of D is the radical of a principal ideal [8, Theorem 2.2] (note that t-dim(D) = 1 means that every prime t-ideal of D has height-one). In [4, Theorem], it was shown that each saturated multiplicative subset S of D is a splitting set (and hence S = N (N (S))) if and only if D is a weakly factorial domain (i.e., each nonzero nonunit of D is a product of primary elements). Proposition 2.5. An integral domain D is a GWFD if and only if S = N (N (S)) for each saturated multiplicative subset S of D. Proof. (⇒) Let S be a saturated multiplicative subset of a GWFD D. Then S = D \ ∪P√α , where the Pα ’s are height-one primes of D since t-dimD = 1. Let each Pα = xα D, and let T = {xα }. Then S ⊆ N (T ) since (xα , s)v = D for all xα and s ∈ S. Also, N (T ) ⊆ S; for if z ∈ N (T ), then (z, xα )v = D for all xα , and hence z ∈ D \ ∪Pα = S. Thus S = N (T ), and hence N (N (S)) = N (N (N (T ))) = N (T ) = S. (⇐) We first show that t-dim(D) = 1. For this, it suffices to show that every prime ideal minimal over a principal ideal is a maximal t-ideal. Let 0 = x ∈ D, and let P be a prime ideal of D minimal over xD (hence P is a t-ideal). Then S = D \ P is a saturated multiplicative subset of D, and hence S = N (N (S)).

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Clearly N (S)  U (D) because S = D∗ . Let a ∈ N (S) \ U (D). Then a ∈ P and for each b ∈ S, √ (a, b)v = D, which implies √ that P is a maximal t-ideal. Next we show that P = aD. Assume that P = aD. Then there is another prime ideal Q of D minimal over aD, and thus for c ∈ Q \ P , D = (a, c)v ⊆ Qt . But this contradicts the fact that √ Q is a t-ideal (note that Q is minimal over aD, and thus a t-ideal). Hence P = aD, and thus D is a GWFD [8, Theorem 2.2]. Remark 2.6. (a) The proof of Proposition 2.5 shows that D is a GWFD if and only if S = N (N (S)) for each S = D \ P , where P is a prime t-ideal of D. (b) Let P be a prime t-ideal of an integral domain D and S = D \ P . Then either N (N (S)) = S or N (N (S)) = D∗ . To see this, suppose that S  N (N (S)). Then choose a ∈ N (N (S)) \ S; so a ∈ P . Then (a, b)v = D for all b ∈ N (S). Thus N (S) ⊆ D \ P = S, and hence N (S) = U (D) by Proposition 2.3(2). Thus N (N (S)) = D∗ . We end this section with a generalization of Proposition 2.5. Proposition 2.7. Let P be a prime t-ideal of an integral domain D and S = D \ P . Then N (N (S)) = S if and only if P is a maximal t-ideal and there is an x ∈ D such that P is the unique maximal t-ideal containing x. Proof. (⇒) Suppose that N (N (S)) = S. We first show that P is a maximal t-ideal of D. Assume that P is not a maximal t-ideal, and let P  Q be a prime t-ideal of D. Choose a ∈ Q \ P . Then a ∈ S and (a, b)v ⊆ Q for all b ∈ P ; so N (S) = U (D) since N (S) \ U (D) ⊆ P by Proposition 2.3(2). Hence N (N (S)) = D∗ . Thus if N (N (S)) = S, then P is a maximal t-ideal. We next show that there is an x ∈ D such that P is the unique maximal t-ideal containing x. Since N (N (S)) = S = D∗ because P is nonzero, N (S) = U (D), and hence N (S) ∩ P = ∅ by Proposition 2.3(2). Thus for any x ∈ N (S) ∩ P , P is the unique maximal t-ideal containing x. (⇐) Since P is the unique maximal t-ideal of D containing x, (x, s)v = D for all s ∈ S; so x ∈ N (S). Thus as x is a nonunit of D, x ∈ N (N (S)) by Proposition 2.3(2), and hence N (N (S)) = S by Remark 2.6 (b).

3

The Relationship Between NA (S) and NB (S)

In this section, we investigate the relationship between NA (S) and NB (S), where A ⊆ B is an extension of integral domains and ∅ = S ⊆ A∗ . In general, there is no containment between the two sets. For example, let A = Z ⊆ B = Z[X] and S = U (Z). Then NA (S) = Z∗ and NB (S) = Z[X]∗ ; so NB (S)  NA (S). Conversely, let A = Q[X, Y ] ⊆ B = Q[X, Y /X] and S = X. Then Y ∈ NA (S), but Y ∈ NB (S). Thus NA (S)  NB (S). Let A ⊆ B be an extension of integral domains and ∅ = S ⊆ A∗ . Then NA (S) ⊆ NB (S) if and only if xB ∩ sB = xsB for all s ∈ S whenever xA ∩ sA = xsA for x ∈ A and all s ∈ S. In particular, NA (S) ⊆ NB (S) when B is LCM-stable over A (recall from [13] that B is LCM-stable over A if (xA ∩ yA)B = xB ∩ yB for all x, y ∈ A). A flat extension of integral domains is clearly LCM-stable. An extension A ⊆ B of integral domains is said to be R2 -stable if aA ∩ bA = cA with a, b, c ∈ A implies aB ∩ bB = cB. It is clear that an LCM-stable extension is R2 -stable and

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that if A ⊆ B is an R2 -stable extension, then NA (S) ⊆ NB (S) for any ∅ = S ⊆ A∗ . But NA (S) ⊆ NB (S) does not imply that A ⊆ B is an R2 -stable extension (for example, let S = U (B) ∩ A). We have already observed that if A ⊆ B is a flat extension of integral domains and ∅ = S ⊆ A∗ , then NA (S) ⊆ NB (S). If in addition, A ⊆ B is faithfully flat, then NB (S) ∩ A = NA (S). This need not hold for a flat extension. For example, let A = Z ⊆ B = Z[ 12 ] and S = 2. Then AS = B, NB (S) = B ∗ , and NA (S) = Z \ 2Z by Example 2.4(a); so NA (S)  NB (S) ∩ A. Let A ⊆ B be an R2 -stable extension of integral domains with B an overring of A and S a saturated multiplicative subset of A. In [9, Theorem 1], the authors showed that if S is a splitting (resp., lcm splitting) set in A, then the saturation of S in B is a splitting (resp., lcm splitting) set in B. (Recall that a splitting set S of D is said to be an lcm splitting set if sD ∩ dD is principal for all s ∈ S and d ∈ D, equivalently, for each s1 , s2 ∈ S, s1 D ∩ s2 D = sD for some s ∈ D [1, Proposition 2.4].) The proof of [9, Theorem 1] also proves the following. Proposition 3.1. Let B be an overring of an integral domain A, S a saturated multiplicative subset of A, and S  the saturation of S in B. If NA (S) ⊆ NB (S) and S is a splitting (resp., lcm splitting) set in A, then S  is a splitting (resp., lcm splitting) set in B and S  = {w ss12 |s1 , s2 ∈ S with ss12 ∈ B and w ∈ U (B)}. Corollary 3.2. (cf. [9, Proposition 1]) Let B be an overring of an integral domain A and p a prime element of A such that ∩pn A = {0} and pB  B. Then p is a prime element of B if and only if B ∩ pApA is a maximal t-ideal of B. Proof. (⇒) Assume that p is a prime element of B. Then ApA ⊆ BpB , and thus ApA = BpB because ApA is a local PID. Hence pB = B ∩ pApA , and thus B ∩ pApA is a maximal t-ideal of B (cf. [11, Proposition 1.3]). (⇐) Assume that B ∩ pApA is a maximal t-ideal of B. Let S = {upn |u ∈ U (A) and n ≥ 0}; then NA (S) ⊆ NB (S). For if t ∈ NA (S), then t ∈ B ∩ pApA , and hence for all upn ∈ S, ((t, upn )B)v = B because B ∩ pApA is the unique prime t-ideal of B containing upn . By Proposition 3.1, S  = {wpn |w ∈ U (B) and n ≥ 0} and S  is a splitting set in B. Thus one can easily verify that p is also prime in B (cf. [2, Corollary 1.4(c)]). We next consider the case of an integral domain of the form T = K + M , where M is a nonzero maximal ideal of T and K is a subfield of T . For D a subring of K, R = D + M is then a subring of T with qf (R) = qf (T ). This construction has proved very useful for constructing examples. In this case, we can relate ND (S) and NR (S) when ∅ = S ⊆ D∗ (cf. [7, Lemma 3.1] for Corollary 3.5). One can easily verify that ND (S) ⊆ NR (S) and NR (S) ∩ D = ND (S); this also follows since D ⊆ R is a faithfully flat extension. Proposition 3.3. Let T = K + M and R = D + M be integral domains, where M is a nonzero maximal ideal of T , K is a subfield of T , and D is a subring of K; and let ∅ = S ⊆ D∗ . 1. If S ⊆ U (D), then NR (S) = R∗ and SNR (S) = R∗ . 2. If S  U (D), then NR (S) = ND (S) + M and SNR (S) = SND (S) + M .

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Proof. (1) This is clear. (2) We show that NR (S) = ND (S) + M ; the second equality is then clear since SM = M . “ ⊆ ” : Let d ∈ S \ U (D). We first show that NR (S) ⊆ R \ M . Suppose that m ∈ NR (S) ∩ M . Then mR ⊆ M ⊆ dR. Thus mR = mR ∩ dR = mdR yields dR = R, and hence d ∈ U (R) ∩ D = U (D), a contradiction. Now let z ∈ NR (S). Then z = x + m for some x ∈ D∗ and m ∈ M by above. We show that x ∈ ND (S). Let s ∈ S. Then zR ∩ sR = zsR yields xD ∩ sD = xsD, and hence x ∈ ND (S). Thus z = x + m ∈ ND (S) + M . “ ⊇ ” : Let z = x+m, where x ∈ ND (S) and m ∈ M . Suppose that z(d1 +m1 ) = s(d2 + m2 ) for some d1 , d2 ∈ D, s ∈ S, and m1 , m2 ∈ M . Then xd1 = sd2 , and hence d1 = sd3 for some d3 ∈ D since x ∈ ND (S). Thus z(d1 +m1 ) = z(sd3 +m1 ) = zs(d3 + ms1 ) with ms1 ∈ M . Hence zR ∩ sR = zsR, and thus z ∈ NR (S). (For this inclusion, we did not need that S contains a nonunit of D.) Corollary 3.4. Let T = K + M and R = D + M be integral domains, where M is a nonzero maximal ideal of T , K is a subfield of T , and D is a subring of K; and let ∅ = S ⊆ D∗ . Then S is a splitting set in R if and only if S = U (R) (so S = U (D) = U (R)). Proof. (⇒) Suppose that S is a splitting set of R. Then S ⊆ U (D) by Proposition 3.3. Since S is saturated in R, we must have S = U (R) = U (D). (⇐) U (R) is always a splitting set. Corollary 3.5. Let T = K + M and R = D + M be integral domains, where M is a nonzero maximal ideal of T , K is a subfield of T , and D is a subring of K; and let S be a saturated multiplicative subset of D∗ which contains a nonunit. Then S is a splitting set of D if and only if SNR (S) = R \ M . Proof. This follows directly from Proposition 3.3(2) since D∗ + M = R \ M . Let X be an indeterminate over an integral domain D. As usual, for f ∈ qf (D)[X], Af is the fractional ideal of D generated by the coefficients of f . One can easily verify that ND (S) ⊆ ND[X] (S) and ND[X] (S) ∩ D = ND (S) for each ∅ = S ⊆ D∗ ; this also follows since D ⊆ D[X] is a faithfully flat extension. We next explicitly determine ND[X] (S). Proposition 3.6. Let D be an integral domain, X an indeterminate over D, and ∅ = S ⊆ D∗ . Then ND[X] (S) = {0 = f ∈ D[X]|(Af , s)v = D for all s ∈ S}. Proof. “ ⊆ ” : Let f ∈ ND[X] (S). Then (f, s)v = D[X] for all s ∈ S, and thus (Af , s)v = D because D[X] = (f, s)v ⊆ ((Af , s)D[X])v = (Af , s)v [X] ⊆ D[X] by [12, Proposition 2.2]. “ ⊇ ” : Let 0 = g ∈ D[X] such that (Ag , s)v = D for all s ∈ S. Suppose that (g, s)v  D[X] for some s ∈ S, and let Q be a maximal t-ideal of D[X] containing (g, s)v . Then Q = (Q ∩ D)[X] and Q ∩ D is a t-ideal of D [11, Proposition 1.1], and hence D = (Ag , s)v ⊆ Q ∩ D, a contradiction. Thus (g, s)v = D[X] for all s ∈ S, and hence g ∈ ND[X] (S).

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Corollary 3.7. ([5, Theorem 2.2]) Let D be an integral domain, X an indeterminate over D, and ∅ = S ⊆ D∗ . Then S is a splitting set in D[X] if and only if S is an lcm splitting set in D. Moreover, S is an lcm splitting set in D[X] if and only if S is a splitting set in D[X]. Proof. (⇒) Suppose that S is a splitting set in D[X]. Let a, b ∈ S and f = a + bX. Since S is a splitting set in D[X], by Proposition 3.6, there are an s ∈ S and g ∈ D[X] with (Ag , s )v = D for all s ∈ S such that f = sg, and so s|a and s|b in D. Since a, b ∈ S and S is saturated, we have as , sb ∈ S, and thus D = (Ag , as )v = ( as , sb )v ; so (a, b)v = sD. Thus S is an lcm splitting set in D [1, Proposition 2.4]. (⇐) Suppose that S is an lcm splitting set in D. Let 0 = f = a0 + a1 X + · · · + an X n ∈ D[X]. Since S is a splitting set in D, each ai = si ti for some si ∈ S and ti ∈ D with (ti , s )v = D for all s ∈ S. Also, since S is an lcm splitting set, (s0 , . . . , sn )v = sD for some s ∈ S (cf. [1, Proposition 2.4]). Let si = ssi for si ∈ D and let f = sg for g ∈ D[X]. Then (Ag , s )v = (s0 t0 , . . . , sn tn , s )v = ((s0 , . . . , sn , s )(t0 , . . . , tn , 1))v = (s0 , . . . , sn , s )v = D [1, Lemma 3.1], and thus g ∈ ND[X] (S) by Proposition 3.6. This implies that D[X] \ {0} = SND[X] (S), and so S is a splitting set in D[X]. The “moreover” statement follows from [1, Proposition 2.4] since s1 D ∩ s2 D = sD with s, s1 , s2 ∈ D if and only if s1 D[X] ∩ s2 D[X] = sD[X]. Acknowledgement The research on this chapter was started while the second author visited the University of Tennessee at Knoxville during 2001-2002. He would like to thank the University of Tennessee for its financial support and hospitality.

Bibliography [1] D.D. Anderson, D.F. Anderson, and M. Zafrullah, Splitting the t-class group, J. Pure Appl. Algebra 74 (1991), 17–37. [2] D.D. Anderson, D.F. Anderson, and M. Zafrullah, Factorization in integral domains, II, J. Algebra 152 (1992), 78–93. [3] D.D. Anderson, T. Dumitrescu, and M. Zafrullah, Almost splitting sets and AGCD domains, Comm. Algebra 32 (2004), 147–158. [4] D.D. Anderson and M. Zafrullah, Weakly factorial domains and groups of divisibility, Proc. Amer. Math. Soc. 109 (1990), 907–913. [5] D.D. Anderson and M. Zafrullah, Splitting sets in integral domains, Proc. Amer. Math. Soc. 129 (2001), 2209–2217. [6] D.F. Anderson, The class group and local class group of an integral domain, in Non-Noetherian Ring Theory, Math. Appl., Kluwer Acad. Publ., Dordrecht 520 (2000), 33–55. [7] D.F. Anderson and G.W. Chang, The class group of integral domains, J. Algebra 264 (2003), 535–552. [8] D.F. Anderson, G.W. Chang, and J. Park, Genealized weakly factorial domains, Houston J. Math. 29 (2003), 1–13. [9] T. Dumitrescu and M. Zafrullah, Lcm-splitting sets in some ring extensions, Proc. Amer. Math. Soc. 130 (2002), 1639–1644. [10] R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972. [11] E. Houston and M. Zafrullah, On t-invertibility, II, Comm. Algebra 17 (1989), 1955–1969. [12] B.G. Kang, Pr¨ ufer v-multiplication domains and the ring R[X]Nv , J. Algebra 123 (1989), 151–170. [13] H. Uda, LCM-stableness in ring extensions, Hiroshima Math. J. 13 (1983), 357–377.

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Chapter 10

Some Remarks on Infinite Products by

Jim Coykendall Abstract In this chapter we will look at some examples of infinite product representations of elements of a commutative ring. Some obstructions to a good general theory of infinite products are explored.

1

Introduction

In the spirit of the recent explosion of interest in the theory of factorization, we wish to consider the possibility of the notion of infinite products. Certainly, notions of infinite products (and sums) have proved quite useful in applied mathematics and analysis as well as some branches of number theory. The usefulness of the notion of infinite products makes considering the possibility of infinite products a “moral imperative” in the setting of general commutative rings. In [3] the present author encountered an application of infinite products in the realm of formal power series. Although the infinite product considered in [3] was mostly a formal notational convention, a similar notion of an infinite product for formal power series is studied slightly more extensively by B. Kang and M. Park in [4]. The context in which the authors of [4] and [3] utilized infinite products did not demand a full formal development of this notion (in the chapter [4] the behavior of the prime spectrum of a power series ring over nondiscrete valuation domains is investigated, and in [3] the dimension theory and SFT stability of power series rings over SFT rings are investigated). The aim of this chapter is to illustrate, via some examples, some possible implications of infinite products and we will show various obstructions to the development of a “clean” general theory. This work will mostly focus on examples, some proposed axioms for general infinite products, some first consequences of such a theory, and directions for further study. 188

Some Remarks on Infinite Products

2

189

A Motivating Example

The paper [1] was a milestone in the study of dimension theory of formal power series rings. A classical result in the study of Krull dimension behavior in commutative rings is that if R is a ring (commutative with identity) such that dim(R) = n then n + 1 ≤ dim(R[x]) ≤ 2n + 1. In [1] a near-Noetherian property termed the SFTproperty (for “strong finite type”) was introduced. We briefly recall that an ideal I ⊆ R is SFT if there is a finitely generated ideal B ⊆ I and a fixed integer N such that xN ∈ B for all x ∈ I. A ring, R, is SFT if all its (prime) ideals have the SFT property. As it turns out, this property is central in the dimension theory of formal power series rings. In [1] it is shown that the SFT property is necessary for the dimension of dim(R[[x]]) to be finite. In [3] the problem of determining if the SFT property (in addition to the property that dim(R) < ∞) is sufficient for dim(R[[x]]) to be finite as well as the question as to whether the SFT property is stable with respect to the adjunction of a power series variable (i.e. if R is SFT then is R[[x]] SFT?) are explored. These questions are both answered in the negative in [3]. In showing that the SFT property is not necessarily preserved in power series extensions, the notion of infinite products arises in a natural fashion (it should also be noted that the investigation of the prime spectra of power series rings over nondiscrete valuation domains in [4] gives rise to the notion of an infinite product in a distinct, but similiar context). As a starting point we will look at a slightly adjusted version of an example that appears in [3]. Example 2.1. Let V be a one-dimensional nondiscrete valuation domain with value group R and residue field F2 . For convenience of computation we will write V := F2 [x; R+ ]M where F2 [x; R+ ] is the monoid ring over F2 with respect to R and M ⊆ F2 [x; R+ ] is the maximal ideal generated by the set {xα }α∈R+ . Let {αi }∞ i=0 be a countable collection of elements from R+ . We define  P∞  ∞  x( i=0 αi ) , if ∞ αi converges; αi x = i=0 ∞ 0, if i=0 αi diverges. i=0 It is easy to verify that this definition is a well-defined notion of an infinite product of any countable collection of powers of the element x ∈ V . We briefly remark here that we are being a bit sloppy about associativity, and indeed, a more formal notion of associativity would require the notion of “infinite” regrouping of parentheses. What we mean by default is that given the infinite product in R ∞ 

πi

i=0

there is a well-defined, deterministic way to associate an element in R. In this ∞ ∞ example, we associate with i=0 xα , the real, positive term-series i i=0 αi . We will deal with associativity a bit more carefully later, but we must first make a definition.

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Definition 2.2. Let V be a valuation domain with value group G and let x ∈ V . A + αi subset {αi }∞ is a well-defined i=0 ⊆ G is called admissible (with respect to x) if x element of V for all i. The motivation for this definition can be found in the previous example. Indeed, the element x is such that xα ∈ V for all α ∈ R+ , but if u is a unit in V , then there is no guarantee of the existence of an arbitrary element of the form (ux)α . With this in mind we make a final preliminary definition. Definition 2.3. Let V be a valuation domain with value group G and maximal ideal M. We say that the element x ∈ M is a pseudo-uniformizer if xg is defined for all g ∈ G+ . Basically, this definition says that if x is a pseudo-uniformizer, then all subsets of G+ are admissible. In constructive examples of valuation domains, pseudouniformizers abound (e.g., in Example 2.1, the element x is a pseudo-uniformizer). We now pause to underscore a couple of observations about the behavior of the infinite product in the more general setting of (1−dimensional) valuation domains. We point out here that the “infinite product” to which the following proposition refers is the natural generalization of the infinite product that appeared in Example 2.1. That is to say, we declare (for a pseudo-uniformizer x and a subset {αi }∞ i=0 ⊆ G+ ) that

∞  i=0

xαi

⎧ P∞ ( αi ) ⎪ ⎨x i=0 , = 0, ⎪ ⎩ undefined,

∞ if i=0 αi converges to an element of G+ ;  if ∞ i=0 αi = ∞; in all other cases.

Proposition 2.4. Let V be a one-dimensional valuation domain with value group G ⊆ R, valuation v and maximal ideal M. If x∈ M is a pseudo-uniformizer, then ∞ the following properties of the infinite product i=0 xαi hold.  αi + is defined for all subsets {αi }∞ if and a) The infinite product ∞ i=0 ⊆ G i=0 x only if G is isomorphic to either Z or R.  ∞ ∞ b) If σ is any permutation of N {0} then i=0 xαi = i=0 xασ(i) . ∞ ∞ ∞ ∞ c) i=0 xβi divides i=0 xαi if and only if i=0 βi ≤ i=0 αi . ∞ + d ) i=0 xαi = 0 for all subsets {αi }∞ i=0 ⊆ G if and only if V is discrete. e) If G ∼ = R, then any element of M may be represented as a nontrivial infinite product. Proof. We first ∞show a) and d) in tandem. If G is isomorphic to Z , then consider the product i=0 xαi . The set {αi }∞ i=0 contains a least element since it is a subset of N and without loss of generality, we will let α0 be aleast element. Clearly since  αi α diverges and hence ∞ = 0. (It should also every αi ≥ α0 , the series ∞ i i=0 i=0 x + ∼ be noted that since G = Z every subset of G is admissible with respect to y for any element y ∈ M.) If G ∼ = R then since x is a pseudo-uniformizer, then xα exists + for all α ∈ R . In particular, the product

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191

∞ 

xαi

i=0

∞

P∞

always exists; it is 0 if the series i=0 αi diverges and is x( i=0 αi ) if the series converges. We now complete the proof Assume that V is nondiscrete but that every  of d). αi + x is 0 for all {αi }∞ infinite product of the form ∞ i=0 ⊆ G . We note that since i=0 + x is a pseudo-uniformizer, every subset of G is admissible. Note that since V is nondiscrete, we can inductively choose ξn ∈ M such that v(ξn ) ≤

v(x) . 2n

We now let α0 = v(x) and inductively αn = v(ξn ) for all n ≥ 1. Consider the infinite product ∞ 

xαi .

i=0

Note that since αn ≤

v(x) 2n ,

we have that

∞

∞

i=0

αi ≤ v(x)

∞

2−n = 2v(x).

i=0

Hence i= αiconverges (say, to β). One of two things occurs at this juncture. If  αi αi = xβ = 0, and if β ∈ / G+ then the infinite product ∞ β ∈ G+ then ∞ i=0 x i=0 x is not defined. In either case, the contradiction finishes the proof of d). To finish the proof of a), we note that if V is nondiscrete and G is a proper subgroup of R then it is well-known that the closure of G (as a subset of R with the standard topology) is all of R. Select an element b ∈ R \ G+ and an increasing + (again with respect to the ordering on R) sequence {an }∞ n=0 of elements of G such that {an } −→ b. Now define α0 = a0 and for all n ≥ 1, define αn = an − an−1 . Note that each αn is positive and since x is a pseudo-uniformizer, xαn is a well-defined element of V for all n. Now consider the infinite product ∞ 

xαn .

n=0

Since, by construction, the infinite sum converges to an element b ∈ / G, we have our desired contradiction. ∞ For part b), note that since each αi is positive, if the series i=0 αi converges, then it converges absolutely. Hence any rearrangement of  the series converges to the same sum. More precisely, if σ is a permutation of N {0} then

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∞

αi =

∞

i=0

ασ(i)

i=0

and hence the infinite products coincide. If the series diverges, then the same argument applies.   βi αi divides ∞ (we are tacitly asFor part c) we first assume that ∞ i=0 x i=0 x suming here that both infinite products exist). So there is a w ∈ V such that ∞ ∞ w i=0 xβi = i=0 xαi . Using the definition of the infinite product and computing values we get that v(w) +

∞

βi =

i=0

∞

αi

i=0

(note that if the right sum diverges, and if the left diverges then so ∞ we are done ∞ does the right). This gives that i=0 βi ≤ i=0 αi (in either case, convergence or divergence). ∞ ∞ ∞ Conversely, that i=0 βi ≤ i=0 αi . Note that if i=0 βi diverges, suppose ∞ then so does i=0 αi and we will discard  this case as degenerate (we will declare ∞ that 0 divides 0 for brevity). Assume that i=0 βi converges and that its sum is b. ∞ If i=0 αi diverges, then we are done, so we will assume that this sum converges to a. Since b ≤ a (and xb and xa are defined since x is a pseudo-uniformizer) then xb must divide xa and we are done. Finally, for part e)  assume that m ∈ M. If m = 0, then we can represent m ∼ as the infinite product ∞ i=0 x. If m = 0 then since G = R, select a positive term ∞ ∞ v(m) infinite series i=0 αi such that i=0 αi = v(x) and note that ∞ 

xαi = x(

P∞

i=0

αi )

= a ∈ V.

i=0

So v(a) = v(m), and hence the proof is complete.

m a

is a unit in V . So we have m = ( m a)

∞ i=0

xαi and

Given an infinite product in a commutative ring R with 1, one can extend this product in a nontrivial way to the power series ring R[[x]] as follows. Example 2.5. Suppose that {ri }∞ i=0 is a subset of a commutative ring with identity, R, such that for any subset of S ⊆ N {0} the infinite product 

ri

i∈S

is an element of R. Then the infinite product ∞  (ri + si xi+1 ) i=0

where each si ∈ R is an element of R[[x]].

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193

By infinite product, we produce a power series via the following algorithm. The constant term is the infinite product of the ri ’s (which is well-defined by assumption). The coefficient of xn is determined via:  ((si1 si2 · · · sik )( rj )). i1 +i2 +···+ik =n

j =i1 ,i2 ,··· ,ik

 Note that any term in the sum exists since subproducts of the form i∈S ri exist. Also the sums are finite since the powers of the polynomial terms are increasing. This example is in the spirit of the previous example where infinite products over valuation domains are considered. In the previous example (because of absolute convergence of convergent positive term series) every subproduct of a well-defined infinite product exists. Also note that this product can be done in more generality. In particular, the terms in the successive polynomials merely have to satisfy the property that for any fixed degree, n, there are only finitely many polynomials of the form r + sxn in the infinite product.

3

Hazards

In the previous sections the approach to the infinite product was more informal. Basically, in all of the results, an algorithm was described to define an infinite product representation of some elements of a commutative ring with identity. In this section we will look more closely at some of the obstructions to a general notion of infinite products. In particular, for infinite products, some problems with associativity and (“infinite”) commutativity naturally arise. We will begin by defining axioms for an infinite product. All products here will be countable. Axioms for the infinite product. Let {ri }∞ i=0 be a subset of a commutative ring, R, and assume that the infinite ∞ product i=0 ri exists. ∞ 1. If any ri = 0 then i=0 ri = 0. ∞ 2. If R has an identity, then i=0 1 = 1.  3. If {n1 , n2 , · · · } and {m1 , m2 · · · } are two partitions of N {0}, then (

n1 

i=0

ri )(

n2 

i=n1 +1

ri ) · · · = (

m1 

i=0

ri )(

m2 

ri ) · · · .

i=m1 +1

∞ 4. Any finite rearrangement of i=0 ri preserves the product; that is,if σ is a  ∞ permutation of N {0} that fixes all but finitely many ri ’s, then i=0 ri = ∞ i=0 rσ(i) . We note the third axiom is associativity in the infinite case. One would also like for any subproduct to exist and for the last axiom to extend to arbitrary permutations (certainly one would wish for this when studying commutative rings), but we will presently see some problems with this.

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Example 3.1. Let V be a 1−dimensional valuation domain with value group R from Example 2.1. We will show that it is impossible to extend the “natural” infinite product to the quotient field, K, of V without incurring some strange difficulties. Indeed consider the following set of elements of K 1

1

{x, x(− 2 ) , x( 3 ) , · · · , x((−1)

n−1 1 n)

, · · · }.

Extending the definition of infinite product to K gives that xln(2) ∈ V . However, the infinite series ∞

(−1)i−1

i=1

∞ i=1

x((−1)

i−1 1 i)

=

1 i

is conditionally convergent. Hence, given any real number α (including α = ±∞), there is a rearrangement of the series that converges to α (one can also rearrange the series so as to diverge without making it approach ±∞). Hence there are rearrangements of this product that take on different values (as well as being undefined). The previous example shows that one cannot hope to extend this infinite product in a meaningful way to powers of x in the quotient field. The key was that infinite products of powers of x in V can be rearranged in any fashion since any infinite series taken from the positive elements of R either converges absolutely or diverges. This example can be extended to nondiscrete valuation domains in general. We now relay an example that is perhaps even more disturbing and shows that the algorithm for defining an infinite product must be carefully defined. Example 3.2. Consider V to be the valuation domain from Example 2.1. We consider the infinite product (defined as in Example 2.5) in V [[t]]: 1

1

n

1

(x + t)(x 2 + t2 )(x 4 + t4 ) · · · (x 2n + t2 ) · · · . As we saw in Example 2.5, this is a well-defined element of V [[t]] (as written, with the algorithm supplied there). But note that n

1

1

n

(x 2n + t2 ) = (x 22n + t)2 since char(V ) = 2. If we write the infinite product 1

1

1

n

(x + t)(x 4 + t)2 (x 16 + t)4 · · · (x 22n + t)2 · · · then we see that this is not associative. In fact, grouping them in the original way gives a well-defined power series, but taking the new factors “one at a time” is not even defined since every term in the product has a linear term. We now produce a couple of general “obstruction” results. Theorem 3.3. Let R be a domain. If {p1 , p2 , · · · } are (not necessarily distinct) primes of R such that the infinite product ∞  i=1

pi = a ∈ R

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195

then there is a localization of T ⊇ R in which there is an infinite product of units that is a nonunit. ∞ Proof. Let i=1 pi = a ∈ R and consider the multiplicative set,
S, generated by the set of primes P := {p1 , p2 , · · · }. Note that the ideal (a) S = ∅. Indeed, if mk 1 ra = pm then there are two cases to consider. If there exists p ∈ P distinct 1 · · · pk mk 1 which is a contradiction. If from pi , 1 ≤ i ≤ k then p divides a but not pm 1 · · · pk p1 , p2 , · · · , pk is an exhaustive list of distinct primes from P , then at least one of them (say p1 without loss of generality) must occur infinitely often as a divisor of mk 1 +1 1 divides a, but not pm a. In particular, pm 1 1 · · · pk . In the localization RS , the ideal (a) survives (so a is a nonunit), but note that in RS each pi is a unit. Hence the nonunit a is an infinite product of units. We now point out a positive result. Proposition 3.4. Any Boolean ring admits an arbitrary, nontrivial commutative infinite product. Proof. We first note that for an arbitrary index set, I, the ring  F2 i∈I

admits a nontrivial infinite product via ∞ 

(j)

{ai }i∈I = {bi }i∈I

j=0 (j)

where {ai }i∈I ∈



F2 and  1 if the ith term of {ai }(j) = 1 for all j bi = 0 otherwise. i∈I

The fact that any Boolean ring admits a nontrivial infinite product follows from the fact that any Boolean ring is a subring of a direct product of the form  i∈I F2 [2]. Proposition 3.5. With the exception of F2 , no field admits a nontrivial general commutative infinite product. Proof. Certainly F2 admits an arbitrary commutative infinite product (as a special case of the previous result). So assume that K = F2 . If K admits an arbitrary infinite product, then we select an x ∈ K such that x is neither 0 nor 1. Consider the product ∞ 

x := y

i=i

and note that xy = y. Since K is a field and x = 1, y must be zero. In particular, the infinite product of any nonzero, nonidentity element is 0. We now consider the infinite product

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Coykendall

∞ 

x−1 := z.

i=i

Since we are assuming that the product is arbitrarily commutative (i.e., any rearrangement preserves an infinite product), then yz = 1 by construction, since ∞ j=0 1 = 1. But y = 0 and this is our contradiction. The main theme to glean from these examples and results is that the hope of creating a general infinite product must be handled with care. Even for the fairly straightforward examples that were produced for 1−dimensional valuation domains, we saw that extending the notion of infinite products to the quotient field induced some noncommutative (albeit, “infinite” noncommutative) behavior when we allowed infinite rearrangements of infinite products. Care must also be taken to avoid problems with associativity as was pointed out in Example 3.2. Of course, in practice, infinite products have been used in an ad hoc way to produce results like the ones in [4] and [3], but the focus in those papers was not to create a general infinite product, but to use specific infinite product representations as a tool to prove theorems. The question remains as to the nature of general infinite products. In particular, it would be nice to classify all rings with a general commutative infinite product (of course one could also include nil rings, which are a distinct class of rings from the Boolean rings). It would also be interesting to quantify the nature of the natural noncommutativity that arises when one considers the infinite product defined earlier for the quotient field of a nondiscrete valuation domain. Finally, it would be good to understand the closure of a domain with respect to an infinite product. More precisely, if V is a 1−dimensional nondiscrete valuation domain with value group a proper subgroup of R, then is the closure of V with respect to our defined infinite products the larger valuation domain with value group R? The answer is “yes” for the constructed Example 2.1. Additionally in this spirit, what is the structure of the domain obtained by adjoining all welldefined infinite products to the ring V [[t]] from Example 2.5? We believe that the answers to these questions could be quite interesting and provoke interesting further directions in factorization theory.

Bibliography [1] J. T. Arnold, Krull dimension in power series rings, Trans. Amer. Math. Soc. 177 (1973), 299–304. [2] N. Bourbaki, Algebra I. Chapters 1–3, Translated from the French, Reprint of the 1989 English translation, Springer, Berlin, 1998. [3] J. Coykendall, The SFT property does not imply finite dimension for power series rings, J. Algebra 256 (2002), no. 1, 85–96. [4] B. G. Kang and M. H. Park, A localization of a power series ring over a valuation domain, J. Pure Appl. Algebra 140(2) (1999), 107–124.

197

Chapter 11

Rings with Prime Nilradical by

Ayman Badawi and Thomas G. Lucas Abstract A commutative ring R is said to be a φ-ring if its nilradical N il(R) is both prime and divided, the latter meaning N il(R) is comparable with each principal ideal of R. Special types include φ-Noetherian (also known as nonnilNoetherian), φ-Mori, φ-chained and φ-Pr¨ ufer. A ring R is φ-Noetherian if N il(R) is a divided prime and each ideal that properly contains N il(R) is finitely generated. If R is a φ-Noetherian ring and X1 , X2 , . . . , Xn are indeterminates, then an ideal I of R[X1 , X2 , . . . , Xn ] which contains a nonnil element of R is finitely generated. Also, for a ring R where N il(R) is a nonzero prime ideal with N il(R)2 = (0), there is a ring A whose nilradical N il(A) is a divided prime such that R embeds naturally in A with R/N il(R) isomorphic to A/N il(A), RNil(R) isomorphic to ANil(A) , and the corresponding total quotient rings, T (R) and T (A), are such that T (R) ⊂ T (A) and A ∩ T (R) = R + N il(T (R)).

1

Introduction

We assume throughout that all rings are commutative with 1 = 0. For such a ring R, we let Z(R) denote the set of zero divisors of R and N il(R) denote the nilradical. We say that N il(R) is divided if it compares with each principal ideal of R (see [9] and [2]). If N il(R) is both divided and a prime ideal, we say that R is a φ-ring. For convenience we let H denote the class of all φ-rings. In [2], [3], [4], [5], and [6] the first-named author investigated this class of rings. In [2] and [4] he introduced the concepts of φ-pseudo-valuation rings and φ-chained rings. Also, D.F. Anderson and the first-named author made further investigation on the class H in [1] and introduced the concepts of φ-Pr¨ ufer rings and φ-B´ezout rings. See Section 4 for definitions of these specific types of φ-rings. The “φ” in the name refers to the canonical map φ : T (R) → RN il(R) from T (R), the total quotient ring of R, to R localized at N il(R) which maps a fraction a/b ∈ T (R) to its image in RN il(R) . 198

Rings with Prime Nilradical

199

An ideal I of a ring R is said to be a nonnil ideal if it is not contained in N il(R). Recall from [5] that a ring R is called a nonnil-Noetherian ring if every nonnil ideal of R is finitely generated. To establish more consistency in nomenclature, we will refer to such rings as φ-Noetherian rings. In the first section of this paper, we will show that many of the properties of Noetherian domains are valid for the nonnil ideals of a φ-Noetherian ring. In Section 3, we study polynomial rings with one or several variables over φ-Noetherian rings. For example, we show that if R is a φ-Noetherian ring and X1 , X2 , . . . , Xn are indeterminates, then an ideal I of R[X1 , X2 , . . . , Xn ] which contains a nonnil element of R is finitely generated. On the other hand, if N il(R) is not finitely generated, then X1 R[X1 , X2 , . . . , Xn ]+N il(R[X1 , X2 , . . . , Xn ]) is not finitely generated. Most, but not all, of our nondomain examples of φ-Noetherian rings are provided by the idealization construction R(+)B arising from a ring R and an R-module B as in Huckaba’s book [13, Chapter VI]. We recall this construction. For a ring R, let B be an R-module. Then the idealization of B over R is the ring R(+)B obtained from taking the product R × B and defining addition and multiplication of elements (r, b) and (s, c) in R×B by (r, b)+(s, c) = (r+s, b+c) and (r, b)(s, c) = (rs, sb + rc). Under these definitions R(+)B becomes a commutative ring with identity. If R is reduced, N il(R(+)B) = (0)(+)B. If R is an integral domain, N il(R(+)B) is prime.

2

Properties of φ-Noetherian Rings

We start with the following characterization of φ-Noetherian rings. It is a combination of several results from [5]. Theorem 2.1. ([5, Corollary 2.3 and Theorems 2.2, 2.4 and 2.6]) Let R ∈ H. Then the following are equivalent. 1. R is a φ-Noetherian ring. 2. R/N il(R) is a Noetherian domain. 3. φ(R)/N il(φ(R)) is a Noetherian domain. 4. φ(R) is a φ-Noetherian ring. 5. Each nonnil prime ideal of R is finitely generated. In the following result, we show that a φ-Noetherian ring is related to a pullback of a Noetherian domain. Theorem 2.2. Let R ∈ H. Then R is a φ-Noetherian ring if and only if φ(R) is ring-isomorphic to a ring A obtained from the following pullback diagram: A −−−−→ ⏐ ⏐ 

S ⏐ ⏐ 

T −−−−→ T /M

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Badawi and Lucas

where T is a zero-dimensional quasilocal ring containing A with maximal ideal M , S = A/M is a Noetherian subring of T /M , the vertical arrows are the usual inclusion maps, and the horizontal arrows are the usual surjective maps. Proof. Suppose φ(R) is ring-isomorphic to a ring A obtained from the given diagram. By Theorem 2.1, it suffices to show that φ(R) is φ-Noetherian. Since T is a zero-dimensional quasilocal ring and M is a prime ideal of both T and A, A ∈ H with N il(A) = Z(A) = M . Since S = A/M is a Noetherian domain, A is a φ-Noetherian ring by Theorem 2.1. Thus R is a φ-Noetherian ring. Conversely, suppose that R is a φ-Noetherian ring. Then, letting T = RN il(R) , M = N il(RN il(R) ), and A = φ(R) yields the desired pullback diagram. A rather nice property of φ-Noetherian rings is that homomorphic images are either Noetherian or φ-Noetherian. Generally speaking this is an exclusive “or”, the exception being when the image is either an integral domain or a local Artinian ring. Lemma 2.3. Let R ∈ H. Then N il(R) is finitely generated if and only if R is either an integral domain or a local Artinian ring with nonzero maximal ideal, N il(R). In particular, if R is Noetherian and not an integral domain, then it is a local Artinian ring with maximal ideal N il(R) = (0). Proof. Obviously, N il(R) is finitely generated if R is either an integral domain or a local Artinian ring. Thus it suffices to prove that if N il(R) is finitely generated and not the maximal ideal of R, then R is an integral domain. Let M be a maximal ideal of R. Since N il(R) is a divided prime ideal of R, N il(R)RM = N il(RM ) is a divided prime of RM . Hence RM is also in H and N il(RM ) is finitely generated. Since N il(RM ) is a divided prime and properly contained in M RM , M N il(RM ) = N il(RM ). Thus N il(RM ) = (0) by Nakayama’s Lemma. As this happens for each maximal ideal of R, N il(R) must be (0) and we have that R is an integral domain. Proposition 2.4. Let R ∈ H be a φ-Noetherian ring and let I = R be an ideal of R. If I ⊂ N il(R), then R/I is a φ-Noetherian ring. If I ⊂ N il(R), then N il(R) ⊂ I and R/I is a Noetherian ring. Moreover, if N il(R) ⊂ I, then R/I is both Noetherian and φ-Noetherian if and only if I is either a prime ideal or a primary ideal whose radical is a maximal ideal. Proof. Suppose that I ⊂ N il(R). Then N il(R/I) = N il(R)/I is a divided prime ideal of R/I. Hence, R/I ⊂ H. Now, let Q be a nonnil prime ideal of R/I. Then Q = P/I for some prime ideal P ⊂ N il(R) of R. Since P is finitely generated, Q is finitely generated. Thus R/I is a φ-Noetherian ring by Theorem 2.1. Now, suppose that I ⊂ N il(R). Then N il(R) ⊂ I since N il(R) is divided. Let Q be a prime ideal of R/I. Then Q = P/I for some nonnil prime ideal P of R such that I ⊂ P . Hence Q is finitely generated since P is finitely generated. Thus R/I is a Noetherian ring since each prime ideal is finitely generated. The third statement follows from Lemma 2.3.

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Corollary 2.5. Let R ∈ H be a φ-Noetherian ring. Then a homomorphic image of R is either a φ-Noetherian ring or a Noetherian ring. An almost Dedekind domain that is not Dedekind is a ring that is locally Noetherian but not Noetherian. If a domain is locally Noetherian and each nonzero element is contained in at most finitely many maximal ideals, then the domain is Noetherian (see, for example, [14, Exercise # 10, page 73]). A similar statement holds for rings in H. Proposition 2.6. Let R ∈ H and suppose that RM is φ-Noetherian for every maximal ideal M of R, and that each nonnil nonunit of R lies in only a finite number of maximal ideals of R, then R is a φ-Noetherian ring. Proof. Set D = R/N il(R) (observe that D is an integral domain). If N il(R) is a maximal ideal of R, then D is Noetherian (being a field), and hence R is φNoetherian by Theorem 2.1. Thus assume that N il(R) is not a maximal ideal of R. Let J be a maximal ideal of D. Then J = M/N il(R) for some maximal ideal M of R. Since RM ∈ H is φ-Noetherian by hypothesis and DJ is ring-isomorphic to RM /N il(R)RM , we conclude that DJ is Noetherian. Since each nonnil nonunit of R lies in only a finite number of maximal ideals of R by hypothesis, we conclude that every nonzero nonunit of D lies in only a finite number of maximal ideals of D. Thus, D is Noetherian by [14, Exercise # 10, page 73]. Hence R is φ-Noetherian by Theorem 2.1. Note that as long as N il(R) is prime and locally divided (i.e., N il(R)RM is a divided prime of RM ), N il(R) is divided. The key to the proof is that for each nonnil element r ∈ R, the conductor of r into N il(R) is N il(R) since N il(R) is prime. Thus no nonnil element can be transformed into a nilpotent element under localization. Hence for each maximal ideal M , rRM will contain N il(R)RM = N il(RM ). For a given nilpotent n ∈ N il(R) and maximal ideal M , there are elements s, t ∈ R with t not in M such that sr = tn. It follows that the ideal (r :R n) = R; i.e., n ∈ rR. Thus the hypothesis in Proposition 2.6 could be superficially weakened to having N il(R) prime and each RM a φ-Noetherian ring (thus a φ-ring) with each nonnil element in only finitely many maximal ideals. However, the assumption that N il(R) is prime cannot be eliminated. For example, let V be a two dimensional valuation domain with principal maximal ideal M and height one prime P . Choose any proper P -primary ideal J and let R = V /J ⊕ V /J. Then the nilradical of R is P/J ⊕ P/J which is neither prime nor divided – simply consider the idempotents (1, 0) and (0, 1) and nilpotent multiples of each. On the other hand, M/J ⊕V /J and V /J ⊕ M/J are the only maximal ideals of R. Localizing at either yields the ring V /J which is a φ-Noetherian ring. Thus R is not φ-Noetherian, but it is locally φ-Noetherian with only finitely many maximal ideals. Note that if J does not contain P 2 , then V /J is definitely not formed by the idealization of a module over an integral domain. On the other hand, if we take V = K + YK[[Y]] + ZK((Y))[[Z]] and J = Z2 K((Y))[[Z]], then V /J is isomorphic to the idealization of K((Y)) over K[[Y]].

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Our next result shows that a φ-Noetherian ring will satisfy the conclusion of the Principal Ideal Theorem (and the Generalized Principal Ideal Theorem). Theorem 2.7. Let R ∈ H be a φ-Noetherian ring and let P be a prime ideal. If P is minimal over an ideal generated by n or fewer elements, then the height of P is less than or equal to n. In particular, each prime minimal over a nonnil element of R has height one. Proof. The ring D = R/N il(R) is a Noetherian domain by Theorem 2.1. Assume P is minimal over the ideal I = (a1 , a2 , . . . , an ). If I ⊂ N il(R), there is nothing to prove since we would have P = N il(R), the prime of height 0. Thus we may assume I is not nilpotent. Since N il(R) is divided, I properly contains N il(R). Thus I/N il(R) can be generated by n (or fewer) elements. Since D is Noetherian, the height of P/N il(R) is less than or equal to n by the Generalized Principal Ideal Theorem ([14, Theorem 152]). Thus P has height less than or equal to n. Other statements about primes of Noetherian rings that can be easily adapted to statements about primes of φ-Noetherian rings include the following. We leave the proofs to the reader. Proposition 2.8. [14, Theorem 145] Let R ∈ H satisfy the ascending chain condition on radical ideals. If R has an infinite number of prime ideals of height 1, then their intersection is N il(R). Proposition 2.9. [14, Theorem 153] Let R ∈ H be a φ-Noetherian ring and P be a nonnil prime ideal of R of height n. Then there exist nonnil elements a1 , . . . , an in R such that P is minimal over the ideal (a1 , . . . , an ) of R, and for any i (1 ≤ i ≤ n), every (nonnil) prime ideal of R minimal over (a1 , . . . , ai ) has height i. Proposition 2.10. [14, Theorem 154] Let R ∈ H be a φ-Noetherian ring and let I be an ideal of R generated by n elements with I = R. If P is a prime ideal containing I with P/I of height k, then the height of P is less than or equal to n + k.

3

Polynomial Rings over φ-Noetherian Rings

By the Hilbert Basis Theorem, a polynomial ring in finitely many indeterminates over a Noetherian ring is Noetherian. The analogous statement cannot be made for a (nontrivial) φ-Noetherian R since a nonzero nilpotent element of R cannot be a multiple of an indeterminate. Also, if N il(R) is not finitely generated, then the ideal XR[X] + N il(R)R[X] is not finitely generated. On the other hand, a local Artinian ring that is not a field is a φ-Noetherian ring. So it is possible for a polynomial ring over a φ-Noetherian ring to be Noetherian. We start this section with dimension related statements. Our first consideration involves the Jaffard property. We will show that if R is an n-dimensional φ-Noetherian ring, then R[X1 , X2 , . . . , Xm ] has dimension n + m for each m > 0.

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Recall that if R is a Noetherian ring, P is a height n prime of R and Q is a prime of R[X] that contracts to P but properly contains P R[X], then P R[X] has height n and Q has height n + 1 [14, Theorem 149]. Proposition 3.1. Let R ∈ H be a φ-Noetherian ring and let P be a height n prime of R. If Q is a prime of R[X] that contracts to P but properly contains P R[X], then P R[X] has height n and Q has height n + 1. Proof. Since N il(R) is the minimal prime of R, N il(R[X]) = N il(R)R[X] is the minimal prime of R[X]. Thus we may shift the setting to D = R/N il(R) and D[X] (identified with R[X]/N il(R[X])). By Theorem 2.1, D is a Noetherian domain. Moreover, P/N il(R) is a height n prime of D and Q/N il(R[X]) is a prime of D[X] that contracts to P/N il(R) and properly contains (P/N il(R))D[X]. Thus the height of (P/N il(R))D[X] and P R[X] is n and the height of Q/N il(R[X]) and Q is n + 1. Similar height restrictions exist for the primes of R[X1 , X2 , . . . , Xm ]. Proposition 3.2. Let R ∈ H be a φ-Noetherian ring and let P be a height n prime of R. If Q is a prime of R[X1 , . . . , Xm ] that contracts to P but properly contains P R[X1 , . . . , Xm ], then P R[X1 , . . . , Xm ] has height n and Q has height at most n + m. Moreover the prime P R[X1 , . . . , Xm ] + (X1 , . . . , Xm )R[X1 , . . . , Xm ] has height n + m. Proof. Let D = R/N il(R). Then D is Noetherian as is each of the rings D[X1 , . . . , Xk ] ≡ R[X1 , . . . , Xk ]/N il(R)[X1 , . . . , Xk ]. Repeated applications of [14, Theorem 149] shows that the image of Q in D[X1 , . . . , Xm ] has height at most n + m. Hence the same restriction holds for the height of Q. Corollary 3.3. If R is a finite dimensional φ-Noetherian ring of dimension n, then dim(R[X1 , . . . , Xm ]) = n + m for each integer m > 0. As noted above, if N il(R) is not finitely generated, then XR[X] + N il(R)[X] is not finitely generated. In Examples 3.6 and 3.7, we will show how to construct a φ-Noetherian ring R that is not Noetherian, but where R[X] has finitely generated primes that contract to N il(R). In our next result we show that each ideal of R[X] that contracts to a nonnil ideal of R is finitely generated. Proposition 3.4. Let R ∈ H be a φ-Noetherian ring. If I is an ideal of R[X1 , X2 , . . . , for which I ∩ R is not contained in N il(R), then I is a finitely generated ideal of R[X1 , X2 , . . . , Xn ]. Xn ]

Proof. The key to the proof is that if I ∩ R is not contained in N il(R), then any single nonnil element in this intersection is enough to generate the nilradical of R[X1 , . . . , Xn ]. Since R/N il(R) is Noetherian, (I/N il(R))[X1 , . . . , Xn ] is finitely generated. Let {f1 , f2 , . . . , fm } ⊂ I generate the image of I modulo N il(R)[X1 , . . . , Xn ]. To get a finite set of generators for I, simply add any single nonnil element r of I ∩ R to the set {f1 , f2 , . . . , fm }. Since rN il(R) = N il(R), the set {r, f1 , . . . , fm } is a finite set of generators for I.

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Since three distinct comparable primes of R[X] cannot contract to the same prime of R, a consequence of Proposition 3.4 is that the search for primes of R[X] that are not finitely generated can be restricted to those of height one. A similar statement can be made for primes of R[X1 , X2 , . . . , Xn ]. Corollary 3.5. Let R ∈ H be a φ-Noetherian ring and let P be a prime of R[X1 , X2 , . . . , Xn ]. If P has height greater than n, then P is finitely generated. The ring in our next example shows that the converse of Proposition 3.4 does not hold even for prime ideals. Example 3.6. Let R = D(+)L be the idealization of L = K((Y))/D over D = K[[Y]]. Then R is a quasilocal φ-Noetherian ring with nilradical N il(R) isomorphic to L. Consider the polynomial g(X) = 1 − YX. Since the coefficients of g generate D as an ideal and g is irreducible, P = gD[X] is a height one principal prime of D[X] with P ∩D = (0). Each nonzero element of L can be written in the form d/y n where n is a positive integer, y denotes the image of Y in L and d = d0 +d1 Y+· · ·+dn−1 Yn−1 with d0 = 0. Given such an element, let f (X) = 1 + y X + · · · + y n−1 Xn−1 ∈ L[X]. Then g(X)(df (X)/y n ) = d/y n since dy n /y n = 0 in L. It follows that g(X)R[X] is a height one principal prime of R[X] that contracts to N il(R). Since the domain D in the above example is a DVR, each prime of D[X] that contracts to (0) is principal and generated by an irreducible polynomial whose coefficients generate D as an ideal. Let h(X) be such a polynomial. If h(X) has the form 1 − YXk(X) for some polynomial k(X) ∈ D[X], then the corresponding prime Q = hD[X] will be such that QR[X] = hR[X] is a height one principal prime of R that contracts to N il(R). The basic scheme of the proof is the same as for g(X) = 1 − YX. Given d = d0 + d1 Y + · · ·+ dn−1 Yn−1 with d0 = 0 and n positive, simply replace f (X) by the polynomial k(X) = 1+y Xk(X)+(y Xk(X))2 +· · ·+(y Xk(X))n−1 . Then, as above, h(X)(dk(X)/y n ) = d/y n . It follows that (h(X), 0) generates a height one principal prime of R[X]. On the other hand, the irreducible polynomial j(X) = Y − X also generates a height one prime of D[X], but it does not generate a height one principal prime of R[X] since its leading coefficient is a unit. A slightly more complicated argument shows that if r(X) is an irreducible polynomial whose coefficients generate D as an ideal, then rR[X] is prime if and only if r(X) is of the form u − YXt(X) for some unit u of D and some nonzero polynomial t(X) ∈ D[X]. Let d(X) ∈ L[X] be such that r(X)d(X) is a nonzero element of L[X]. Since D is a valuation domain, there is a largest positive integer m such that some coefficient of d(X) is of the form v/y m where v is (the image of) a unit of D. Let q be the largest integer such that q X has a coefficient of this form. Also let p be the largest integer such that rp , the coefficient on Xp , is a unit of D. In the product r(X)d(X) the coefficient on Xp+q has the form r0 dp+q + · · · rp−1 dq+1 + rp dq + rp+1 dq−1 + · · · rp+q d0 . Since the exponents on Y in r0 , . . . , rp−1 are nonnegative and those on y in dp+q , . . . , dq+1 are larger than −m, the resulting powers on y in the corresponding products are all strictly larger than −m. For i > p, the coefficient on Y in ri is positive and that on y in the corresponding dj is greater than or equal to −m. Thus the coefficient on Xp+q is

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a unit multiple of y −m . It follows that a necessary condition for rR[X] to contract to N il(R) is that r(X) have the form u − YXt(X) for some unit u of D and some nonzero polynomial t(X) ∈ D[X]. There are non-Noetherian φ-Noetherian rings with height one principal primes that contract to the nilradical but are not generated by a polynomial whose constant term is a unit nor do the coefficients generate the entire ring as an ideal. Example 3.7. Let D = K[Y, Z] and let L = K(Y, Z)/DP where P is the principal prime YD. Then the ring R(+)L is a φ-Noetherian ring. Since D is a UFD, each height one prime of D[X] that contracts to (0) is principal and generated by an irreducible polynomial whose coefficients generate an ideal of D whose inverse is D. Let Q = g(X)D[X] be such a prime and suppose that g(X) = a − YXt(X) where a ∈ D\YD (and t(X) ∈ D[X] is nonzero). Since DP is a DVR with maximal ideal YDP , each nonzero element of L can be written as a quotient d/y n for some d ∈ DP \P DP and some positive integer n. As in the previous example, let n be a positive integer and let f (X) = an−1 + an−2 y Xt(X) + · · · y n−1 Xn−1 t(X)n−1 and d/y n ∈ L with d ∈ DP \P DP . Since a is a unit of DP , a−n d/y n is an element of L. Hence we have (a − YX)(f (X)a−n d/y n ) = d/y n ∈ g(X)R[X]. It follows that g(X)R[X] is a principal height one prime of R[X] that contracts to N il(R). The argument given above characterizing the prinicipal height one primes of the polynomial ring in Example 3.6 can be adapted to this ring. Let r(X)D[X] be a height one prime of D[X] that contracts to (0) where r(X) is an irreducible polynomial whose coefficients generate an ideal of D with inverse equal to D. Let d(X) ∈ L[X] be such that r(X)d(X) is a nonzero element of L[X]. As above let p be the largest integer where the coefficient on Xp in r(X) is not in YD and let m and q be such that m is the largest integer such that some coefficient of d(X) is of the form dk /y m for some unit dk of D and let q be the largest value of k where this maximum occurs. Similar analysis of the coefficient on Xp+q shows that this coefficient is of the form e/y m for some unit e of DP . It follows that the height one principal primes of R[X] that contract to N il(R) are of the form r(X)R[X] where r(X) = r0 − YXk(X) is an irreducible polynomial with r0 ∈ D\YD and k(X) nonzero. Moreover, each such polynomial generates such a prime. We end this section with the following problem. Problem. Let R ∈ H be a φ-Noetherian ring. Characterize the finitely generated height one primes of R[X].

4

Embedding R into a φ-Ring

We begin this section with a question. Question. Let R be a ring whose nilradical N il(R) is prime. Is it possible to embed R into a φ-ring A with nilradical N il(A) in such a way that all of the following hold? (I) AN il(A) is isomorphic to RN il(R) , (II) A/N il(A) is isomorphic to R/N il(R), and R embeds in A in such a way that

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(III) Z(R) ⊂ Z(A), (IV) T (R) ⊂ T (A), (V) A ∩ T (R) = R + N il(T (R)), and (VI) A = R + N il(A). For all but (I), it is easy to answer “Yes” if R is formed by taking the idealization of a module over an integral domain. For a D-module B, we let ZD (B) = {r ∈ D | rb = 0 for some nonzero b ∈ B}. If D is an integral domain, then for R = D(+)B, Z(R) = {(r, c) | r ∈ ZD (B), c ∈ B} [13, Theorem 25.3]. Lemma 4.1. Let D be an integral domain and let B be a D-module. Then there is a D-module C containing B which is a divisible D-module such that ZD (C) = ZD (B). Proof. Let S = D\ZD (B) and let U (DS ) denote the units of DS . Then BS is both a D-module and a DS -module and B naturally embeds as a D-submodule of BS . Also, each nonunit of DS is a zero divisor on BS , so ZD (BS ) = ZD (B) and ZDS (BS ) = ZD (B)S = DS \U (DS ). By [10, Proposition VII.1.4], there is a divisible DS -module C containing BS . Since each element of S is a unit of DS , ZD (C) = ZD (B). We also have that C is a divisible D-module. An immediate consequence is the following theorem. Theorem 4.2. Let R = D(+)B where D is an integral domain and B is a Dmodule. Then there is a divisible D-module C such that A = D(+)C is a φ-ring for which (i) A/N il(A) is isomorphic to R/N il(R), (ii) Z(R) ⊂ Z(A), (iii) T (R) ⊂ T (A), (iv ) A ∩ T (R) = R + N il(T (R)), and (v ) A = R + N il(A). Proof. As in Lemma 4.1, let S = D\ZD (B) and let C be a divisible DS -module that contains BS . Set A = D(+)C. Then N il(R) = (0)(+)B and N il(A) = (0)(+)C. Thus R/N il(R) = D = A/N il(A) and A = R + N il(A). As in the proof of Lemma 4.1, we have ZD (B) = ZD (C) and ZDS (BS ) = ZDS (C) = DS \U (DS ) where U (DS ) is the set of units of DS . With regard to zero divisors, we have Z(R) = ZD (B) × B and Z(A) = ZD (C) × C. From this it is easy to see that T (R) can be identified with DS (+)BS and T (A) with DS (+)C. Thus A ∩ T (R) = R(+)BS . To see that N il(A) is divided, consider (0, c) ∈ N il(A) and (r, b) ∈ A\N il(A). Since C is a divisible D-module, there is an element f ∈ C such that rf = c. Clearly, (r, b)(0, f ) = (0, c) and therefore N il(A) is divided.

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What is missing in the conclusions in Theorem 4.2 is the statement about the localizations RN il(R) and AN il(A) . In some sense, these were not simply ignored. The module C could be too large (since C ⊕ V is divisible for each vector space V over the quotient field of D). Rather than showing that it is possible to cut C down to a divisible module C  for which A = D(+)C  would satisfy all six of the desired properties, we will establish the existence of a φ-ring satisfying all six under the somewhat more general assumption that the square of the nilradical is the zero ideal (no matter whether the original ring R is formed by idealization or not). In Chapter VII of [10], L. Fuchs and L. Salce present a useful technique for constructing a divisible module of projective dimension one over a given integral domain. They then use this module to generate a divisible module containing a specific D-module B. We make use of the general notion of their construction to establish the following theorem. Theorem 4.3. Let R be a ring with nilradical N il(R). If N il(R) is prime and N il(R)2 = (0), then there is a ring A with nilradical N il(A) such that (i) N il(A) is a divided prime of A, (ii) AN il(A) is isomorphic to RN il(R) , (iii) A/N il(A) is isomorphic to R/N il(R), and R embeds in A in such a way that (iv) Z(R) ⊂ Z(A), (v) T (R) ⊂ T (A) and (vi) A ∩ T (R) = R + N il(T (R)), and (vii) A = R + N il(A). Proof. Assume N il(R) is a prime ideal of R with N il(R)2 = (0). Then N il(T (R)) is a common prime ideal of T (R) and R + N il(T (R)) whose square is zero. Note that there are natural isomorphisms between R/N il(R) and [R + N il(T (R))]/N il(T (R)) and between the localizations RN il(R) and [R + N il(T (R))]N il(T (R)) . Thus we may assume N il(R) = N il(T (R)). We start by constructing a ring A such that (i) N il(A ) is a divided prime of A , (ii) AN il(A
) is isomorphic to T (R)N il(R) , (iii) A /N il(A ) is isomorphic to T (R)/N il(R), and T (R) embeds in A (and thus in T (A )) in such a way that (iv) A can be identified with T (R) + N il(A ) and (v) the image of T (R) in AN il(A
) is the same as the image of A . Given such a ring A , the ring A resulting from taking the pullback of R/N il(R) along N il(A ) will satisfy conditions (i)–(vii). Pictorially, A is the subring obtained from the following pullback diagram with i the inclusion map and σ the restriction of the isomorphism from T (R)/N il(R) to A /N il(A ). A −−−−→ R/N il(R) ⏐ ⏐ ⏐ ⏐σ i  A −−−−→ A /N il(A ) Let S = R\Z(R), N = N il(R)\{0} and C = Z(R)\N . If C contains only 0 (equivalently, Z(R) = N il(R)), there is nothing to prove as N il(R) will be a divided prime (since we have assumed N il(R) = N il(T (R))). So we may assume Z(R) properly contains N il(R). Let F denote the subset of N × C N consisting of those elements of the form α = (α0 , α1 , α2 , . . . ) such that there is a positive integer nα where αj = 0 for each j ≤ nα and αk = 0 for each k > nα . Let E = T (R)[X ] where X = {Xα }

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is a set of indeterminates indexed over F . For each α ∈ F with nα > 1, let α−k = (α0 , α1 , . . . , αnα −k , 0, . . . ) for 1 ≤ k < nα . Let J be the ideal of E generated by (i) the products of the form Xα Xβ , (ii) the products of the form mXα for each m ∈ N il(R), (iii) the elements of the form αnα Xα − Xα−1 for those α with nα > 1, and (iv) the elements of the form α1 Xα − α0 for those α with nα = 1. By (i), the square of each Xα is in J. Also, J is contained in the ideal Q = N il(R) + X T (R)[X ], the set of polynomials with constant term a nilpotent of T (R) (and R). Since N il(R) is a prime ideal of T (R), Q is a prime ideal of E. As a power of each element of Q is contained in J, Q/J is the nilradical of A = E/J. Moreover, by the construction of J, J contains Q2 . Thus (Q/J)2 = (0) and we have that N il(A )2 = (0). We next show that J ∩ T (R) = (0) and no single Xα is in J. Note that the former implies that T (R) embeds naturally into A . Moreover, since N il(A ) = Q/J with Q = N il(R) + X T (R)[X ], we may view A as T (R) + N il(A ) and we have that there is a natural isomorphism between A /N il(A ) and T (R)/N il(R). Since J is contained in N il(R) + X T (R)[X ], J ∩ T (R) is contained in N il(R). By way of contradiction assume that some nonzero nilpotent of T (R) is contained in J. Based on the generating set described above, an arbitrary element of J can    qγ (γnγ Xγ − be written as a finite sum of the form pα (α1 Xα − α0 ) Xβ +      Xγ−1 ) X + r Xτ + yλ Xρ where nα = 1 for each α, the products  µ  σ     Xβ , Xµ , Xτ and Xρ are finite with Xβ and Xµ arbitrary (including   Xγ , at least two in each Xρ “empty products”), at least one term in each and pα , qγ , rσ , yλ ∈ T (R) with each rσ nilpotent. To have such a sum reduce  Xβ are empty. To then to a nonzero nilpotent of T (R), it must be that some “cancel out” the terms pα α1 Xα , not all of the remaining sums can contain only products of two or more variables. The part of the expression that contributes nonzero constant terms and constant multiples of single variables has the form     uα (α1 Xα − α0 ) + vα (α1 Xα − α0 )Xβ + zγ (γnγ Xγ − xγ−1 ) + wσ Xτ . All others involve only products of two or more indeterminates. Note that each wσ is nilpotent. If each uα is nilpotent, then there is no nonzero constant term since N il(R)2 = (0). So we may assume some uα is not nilpotent. For each such nonnil uα , uα α1 is not in N il(R) since N il(R) is prime and α1 is not nilpotent. In the   sums vα (α1 Xα − α0 )Xβ and wσ Xτ all α0 and wσ are nilpotent, so none of these can cancel with the nonnil uα α1 ’s. The same is true for each zγ that is nilpotent. Thus there are nonnil zγ ’s. Since nγ > 1 for each γ, there must be γ’s with maximal values of nγ and zγ not nilpotent. Such terms cannot be cancelled in the sum. Thus no nonzero element of J is contained in N il(R). Note that if Xψ is in J for some ψ, then so is Xψ−1 = ψnψ Xψ − (ψnψ Xψ − Xψ−1 ) (or ψ0 if nψ = 1). By continuing, we would eventually have ψ0 in J, a contradiction. For the remainder of the proof we let xα denote the image of

Xα

in A .

Let t be a nonnil element of A . Then there is a nonnil element r ∈ T (R)\N il(R) and a nilpotent element g ∈ N il(A ) such that t = r − g. Since g 2 = 0, t(r + g) = r2 . Thus to show that each nonnil element of A divides each nilpotent element, it suffices to show that each nonnil element of T (R) divides each nilpotent in A . There is nothing to prove for the units of T (R) as each remains a unit under

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the embedding into A . So we may further reduce the problem to showing that each nonnil element r ∈ C divides each nilpotent of A . For a nonzero nilpotent b ∈ N il(R), let α = (b, r, 0, . . . ). Then rxα = b since rXα − b is one of the generators of J. Similarly, for Xα with nα > 1, let β = (α0 , α1 , . . . , αn , r, 0, . . . ). Then rxβ = xα since rXβ − Xα is in J. Since N il(R) and the set {xα } generate N il(A ), N il(A ) is a divided prime. It is only slightly more complicated to take care of statement (ii) dealing with the localizations T (R)N il(R) and AN il(A
) . The easiest thing to prove is that T (R)N il(R) embeds naturally in AN il(A
) . For this consider two equivalent fractions t/s and v/q of AN il(A
) where s, t, q, v ∈ T (R). To be equivalent, there must be a nonnil element z ∈ A such that z(sv − qt) = 0. Since A = T (R) + N il(A), there are elements u ∈ T (R) and f ∈ N il(A ) such that z = u + f . The product z(u − f ) = u2 ∈ T (R)\N il(R) since f 2 = 0. It follows that u2 (sv − qt) = 0 and the fractions t/s and v/q are also equivalent in T (R)N il(R) . To complete the proof that this embedding is surjective it suffices to show that each member of the set {xα } is equivalent to a quotient of the form n/q for some nilpotent n ∈ T (R) and nonnil q ∈ T (R). This is quite simple. Let xσ ∈ {xα } and  consider the product q = σi with i ranging from 1 to nσ (the last nonzero σj ). This is a product of nonnil elements of T (R) so q is nonnil. By the construction of J, σn xσ = xσ−1 , σj xσ−k = xσ−(k+1) for k = nσ − j and 1 < j < nσ , and σ1 xσ−k = σ0 for k = nσ − 1. Thus qxσ = σ0 . In the localization T (R)N il(R) , we have xσ = σ0 /q. Therefore we may view T (R)N il(R) and AN il(A
) as the same ring.

As mentioned in the introduction, several specific types of φ-rings have been studied. A general definition for a φ-BLANK ring is that a φ-ring R is a φ-BLANK ring if R/N il(R) is a BLANK domain. For each of the types that have been studied so far, the specific definition involves properties of the nonnil ideals of R, sometimes with regard to the image of such ideals in φ(R). In each case, a consequence of the definition is that R is a φ-BLANK ring if and only if R/N il(R) is a BLANK domain. For example, given a φ-ring R, it is (i) a φ-chained ring if the nonnil ideals are linearly ordered – equivalent to R/N il(R) being chained (i.e., a valuation domain); (ii) a φ-pseudo-valuation ring if each nonnil prime is strongly prime with respect to T (R) (meaning if rt ∈ P for some r, t ∈ T (R), then either r ∈ P or t ∈ P ) – equivalent to R/N il(R) being a pseudo-valuation domain; (iii) a φ-Pr¨ ufer ring if for each finitely generated nonnil ideal I, φ(I) is an invertible ideal of φ(R) – equivalent to R/N il(R) being a Pr¨ ufer domain; (iv) a φ-B´ezout ring if each finitely generated nonnil ideal is principal – equivalent to R/N il(R) being a B´ezout domain; and (v) a φ-Mori ring if it satisfies the ascending chain condition on those nonnil ideals I for which φ(I) is divisorial as an ideal of φ(R) – equivalent to R/N il(R) being a Mori domain (a domain with a.c.c. on divisorial ideals). [For references see [4], [2], [3], [1], and [8].] For each of these special types of φ-rings, we have the following corollary to Theorem 4.3.

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Corollary 4.4. Let R be a ring with nonzero nilradical N il(R). For BLANK any one of “chained”, “pseudo-valuation”, “Pr¨ ufer”, “B´ezout”, “Mori”, and “Noetherian”, if N il(R) is prime with R/N il(R) a BLANK domain and N il(R)2 = (0), then there is a φ-BLANK ring A for which statements (I)–(VI) hold for the pair R and A. Note that with regard to idealization, if D is an integral domain and B is a D-module, then the process used in the proof of Theorem 4.3 can be used to build a divisible D-module C such that the pair R = D(+)B and A = D(+)C satisfy statements (I)–(VI). It follows that if D is Noetherian, the resulting ring A is a φ-Noetherian ring. We have been unable to extend Theorem 4.3 to rings with prime nilradicals whose squares are not zero. For some such rings it is not hard to show that it is impossible to find a φ-ring A that satisfies conditions (I) through (VI). In particular, one necessary condition for the existence of A is that each nonzero nilpotent with a nonnil annihilator must annihilate N il(R). To see this assume rn = 0 for some nonnil element r ∈ R and nonzero nilpotent n. If N il(A) is divided, then for each b ∈ N il(R), there is a nilpotent m ∈ N il(A) such that rm = b. Since nr = 0, we must have nb = 0 as well. Hence nN il(R) = (0). A specific example of a ring with prime nilradical which cannot be embedded in a φ-ring is the ring R = K[Y, Z]/(Y3 , YZ). The nilradical of R is the prime ideal yK[y, z] (where the lower case letters represent the images of the indeterminates in R) and y has a nonnil annihilator, namely the element z. As y 2 = 0, it is impossible to embed R in a ring whose nilradical is divided.

Bibliography [1] D. F. Anderson and A. Badawi, On φ-Pr¨ ufer rings and φ-B´ezout rings, to appear in Houston J. Math. [2] A. Badawi, On φ-pseudo-valuation rings, Advances in Commutative Ring Theory (Fez, Morocco, 1997), Lecture Notes Pure Appl. Math., Marcel Dekker 205 (1999), 101–110. [3] A. Badawi, On φ-pseudo-valuation rings II, Houston J. Math. 26 (2000), 473– 480. [4] A. Badawi, On φ-chained rings and φ-pseudo-valuation rings, Houston J. Math. 27 (2001), 725–736. [5] A. Badawi, On nonnil-Noetherian rings, Comm. Algebra 31 (2003), 1669– 1677. [6] A. Badawi, On divided rings and φ-pseudo-valuation rings, Int. J. of Commutative Rings 1 (2002), 51–60. [7] A. Badawi, On divided commutative rings, Comm. Algebra 27 (1999), 1465– 1474. [8] A. Badawi and T. Lucas, On φ-Mori rings, preprint. [9] D. E. Dobbs, Divided rings and going-down, Pacific J. Math. 67 (1976), 353– 363. [10] L. Fuchs and L. Salce, Modules over Non-Noetherian Domains, Mathematical Surveys and Monographs, Vol 84, American Mathematical Society, Providence, 2001. [11] R. Gilmer, Multiplicative Ideal Theory, Queen’s Papers Pure Appl. Math. Vol 90, Queen’s University Press, Kingston, 1992. [12] R. Gilmer, W. Heinzer, and M. Roitman, Finite generation of power of ideals, Proc. Amer. Math. Soc. 127 (1999), 3141-3151. [13] J. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker, New York/Basel, 1988. 211

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[14] I. Kaplansky, Commutative Rings – rev. ed., The University of Chicago Press, Chicago, 1974.

Chapter 12

On the Ideal Generated by the Values of a Polynomial by

Jean-Luc Chabert and Sabine Evrard Abstract Pn k ∈ K[X] where Let D be a Dedekind domain and let f (X) = k=0 ak X K denotes the quotient field of D. According to P´ olya, for every maximal + wq (deg f ) ideal m of D with norm q, one has: vm (f (D)) ≤ vm (f (X)) P where vm denotes the valuation on K associated to m, wq (n) = k≥1 [n/q k ], vm (f (D)) = inf{vm (f (b)) | b ∈ D}, and vm (f (X)) = inf{vm (ak ) | 0 ≤ k ≤ n}. Vˆ ajˆ aitu gives another bound for vm (f (D)) in the case where D = Z. We extend this bound to more general cases and, in particular, to Dedekind domains: as soon as deg(f ) ≤ p(q − 1) where p = char(D/m), one has vm (f (D)) < vm (f (X)) + νm (f ) with νm (f ) = Card {k | vm (ak ) = vm (f (X))}.

1

Introduction

Let D be a Dedekind domain with quotient field K and let f=

n

ak X k ∈ K[X]

(12.1)

k=0

be a polynomial of degree n. We denote by C(f ) the content of f , that is, the ideal of D generated by the coefficients of f and by D(f ) the divisor of f , that is, the ideal generated by the values of f on D. In this introduction we assume that f is primitive, that is, C(f ) = D. When f ∈ Z[X], it is well-known that the gcd of the values of f on Z divides n!. For Dedekind domains, this result was generalized by P´ olya [8, S4] in the following way (see also [5, II.3.3]): the ideal D(f ) divides the nth factorial ideal n!D where n!D is defined by  mwN (m) (n) (12.2) n!D = m∈max(D), N (m)≤n

213

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with N (m) = Card(D/m)

(12.3)

$n% . ql

(12.4)

and wq (n) =

l≥1

Writing the ideal D(f ) in the following form  D(f ) = mdm (f ) ,

(12.5)

m∈max(D)

this divisibility relation may be written as inequalities. For each maximal ideal m of D, one has: (12.6) dm (f ) ≤ wN (m) (n). The aim of this chapter is to state another divisibility relation making use of the number of coefficients of f not belonging to m instead of the degree of f . More precisely, let / m}; (12.7) µm (f ) = Card {ak | ak ∈ we are going to prove that dm (f ) < µm (f ).

(12.8)

But this inequality holds only for small values of n = deg(f ), namely: deg(f ) ≤ char(D/m) × (N (m) − 1).

(12.9)

Vˆajˆ aitu [9, Theorem 2] proved Inequality (12.8) when D = Z. Here, we generalize it to every Dedekind domain D (Proposition 4.1). Then, we extend it to the ideal D(f, E) generated by the values of f on a subset E of D when D = V is a discrete valuation domain (Proposition 5.4).

2

Technical Preliminaries

Notation. For every polynomial f , we denote by µ(f ) the number of nonzero coefficients of f . The following technical result is implicitly contained in the proof of [9, Thm 2]. Lemma 2.1. Let k be a field with characteristic p > 0 such that k p = k (for instance, a finite field). Let f ∈ k[X] be a nonzero polynomial and let z ∈ k be a nonzero root of f with multiplicity m. If m < p, then m < µ(f ). In order to prove this lemma we associate to every polynomial f an integer s(f ) < µ(f ) defined by means of the following algorithm: Algorithm A. s ← 0 , fs ← f

On the Ideal Generated by the Values of a Polynomial

215

while µ(fs ) > 1 do begin s ← s + 1  fs ← X vXfs(fs ) end s(f ) = s where vX (g) denotes the least degree of the monomials of g and the symbol  denotes the formal derivation of polynomials. It is clear that the procedure is finite since, for every g = 0, one has:   g  µ < µ(g). X vX (g) Consequently, s(f ) < µ(f ). Proof. of Lemma 2.1. Assume that m ≥ µ(f ). Then z is a root of all the polynomials fs constructed in Algorithm A with multiplicity m − s ≥ m − s(f ) > m − µ(f ) ≥ 0. By definition of s(f ), one has µ(fs(f ) ) ≤ 1. If µ(fs(f ) ) = 1, then fs(f ) is of the form bX h and z cannot be a root of fs(f ) . Consequently, fs(f ) = 0, and hence, fs(f )−1 (X) = b1 X h1 + . . . + bl X hl with l ≥ 2, bj ∈ k ∗ , h1 < h2 < · · · < hl , hj − h1 = pmj with mj ∈ N∗ . By hypothesis, for j = 1, . . . , l, bj = cpj with cj ∈ k ∗ . Thus, fs(f )−1 (X) = X h1

l

cpj X pmj

⎛ ⎞p l = X h1 ⎝ cj X m j ⎠ .

j=1

j=1

 Since z is a root of fs(f )−1 , z is a root of lj=1 cj X mj . Consequently, the multiplicity of z as a root of fs(f )−1 is a nonzero multiple of p. But this multiplicity is m−s(f )+1, so m ≥ p, which contradicts the hypothesis. Corollary 2.2. Let k be a field of characteristic p > 0 such that k p = k and let f be a nonzero polynomial in k[X]. If x ∈ k is a root of f with multiplicity mx < p then, for every y ∈ k, y = x, one has : mx < µ(f (X + y)). Proof. It suffices to use Lemma 2.1 with the polynomial g(X) = f (X + y) and its root z = x − y.

216

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Polynomials with Coefficients in a Discrete Valuation Domain

Hypotheses and notation for Section 3. Let V be a discrete valuation domain with finite residue field. Denote by K the quotient field of V , v the corresponding valuation of K, m the maximal ideal of V , π a generator of m, k = V /m the residue field, p the characteristic of k, and q = pf its cardinality. For every nonzero polynomial f (X) =

n

ai X i ∈ K[X],

(12.10)

i=0

we consider the following integers: v(f ) = inf

v(ai ),

(12.11)

d(f ) = inf v(f (a)),

(12.12)

ν(f ) = #{i | v(ai ) = v(f )}.

(12.13)

0≤i≤n

a∈V

Note that ν(f ) = µ(f˜) where f˜ denotes the image of

1 f π v(f )

in k[X].

Clearly, v(f ) ≤ d(f ),

(12.14)

and we also know that (see for instance [8] or [5, Corollary II.2.13]) d(f ) ≤ v(f ) + wq (deg(f )) where wq is defined by: wq (n) =

$n% . ql

(12.15)

(12.16)

l≥1

In particular, if deg(f ) < q, then d(f ) = v(f ). Here we prove another inequality for d(f ): Proposition 3.1. With the previous hypotheses and notation, for every nonzero polynomial f in K[X] such that deg(f ) ≤ p(q − 1) + 1,

(12.17)

v(f ) ≤ d(f ) < v(f ) + ν(f ).

(12.18)

one has: Proof. We may replace f by π −v(f ) f and assume that f is primitive in V [X], that is, v(f ) = 0. Note that ν(f ) ≥ 1 and, if ν(f ) = 1, then necessarily d(f ) = 0. Consequently, one may also assume that d(f ) ≥ 2.

On the Ideal Generated by the Values of a Polynomial

217

First, we recall some classical results concerning the values of a polynomial. Let u0 = 0, u1 , . . . , uq−1 be a complete system of representatives of V modulo m. We extend the sequence ur in the following way: for r = r0 + r1 q + . . . + rl q l

where 0 ≤ ri < q,

we let u r = u r0 + u r1 π + . . . + u rl π l . Clearly, the following sequence of polynomials gi (X) =

i−1 

(X − uj ) , i ∈ N

j=0

is a basis of the V -module V [X]. Then let f (X) =

n

bi gi (X) with bi ∈ V.

i=0

We know, and this is easy to check, that the ideal generated by the values of f on V is equal to the ideal generated by the values of f on u0 , u1 , . . . , un where  n = deg(f ) [5, Corollary II.2.9], that is, the ideal generated by the bi j ν(f ) − 1 as soon as p ≥ 5 ). Let us introduce another notation: for each a ∈ V , let νa (f ) = ν(f (X + a)). In particular, ν0 (f ) = ν(f ). Let ν˜ (f ) = inf νa (f ) = inf ν(f (X + a)). a∈V

a∈V

Of course, v(f (X)) = v(f (X + a)) and d(f (X)) = d(f (X + a)). Consequently, Corollary 3.3. If deg(f ) ≤ p(q − 1) + 1, then v(f ) ≤ d(f ) < v(f ) + ν˜(f ).

(12.19)

Example 3.4. Let p be a prime number ≥ 5, let V = Z(p) , and let f (X) = (X − 1)p−1 − 1. Then, on the one hand, for every a ∈ Z, f (X + a) ≡ (X + (a − 1))p−1 − 1

(mod p).

If a ≡ 1 (mod p), then νa (f ) = p − 1. If a ≡ 1 (mod p), then νa (f ) = 2. On the other hand, f (1) = −1, and hence, d(f ) = 0. Finally, d(f ) = 0 < ν˜ (f ) − 1 = 1 < ν(f ) − 1 = p − 2. Thus, we may have strict inequalities. Nevertheless: Remark. For n ≤ p(q − 1) < q 2 , one has wq (n) = [ nq ] ≤ p − 1, and hence, it follows from Inequalities (12.15) and (12.18) that, if f ∈ V [X] is primitive of degree

then

n ≤ p(q − 1) + 1,

(12.20)

$ %  n d(f ) ≤ min , ν(f ) − 1 . q

(12.21)

Moreover, both inequalities are sharp. Inequality (12.20) is sharp as shown by the following example: f (X) = X 2 (X q−1 − 1)p , deg(f ) = p(q − 1) + 2 , ν(f ) ≤ 2 , d(f ) = 2.

On the Ideal Generated by the Values of a Polynomial

219

Inequality (12.21) is sharp in the following sense: for every integer ν between 1 and p, there exists a polynomial f primitive in V [X] of degree n ≤ p(q − 1) + 1 such that d(f ) = ν(f ) − 1 = ν − 1: For 0 ≤ k ≤ p − 1, the polynomial fk (X) = (X q − X)k satisfies d(fk ) = k and ν(fk ) = k + 1 with deg(fk ) = kq ≤ (p − 1)q ≤ p(q − 1) + 1.

4

Polynomials with Coefficients in a Dedekind Domain

Now we globalize the previous results. Hypotheses and notation for section 4. Let D be a Dedekind domain with quotient field K. For every maximal ideal m of D, we denote by vm the corresponding valuation of K, by pm the characteristic of the residue field D/m, and by qm its cardinality (finite or infinite). For every polynomial f=

n

ai X i ∈ K[X],

(12.22)

i=0

C(f ) denotes the content of f , that is, the fractional ideal of D generated by the coefficients of f and D(f ) denotes the divisor of f , that is, the fractional ideal generated by the values of f on D. For every maximal ideal m of D, we introduce the following integers: vm (f ) = inf

vm (ai ),

(12.23)

dm (f ) = inf vm (f (a)),

(12.24)

νm (f ) = #{i | vm (ai ) = vm (f )}.

(12.25)

0≤i≤n

a∈D

Obviously, C(f ) =



mvm (f ) ,

(12.26)

mdm (f ) .

(12.27)

m∈max(D)

D(f ) =



m∈max(D)

Clearly, the ideal C(f ) divides the ideal D(f ) and it is known [5, Proposition II.3.3] that D(f ) divides C(f ) × n!D where the ideal n!D is defined by Formula (12.2), in other words, for every maximal ideal m of D, analogously to Formulas 12.14 and 12.15, one has the inequalities: vm (f ) ≤ dm (f ) ≤ vm (f ) + wN (m) (deg(f )) where N (m) and wN (m) are defined by Formulas (12.3) and (12.4). Proposition 3.1 may be globalized in the following way:

(12.28)

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Proposition 4.1. Let D be a Dedekind domain with quotient field K and let f ∈ K[X] be a nonzero polynomial of degree n. With the previous notation, for every maximal ideal m of D such that n ≤ pm (qm − 1) + 1,

(12.29)

dm (f ) < vm (f ) + νm (f ).

(12.30)

one has: This proposition is the extension of Theorem 2 of Vˆajˆ aitu [9] from Z to every Dedekind domain D. Theorem 3 of [9] corresponds to the first example below. Examples 4.2. In these three examples, f denotes a polynomial of degree n primitive in D[X]. 1) Let D = Z and denote by P the set of prime integers. Then, a generator of D(f ) divides the integer   n pwp (n) × pmin([ p ],νp (f )−1) . √ 3 1 √ 3 1 p∈P, p
1 =ht(J) . Also, J is not wB-unmixed since Q ∈Min(J) ⊆ Assf (J) , so Q, P ∈Assf (J) , but Q  P. Therefore, R is neither wB-ht-unmixed nor wB-unmixed. As a corollary to theorem 3.1 we get a complete classification of the unmixedness of Pr¨ ufer domains. Pr¨ ufer domains have been described by Gilmer [7, Chapter IV] as playing a central role in multiplicative ideal theory; so having this characterizations is important to making this unmixedness theory useful. Corollary 3.2. A Pr¨ ufer domain R is wB-ht-unmixed/wB-unmixed if and only if dim (R) ≤ 1. Proof. One direction is an immediate consequence of the definition of unmixedness from which it follows that all zero-dimensional rings and all one-dimensional domains are unmixed (see [1]). Now, suppose R is an unmixed Pr¨ ufer domain. By theorem 3.1, R must satisfy PIT. In [8] it was shown that a Pr¨ ufer domain R satisfies PIT if and only if dim (R) ≤ 1. Therefore, dim (R) ≤ 1. A more general form of the PIT condition is the GPIT (generalized principal ideal theorem) condition. A ring satisfies GPIT if whenever P is a prime ideal which is minimal over a (proper) ideal generated by n elements, ht(P ) ≤ n. Every Noetherian ring satisfies GPIT. This fact is often referred to as Krull’s generalized principal ideal theorem. One good source for more information on these conditions is the paper by Anderson, Dobbs, Eakin and Heinzer [9]. Our next theorem demonstrates

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that within the class of rings which satisfy GPIT the concepts of wB-ht-unmixed and wB-unmixed are equivalent. Theorem 3.3. If R satisfies GPIT, then R is wB-ht-unmixed if and only if R is wB-unmixed. Proof. Suppose R is a ring which satisfies GPIT. ( =⇒ ) Suppose R is wB-ht-unmixed and let I be a height-generated ideal in R. In a ring which satisfies GPIT every ideal I satisfies ht(I) ≤ µ (I) where µ (I) denotes the minimal number of generators of I. Thus, in a ring with GPIT, I is height-generated if and only if ht(I) = µ (I) < ∞. To show that I is wB-unmixed, suppose P, Q ∈ Assf (I) with P ⊆ Q. Since R is wB-ht-unmixed we have that I is wB-ht-unmixed. Thus, ht(P ) =ht(Q) = ht(I) < ∞. So, P and Q are prime ideals with P ⊆ Q and ht(P ) =ht(Q) < ∞. Thus, P = Q and so I is wB-unmixed. Therefore, R is wB-unmixed. (⇐=) Suppose R is wB-unmixed and let I be a height-generated ideal in R. As in the first part of this proof, we have ht(I) = µ (I) < ∞. Let n =ht(I) = µ (I) . Note that, since R satisfies GPIT, we have ht(P ) ≤ n for all P ∈ Min(I) . However, since ht(I) ≤ ht(P ) for all P ∈Min(I) and ht(I) = n, we have ht(P ) = n for all P ∈ Min(I) . Since I is wB-unmixed, we have Assf (I) = Min(I) . Therefore, ht(P ) = n for all P ∈ Assf (I) and thus, I is wB-ht-unmixed. Therefore, R is wB-ht-unmixed. The following result and corollary follow from applying this theorem, and the fact that the condition GPIT localizes, to a result from [1, Theorem 3]. Theorem 3.4. Let R be a ring with GPIT. If RM is a wB-ht-unmixed for every maximal ideal M in R, then R is wB-ht-unmixed. Corollary 3.5. If R is a ring such that RM is a wB-ht-unmixed ring satisfying GPIT (so RM is also wB-unmixed by theorem 3.3) for every maximal ideal M in R, then R is wB-ht-unmixed (and wB-unmixed). Using this corollary we can now show that every locally Cohen-Macaulay ring is both wB-ht-unmixed and wB-unmixed. A ring R is said to be locally Noetherian if RM is Noetherian for every maximal ideal M in R. Locally Noetherian rings were studied extensively by Heinzer and Ohm [10]. Since GPIT localizes and Noetherian rings satisfy GPIT, we have that every locally Noetherian ring satisfies GPIT. Example 2.2 from [10] gives an example of a non-Noetherian locally Noetherian domain D whose localizations DP are all Noetherian valuation domains. Noetherian valuation domains are Cohen-Macaulay; so this is an example of a non-Noetherian

Unmixedness and the Generalized Principal Ideal Theorem

287

ring which is locally Cohen-Macaulay where locally Cohen-Macaulay is defined analogously to locally Noetherian. Corollary 3.6. Every locally Cohen-Macaulay ring is wB-ht-unmixed. Proof. Suppose R is locally Cohen-Macaulay. Then RM is Cohen-Macaulay for every maximal ideal M in R. Cohen-Macaulay rings are Noetherian; therefore each localization RM is Noetherian and, therefore, satisfies GPIT. Cohen-Macaulay rings are wB-ht-unmixed (see [2]). So each localization RM is wB-ht-unmixed. Apply corollary 3.5 and we have that R is wB-ht-unmixed. Note that locally Cohen-Macaulay rings are also wB-unmixed (by theorem 3.3) since they satisfy GPIT. Applying this corollary to the example from [10] cited above we see that the example is wB-ht-unmixed. In this case, however, this corollary was unnecessary since the ring D given in the example is a one-dimensional domain and it can easily be shown that, in general, one-dimensional domains are wB-htunmixed.

4

Unmixedness of the Ring k [x1 , x2, . . .]

We would like to have the non-Noetherian ring k [X1 , X2 , . . .] on our list of nonNoetherian Cohen-Macaulay rings. One reason for this is that, in his work that led to the theory of Cohen-Macaulay rings, Macaulay was studying polynomial rings (in finitely many variables) over a field. In [1] it was shown that this ring is wB-unmixed by showing that R [X1 , X2 , . . .] is wB-unmixed for every Cohen-Macaulay domain R. In the proof of this fact (see [1, theorem 4]), the only place where the fact that R is a domain was used was in part (2) of lemma 6. Lemma 4.1 below shows that the assumption that R is a domain was unnecessary. Therefore, we have R [X1 , X2 , . . .] is wB-unmixed for every Cohen-Macaulay ring R. Lemma 4.1. Let R be a Noetherian ring and let S = R [X1 , X2 , . . .] . For any prime ideal P in R we have ht(P ) = ht(P S) where ht(P S) refers to the height of the ideal P S in S. Note that the proof of this lemma depends only on the weaker condition that R is a strong S-ring (see [6] for more information on strong S-rings). It is not necessary for the ring to be Noetherian. Proof. First note that we have trivially that ht(P ) ≤ht(P S) since the extensions of a chain of distinct prime ideals in R is a chain of distinct prime ideals in S. For i ≥ 1, let Ri = R [X1 , X2 , . . . , Xi ] so S = lim Ri . Let Pi = P Ri . Since R −→

is Noetherian (and thus a strong S-ring) we have ht(P ) = ht(Pi ) [6, theorem 149, page 108].

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Now, suppose ht(P S) > n where n = ht(P ) . Then there is a chain of prime ideals Q0 ⊂ Q1 ⊂ · · · ⊂ Qn+1 = P S in S. For 1 ≤ i ≤ n + 1, choose xi ∈ Qi \ Qi−1 . Since S = lim Ri , there is a positive integer j such that {x1 , x2 , . . . , xn+1 } ∈ Rj . For 0 ≤ i ≤ n + 1, let Ti = Qi ∩ Rj . Then

−→

T0 ⊂ T1 ⊂ · · · ⊂ Tn+1 is a chain of prime ideals in Rj . So, ht(Tn+1 ) ≥ n+1. However, Tn+1 = Qn+1 ∩Rj = P S∩Rj = Pj and we have already noted that ht(Pj ) = n so we have a contradiction. Therefore, ht(P S) = ht(P ) . In [8, Proposition 6.4] it was shown that R [X1 , X2 , . . .] satisfies GPIT (if R is a Noetherian ring). The statement of this fact in [8] actually makes the assumption that R is a domain; however, the fact that R is a domain, is not necessary in the proof given in [8], so we will use the more general result. By applying theorem 3.3 to this result, we get the following theorem. Theorem 4.2. Let R be a Cohen-Macaulay ring. Then R [X1 , X2 , . . .] is wB-htunmixed. The desired result regarding the ring k [X1 , X2 , . . .], where k is a field, follows as an easy corollary to this since every field is a Cohen-Macaulay ring. Corollary 4.3. For any field k, the ring k [X1 , X2 , . . .] is a wB-ht-unmixed ring.

5

Unmixedness and Direct Sums

An interesting result about wB-ht-unmixedness (resp. wB-unmixedness) is that wB-ht-unmixed (resp. wB-unmixed) rings which can be written as a direct sum will have wB-ht-unmixed (resp. wB-unmixed) components. Theorem 5.1. Let R be a wB-ht-unmixed (resp. wB-unmixed) ring and assume that R is the direct sum of two rings A and B, so R = A ⊕ B. Then, both A and B are wB-ht-unmixed (resp. wB-unmixed) rings. To prove this theorem we will need some preliminary lemmas. Lemma 5.2. Let A and B be commutative rings and let R = A ⊕ B. Then, for any ideal I in A we have ht (J) = ht (I) where J = I ⊕ B. Lemma 5.2 follows immediately from the fact that any prime ideal in R is of the form P ⊕ B for some prime ideal P in A or of the form A ⊕ Q for some prime ideal Q in B.

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Lemma 5.3. Let A and B be commutative rings and let R = A ⊕ B. Then, for any ideal I in A and any P ∈ Assf (I) , the prime ideal Q = P ⊕ B is in Assf (J) where J = I ⊕ B. Proof. Suppose I is an ideal in A and P ∈ Assf (I) . Let Q = P ⊕ B and J = I ⊕ B. Since P ∈ Assf (I) there is some element a ∈ R such that P ∈ Min(I : a) . We will show that Q ∈ Min(J : (a, 1)) , so Q ∈ Assf (J) . To show that Q ∈ Min(J : (a, 1)) we must first verify that (J : (a, 1)) ⊆ Q. It is easily verified that (J : (a, 1)) = (I : a) ⊕ B. So, since (I : a) ⊆ P, we have (J : (a, 1)) = (I : a) ⊕ B ⊆ P ⊕ B = Q. Now, suppose there is a prime ideal T in R such that (J : (a, 1)) ⊆ T ⊆ Q. Then (I : a) ⊕ B ⊆ T ⊆ Q. Any prime ideal in R is of the form S ⊕ B for some prime ideal S in A or of the form A ⊕ S for some prime ideal S in B. Since T is a prime ideal and (I : a) ⊕ B ⊆ T, we have T = S ⊕ B for some prime ideal S in A. Note that (I : a) ⊆ S. Also, since S ⊕ B = T ⊆ Q = P ⊕ B we have S ⊆ P. Therefore, S = P since P is minimal among prime ideals containing (I : a) . Thus, T = S ⊕ B = P ⊕ B = Q, so Q ∈ Min(J : (a, 1)) and, therefore, Q ∈ Assf (I) . Now, we are ready to prove theorem 5.1. Proof. Clearly it is sufficient to prove that if R = A ⊕ B is wB-ht-unmixed (resp. wB-unmixed), then A is wB-ht-unmixed (resp. wB-unmixed). Let I be a height-generated ideal in A. Then I = (a1 , a2 , . . . , an ) for some a1 , a2 , . . . , an ∈ A where ht(I) ≥ n. Let J = I ⊕ B. Then J is generated in R by the elements (a1 , 1) , (a2 , 1) , . . . , (an , 1) . Also, by lemma 5.2, ht(J) = ht(I) . Therefore, J is height-generated. Let P, Q ∈ Assf (I) with P ⊆ Q. Then, by lemma 5.3, P ⊕ B, Q ⊕ B ∈ Assf (J) . (Also, of course, P ⊕ B ⊆ Q ⊕ B.) • If we are assuming that R is wB-ht-unmixed, then, since J is height-generated in R and since P ⊕ B ∈ Assf (J) , we have ht(P ⊕ B) = ht(J) . By lemma 5.2, ht(P ) = ht(P ⊕ B) . Therefore, ht(P ) = ht(P ⊕ B) = ht(J) = ht(I) . So, A is wB-ht-unmixed. • On the other hand, if we are assuming that R is wB-unmixed, then, since J is height-generated in R and P ⊕ B, Q ⊕ B ∈ Assf (J) with P ⊕ B ⊆ Q ⊕ B, we have P ⊕ B = Q ⊕ B. Therefore, P = Q. So, A is wB-unmixed.

The converse of this theorem (theorem 5.1) is true if the ring R is assumed to satisfy GPIT. It is easily seen that for R = A ⊕ B, we have R satisfies GPIT if and only if A and B satisfy GPIT. Thus, it is sufficient to assume that R satisfies GPIT or that A and B satisfy GPIT to get the converse. Also, since R, A and B all satisfy GPIT in this theorem, wB-ht-unmixed is equivalent to wB-unmixed, so we will not have to consider two cases as we did in the proof of theorem 5.1 above.

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Theorem 5.4. Suppose R is the direct sum of two rings A and B, so R = A ⊕ B. Assume R satisfies GPIT (or equivalently that both A and B satisfy GPIT). Then R is wB-ht-unmixed (resp. wB-unmixed) if and only if A and B are both wB-htunmixed (resp. wB-unmixed). Proof. One direction of this theorem follows immediately from theorem 5.1. Also, as was mentioned above, it is sufficient to prove the wB-ht-unmixed version of this theorem. We need to show that if A and B are wB-ht-unmixed rings, then R = A ⊕ B is wB-ht-unmixed. Suppose A and B are wB-ht-unmixed and that R = A ⊕ B satisfies GPIT (so A and B also satisfy GPIT). Throughout the following we will make use of the following notation. If J is an ideal in R, we define JA = {a ∈ A : (a, b) ∈ J for some b ∈ B} and JB = {b ∈ B : (a, b) ∈ J for some a ∈ A}. It is easy to see that J ⊆ JA ⊕ JB and, in fact, J ⊆ S ⊕ T (where S is an ideal in A and T is an ideal in B) if and only if JA ⊆ S and JB ⊆ T. Let I be a height-generated ideal in R of height n. Then I = ((a1 , b1 ) , (a2 , b2 ) , . . . , (an , bn )) for some a1 , a2 , . . . , an ∈ A and some b1 , b2 , . . . , bn ∈ B. It follows easily that IA = (a1 , a2 , . . . , an ) and IB = (b1 , b2 , . . . , bn ) . Recall that every prime ideal in R is of the form P ⊕ B or A ⊕ Q where P ∈ Spec(A) and Q ∈ Spec(B) , so n = =

ht (I) = min {ht (T ) : T ∈ Spec (R) and I ⊆ T } min {min {ht (P ⊕ B) : P ∈ Spec (A) and I ⊆ P ⊕ B} , min{ht (A ⊕ Q) : Q ∈ Spec (B) and I ⊆ A ⊕ Q}}.

It is easily seen (see lemma 5.2) that, for P ∈ Spec(A) and Q ∈ Spec(B), we have ht(P ⊕ B) = ht(P ) and ht(A ⊕ Q) = ht(Q) . Also, I ⊆ P ⊕ B if and only if IA ⊆ P and I ⊆ A ⊕ Q if and only if IB ⊆ Q. Therefore, n = =

min {min {ht (P ) : P ∈ Spec (A) and IA ⊆ P } , min{ht (Q) : Q ∈ Spec (B) and IB ⊆ Q}} min {ht (IA ) , ht (IB )} .

This gives us that n ≤ ht(IA ) and n ≤ ht(IB ) . However, IA and IB are both ideals which can be generated by n elements in the GPIT satisfying rings A and B respectively. So, ht(IA ) ≤ n and ht(IB ) ≤ n. Therefore, IA and IB are heightgenerated ideals in their respective rings of height n. Thus, since A and B are both assumed to be wB-ht-unmixed rings we have ht(P ) = n for all P ∈ Assf (IA ) and ht(Q) = n for all Q ∈ Assf (IB ). Suppose T ∈ Assf (I) . We need to show ht(T ) = n. Since T is a prime ideal T = P ⊕ B for some P ∈ Spec(A) or T = A ⊕ Q for some Q ∈ Spec(B) . Without loss of generality, suppose T = P ⊕ B for some P ∈

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291

Spec(A) . Since T ∈ Assf (I) , we have T ∈ Min(I : (a, b)) for some (a, b) ∈ R. So, P ⊕ B ∈ Min(I : (a, b)) . This will be true if and only if P ∈ Min(I : (a, b))A . Claim: (I : (a, b))A = (IA : a) • Note that (I : (a, b)) = ⊆ = =

{(x, y) : (ax, by) ∈ I} {(x, y) : ax ∈ IA and by ∈ IB } {(x, y) : x ∈ (IA : a) and y ∈ (IB : b)} (IA : a) ⊕ (IB : b) .

Therefore, (I : (a, b))A ⊆ (IA : a) . • Also note that x ∈ (IA : a)

=⇒ =⇒ =⇒ =⇒ =⇒ =⇒

ax ∈ IA (ax, y) ∈ I for some y ∈ B (ax, y) (1, b) ∈ I for some y ∈ B (ax, by) ∈ I for some y ∈ B (x, y) ∈ (I : (a, b)) for some y ∈ B x ∈ (I : (a, b))A .

Therefore (IA : a) ⊆ (I : (a, b))A . Thus, the claim has been proven. So, P ∈ Min(IA : a) and therefore, P ∈ Assf (IA ) . Thus, ht(P ) = ht(IA ) = n, which give us ht(T ) = ht(P ) = n = ht(I) . Thus, I is wB-ht-unmixed. Therefore, R is wB-ht-unmixed.

Bibliography [8] David F. Anderson, Valentina Barucci, and David E. Dobbs, Coherent Mori domains and the principal ideal theorem, Comm. Algebra 15(6) (1987), 1119– 1156. [9] David F. Anderson, David E. Dobbs, Paul M. Eakin, Jr., and William J. Heinzer, On the generalized principal ideal theorem, Pacific J. Math. 146(2) (1990), 201–215. [7] Robert W. Gilmer, Multiplicative ideal theory, volume 12 of Pure and Applied Mathematics, Marcel Dekker, Inc., 1972. [3] Sarah Glaz, Coherence, regularity and homological dimensions of commutative fixed rings, in Ngˆ o Viˆet Trung, Aron Simis and Giuseppe Valla, editors, Commutative Algebra, Workshop on Commutative Algebra, pages 89–106, World Scientific, September 1992. [4] Sarah Glaz, Homological dimensions of localizations of polynomial rings, David F. Anderson and David E. Dobbs, editors, Zero-dimensional Commutative Rings, Lecture Notes in Pure and Applied Mathematics, John H. Barrett Memorial Lectures and Conference on Commutative Ring Theory, Marcel Dekker 171 (1994), 209–222. [2] Tracy Dawn Hamilton, Weak Bourbaki height-unmixed rings: Another step towards non-Noetherian Cohen-Macaulayness, submitted for publication. [1] Tracy Dawn Hamilton, Weak Bourbaki unmixed rings: A step towards nonNoetherian Cohen-Macaulayness, Rocky Mountain J. Math. 34(2) (2004). [10] William Heinzer and Jack Ohm, Locally Noetherian commutative rings, Trans. Amer. Math. Soc. 158(2) (1971), 273–284. [5] Juana Iroz and David E. Rush, Associated prime ideals in non-Noetherian rings, Canad. J. of Math. XXXVI(2) (1984), 344–360. [6] Irving Kaplansky, Commutative rings, Alyn and Bacon, 1970.

292

Chapter 19

A Note on Sets of Lengths of Powers of Elements of Finitely Generated Monoids by

Wolfgang Hassler1 Abstract Let H be a finitely generated commutative, cancellative monoid. Then every nonunit a ∈ H decomposes (in general in a highly nonunique way) into a product (19.1) a = u1 · . . . · un of irreducible elements (atoms) of H. The integer n is called the length of the factorization (19.1). We call L(a) = {n ∈ N | a has a factorization into n irreducible elements of H} the set of lengths of a. It is known that sets of lengths of elements of H have the following structure: they are, up to bounded initial and final segments, a union of arithmetical progressions with bounded distance. We say that two sets of lengths are of the same type if their initial and final segments coincide (up to a shift) and if their central parts have the same period. Clearly, the set of equivalence classes with respect to this equivalence relation is finite. In the present note we examine the structure of sets of lengths of powers of elements of H. We prove a theorem which asserts that there exist constants N, B ∈ N (which only depend on H) such that the following holds: if a ∈ H and k, l ≥ B with k ≡ l mod N , then L(ak ) and L(al ) are of the same type. This result carries over to Krull monoids with finite divisor class group, e.g. rings of integers of algebraic number fields. In a weaker form it is true for every Krull monoid.

2000 Mathematics Subject Classification. 11R27, 13A05, 20D60. Key words and phrases. factorization, irreducible element, set of lengths, finitely generated monoid. 1 This research was supported by a grant from the Fonds zur F¨ orderung der wissenschaftlichen Forschung, project number P16770–N12.

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Introduction and Preliminaries

By a monoid we always mean a (usually multiplicatively written) commutative semigroup H with identity element for which the cancellation law holds (i.e. ab = ac implies b = c for all a, b, c ∈ H). Let H be a monoid. An element u ∈ H\H × (where H × denotes the group of invertible elements of H) is called irreducible (or an atom) if u = ab implies that either a ∈ H × or b ∈ H × for all a, b ∈ H. The monoid H is called atomic if every nonunit a ∈ H possesses a factorization a = u1 · . . . · un

(19.2)

into irreducible elements ui of H. The integer n is called the length of the factorization (19.2). We call L(a) = LH (a) = {n ∈ N | n is length of some factorization of a} the set of lengths of a. During the last years, much effort was made to determine the structure of sets of lengths for certain classes of domains and monoids, e.g. finitely generated monoids, orders of global fields, Krull monoids, Congruence monoids, one-dimensional local domains and higher dimensional algebras over perfect fields. The reader is referred to [1], [5], [6], [12], [14] and [13] for the most recent results in this area. It turned out that sets of lengths in all mentioned classes of rings and monoids (in some cases under additional finiteness assumptions) have the following special structure: they are, up to bounded initial and final segments, a union of arithmetical progressions with bounded distance (see Definition 1.2 and Theorem 1.3 for the detailed statement in the case of finitely generated monoids). As a consequence of this structure theorem, one can classify sets of lengths by their initial and final segment and the period of their central part. Obviously, there are only finitely many classes (called “types”, see Definition 2.1). Very little is known about the behavior of the type of sets of lengths, yet. For example, given two elements a and b, what can be said about the type of L(ab) if the type of L(a) and L(b) is known? What kind of additional information on a and b is needed to determine L(ab)? In view of efficient methods for the computation of sets of lengths (and for theoretical reasons) it would be desirable to have “simple” criteria to determine the type of the sets of lengths of (at least certain classes) of elements. In this note we provide such a criterion for powers of elements of a finitely generated monoid H. Our research was initiated by the following question raised by F. Halter-Koch: let a ∈ H\H × (assuming that H is finitely generated) and consider the sequence L(an ) of sets of lengths. Do the initial and final segments of L(an ) repeat periodically if n grows? We answer this question positively by proving (cf. Theorem 2.2) that there exist positive integers N and B (which only depend on H) such that for every a ∈ H and for all integers k ≥ B and l ≥ B the congruence k ≡ l mod N already implies that L(ak ) and L(al ) are of the same type. This result immediately carries over to Krull monoids with finite divisor class group, e.g. rings of integers of algebraic number fields (Corollary 2.3). In a weaker form, the Theorem holds for every Krull monoid (see Lemma 2.4 and Corollary 2.5).

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295

We denote by N resp. N0 the set of positive resp. nonnegative integers. If r ∈ N we set N≥r = {x ∈ N | x ≥ r} (resp. N≤r = {x ∈ N | x ≤ r}). For a, b ∈ Z we set [a, b] = {x ∈ Z | a ≤ x ≤ b}. For sets A, B ⊂ Z we write A + B = {a + b | a ∈ A, b ∈ B} and if b ∈ Z, we set A + b = A + {b}. Let H be a monoid. Since H satisfies the cancellation law, we can form the quotient group of H. We denote it by Q(H). A submonoid H0 ⊂ H is called a saturated submonoid of H if a |H b implies a |H0 b for all a, b ∈ H0 . A monoid H is called reduced if H × = {1}. For a reduced atomic monoid H we denote by A(H) the set of atoms of H. The free monoid F (A(H)) generated by A(H) (which is isomorphic to the coproduct of |A(H)| copies of (N0 , +)) is called the factorization monoid of H. We denote it by Z(H). We have a canonical homomorphism πH : Z(H) → H which sends every (formal) product of atoms in Z(H) to the corresponding product in −1 (a) the set of factorizations of a. If H. For some a ∈ H we denote by Z(a) = πH  nq z = q∈A(H) q ∈ Z(H) we set vq (z) = nq for some q ∈ A(H). For y ∈ Z(H) we set Supp(y) = {q ∈ A(H) | vq (y) > 0}. The length function | · | : Z(H) → N0 and the distance function d : Z(H) × Z(H) → N0 (see for instance [3, section 2]) are defined as follows: If x, y, z ∈ Z(H),  then |z| = q∈A(H) vq (z) and     y x     d(x, y) = max  ,   . gcd(x, y) gcd(x, y) Here “gcd” denotes the greatest common divisor of x and y in the free monoid Z(H). Let H ⊂ D be monoids. We say that H is a divisor closed submonoid of D if d |D h implies d ∈ H for all h ∈ H, d ∈ D. For a ∈ D we denote by [[a]]D ⊂ D the smallest divisor closed submonoid of D containing a. It is easy to see that [[a]]D = {d ∈ D | d | ak for some k ∈ N}. Let H ⊂ D be a divisor closed submonoid of some atomic monoid D. Then every factorization of an element h ∈ H into atoms of D is a factorization into atoms of H and vice versa. In particular, H is atomic and LH (h) = LD (h) for every h ∈ H. Definition 1.1. i.) Let T ⊂ Z. Two elements k, l ∈ T are called successive elements of T if k = l and T ∩ [min{k, l}, max{k, l}] = {k, l}. ii.) We call ∆(T ) = {|k − l| | k and l are successive elements of T } ⊂ N the set of differences of T . (Observe that ∆(T ) = ∅ if and only if |T | ≤ 1.) iii.) Let H be an atomic monoid. We call  ∆(H) = ∆(L(a)) ⊂ N a∈H

the set of differences of H.

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The set of differences measures the size of “gaps” which occur in the sets of lengths of an atomic monoid. Definition 1.2. A nonempty, finite set L ⊂ Z is called an almost arithmetical multiprogression bounded by M ∈ N, if there exists some d ∈ [1, M ], some subset D ⊂ [0, d − 1] and a decomposition L = L1 ∪ L∗ ∪ L2 such that ∅ = L∗ , L1 ⊂ [−M, −1] + min L∗ , L2 ⊂ max L∗ + [1, M ] and L∗ = [min L∗ , max L∗ ] ∩ (D + dZ). The integer d is called the period length of L. If D is a singleton, then L is called an almost arithmetical progression bounded by M . Theorem 1.3 (Structure Theorem for sets of lengths). Let H be a finitely generated monoid. Then there exists some M ∈ N such that L(a) is an almost arithmetical multiprogression with bound M for every a ∈ H. Proof. See [5, Proposition 8.1]. For a geometric approach see [9, section 3]. The following remark is in order. Remark 1.4. Let H be a finitely generated monoid and a ∈ H. Then [[a]]H ⊂ H is finitely generated, too (cf. [7, Proposition 6.2]). By [3, Proposition 3.3] and [3, Proposition 3.4], every finitely generated monoid is locally tame and has finite catenary degree (see [3, Definition 2.3] and [3, Definition 3.1] for the relevant definitions.) From [4, Theorem 5.1] and [4, Corollary 5.2] it then follows that the sets L(an ) are almost arithmetical progressions (rather than multiprogressions) for all sufficiently large n ∈ N. Furthermore, their period length is min ∆([[a]]H ). Let H be an atomic monoid and a ∈ H\H × . Two integers k and l are called successive lengths of a if k = l and k and l are successive elements of L(a) (cf. Definition 1.1). A. Foroutan showed only recently (cf. [2, Theorem 3.9]) that for a finitely generated monoid H there exists some K ∈ N such that for all a ∈ H\H × and for all successive lengths k, l of a the following holds: if z ∈ Z(a) with |z| = k, then there exists some z  ∈ Z(a) with |z  | = l and d(z, z  ) ≤ K. The smallest possible K with this property is called the strong successive distance of H. It is denoted by δ(H).

2

Sets of Lengths of Powers

We begin with the classification of sets of lengths by their initial and final part and by the period of their central part. Definition 2.1. We say that two almost arithmetical multiprogressions L and L bounded by M are of the same type (with respect to M ) if (L − min L) ∩ [0, 2M ] = (L − min L ) ∩ [0, 2M ] and (L − max L) ∩ [−2M, 0] = (L − max L ) ∩ [−2M, 0]. Note that if two almost arithmetical multiprogressions are of the same type then they differ only by a shift and by the length of their central parts. Our aim is to show the following Theorem.

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Theorem 2.2 (Main Theorem). Let H be a monoid for which H/H × is finitely generated. Let M ∈ N be a constant such that every set of lengths of H is an almost arithmetical multiprogression bounded by M . Then there exist constants N, B ∈ N with the following property: if a ∈ H and m, n ∈ N≥B with m ≡ n mod N , then the sets of lengths L(am ) and L(an ) are of the same type (with respect to M ). This result immediately carries over to Krull monoids with finite divisor class group (see [10] for a comprehensive treatment of Krull monoids). To see this, we briefly sketch the theory of transfer homomorphisms and block monoids. For details see [1, section 3.1] or [9, section 5]. Let H be a Krull monoid with divisor class group G. Let G0 ⊂ G denote the set of classes containing prime divisors and let      B(G0 ) = g ng ∈ F(G0 )  ng g = 0 g∈G0

g∈G0

be the block monoid of G0 . Then there exists a (natural) monoid epimorphism β : H → B(G0 ) such that LH (h) = LB(G0 ) (β(h)) for all h ∈ H. In particular, {LH (h) | h ∈ H} = {LB(G0 ) (b) | b ∈ B(G0 )}. Corollary 2.3. Let H be a Krull monoid with finite divisor class group. Then there exist constants N, B ∈ N with the following property: if a ∈ H and m, n ∈ N≥B with m ≡ n mod N , then the sets of lengths L(am ) and L(an ) are of the same type. Proof. Let G be the divisor class group of H and let G0 ⊂ G denote the set of classes containing prime divisors. Clearly, the block monoid of G0 is finitely generated. Thus the assertion follows from Theorem 2.2 and from the theory of transfer homomorphisms described above. Corollary 2.3 in particular holds for the multiplicative monoid D\{0} of some Dedekind domain D with finite Picard group, e.g. for the ring of integers of some algebraic number field. Let H be a monoid. Following [8] we say that H is an SFF-monoid (strong finite factorization monoid) if [[a]]H /[[a]]× H is a finitely generated monoid for every a ∈ H. It immediately follows from the definition (and from the fact that finitely generated monoids are atomic) that SFF-monoids are atomic. Important examples for this class of monoids are Krull monoids. This seems to be well-known. However, we include a short proof of this fact for sake of completeness. Lemma 2.4. Let H be a Krull monoid. Then H is an SFF-monoid. Proof. Without restriction we may assume that H is reduced. Then H is a saturated submonoid of some free monoid F (cf. for instance [10] or [1]). If a ∈ H, then [[a]]H ⊂ [[a]]F is a saturated submonoid and [[a]]F is finitely generated (and indeed free). But then [[a]]H is finitely generated, too (cf. [7, Proposition 6.2]). Let H be an SFF-monoid. Then Theorem 1.3 immediately implies that for every a ∈ H there exists some M (a) ∈ N such that for every n ∈ N the set of lengths

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LH (an ) is an almost arithmetical multiprogression bounded by M (a). This makes it possible to say that the sets of lengths of two powers am and an are of the same type. Theorem 2.2 now implies: Corollary 2.5. Let H be an SFF-monoid. Then for every a ∈ H there exist constants N (a), B(a) ∈ N such that for all m, n ∈ N≥B(a) the congruence m ≡ n mod N (a) implies that L(am ) and L(an ) are of the same type. In order to prove Theorem 2.2, we need two preliminary Lemmas. Lemma 2.6 follows from [11, Lemma 6.6] and [3, Proposition 3.4]. Lemma 2.6. Let H be a finitely generated monoid and c ∈ H. Then the sets {| min L(cd) − min L(d)| | d ∈ H} and {| max L(cd) − max L(d)| | d ∈ H} are bounded. Definition 2.7. Let H be a reduced atomic monoid. A set S ⊂ A(H) is called universally minimal (resp. universally maximal) if |ξ| = min L(πH (ξ)) (resp. |ξ| = max L(πH (ξ))) holds for every ξ ∈ Z(H) with Supp(ξ) ⊂ S. The following Lemma is crucial for our proof of Theorem 2.2. Lemma 2.8. Let H be a finitely generated monoid. Then there exist constants N, C ∈ N such that for all a ∈ H and y, z ∈ Z(a) with |y| = min L(a) and |z| = max L(a) there exist factorizations ymin, zmax ∈ Z(aN ) with the following properties: Supp(ymin ) resp. Supp(zmax ) are universally minimal resp. maximal, {q ∈ A(H) | vq (y) ≥ C} ⊂ Supp(ymin ) and {q ∈ A(H) | vq (z) ≥ C} ⊂ Supp(zmax ). Proof. Without restriction we may assume that H is reduced. Set M = {(w, w ) ∈ Z(H) × Z(H) | πH (w) = πH (w ) and |w| < |w |}. A(H) A(H) Since Z(H) ∼ × = N0 , M is partially ordered by the induced partial order of N0 A(H) N0 . By Dickson’s Theorem (see [15, Satz 12]), the set Min(M) of minimal points of M is finite, say Min(M) = {(w1 , w1 ), . . . , (wr , wr )}. Furthermore, for every (w, w ) ∈ M there exists some i ∈ [1, r] with (wi , wi ) ≤ (w, w ). We proceed with the following construction: we show that there exists some C0 ∈ N such that for all b ∈ H and u1 , u2 ∈ Z(b) with |u1 | = min L(b) and |u2 | = max L(b) there exist u1 , u2 ∈ Z(b2 ) with |u1 | = min L(b2 ), |u2 | = max L(b2 ), d(u21 , u1 ) ≤ C0 and d(u22 , u2 ) ≤ C0 . Put

S = max{vq (wi ), vq (wi ) | i ∈ [1, r] , q ∈ A(H)}. Let b ∈ H and u1 , u2 ∈ Z(b) with |u1 | = min L(b) and |u2 | = max L(b). Set Ji = {q ∈ A(H) | vq (ui ) > S} and define   ξi = q vq (ui ) and ωi = q vq (ui ) q∈Ji

q∈A(H)\Ji

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for i ∈ [1, 2]. Set ci = πH (ξi ) and di = πH (ωi ). We claim that |ξ12 | = min L(c21 ) and |ξ22 | = max L(c22 ).

(19.3)

Assume on the contrary that there exists some σ ∈ Z(c21 ) whose length is strictly smaller than |ξ12 |. Then (σ, ξ12 ) ∈ M. Hence there exists some i ∈ [1, r] such that (wi , wi ) ≤ (σ, ξ12 ). Thus wi |Z(H) ξ12 . Since vq (wi ) ≤ S < vq (ξ1 ) for all q ∈ J1 , we infer that wi |Z(H) ξ1 . This shows that min L(c1 ) < |ξ1 |. But this is a contradiction since we assumed that u1 = ξ1 ω1 is the shortest factorization of b. The equation |ξ22 | = max L(c22 ) is proved by the same arguments. Now let D ∈ N by Lemma 2.6 be an upper bound for the sets {| min L(gh) − min L(h)| | h ∈ H} and {| max L(gh) − max L(h)| | h ∈ H}, where g ∈ {πH (z) | z ∈ Z(H) , vq (z) ≤ 2S for all q ∈ A(H)}. Then | min L(c21 d21 ) − |ξ12 || ≤ D and | max L(c22 d22 ) − |ξ22 || ≤ D. With b2 = c21 d21 we thus get | min L(b2 ) − |u21 || = | min L(b2 ) − |ξ12 | − |ω12 || ≤ D + |ω12 | ≤ D + 2S|A(H)|. Similarly, we get | max L(b2 ) − |u22 || ≤ D + 2S|A(H)|. Since the strong successive distance δ(H) of H is finite, there exists some C0 ∈ N such that for all h ∈ H, u ∈ Z(h) and l ∈ L(h) with ||u| − l| ≤ D + 2S|A(H)| there exists some factorization u ∈ Z(h) with |u | = l and d(u, u ) ≤ C0 . Thus there exist factorizations u1 , u2 ∈ Z(b2 ) such that |u1 | = min L(b2 ), |u2 | = max L(b2 ), d(u21 , u1 ) ≤ C0 and d(u22 , u2 ) ≤ C0 . Now we get to the core of the proof. Put C = max{C0 + 1, S} and let N = lcm {2γ − 2δ | 0 ≤ δ < γ ≤ (C + 1)|A(H)| } ∈ N. In the following we prove only the existence of a factorization ymin with the required properties. The existence of zmax is shown by completely analogous arguments. Let a ∈ H and y ∈ Z(H) with |y| = min L(a). We construct inductively a i i sequence (vi )i∈N0 of factorizations vi ∈ Z(a2 ) with |vi | = min L(a2 ) as follows: put v0 = y and assume that v0 , . . . , vi are already constructed. By the above i+1 i+1 construction, let vi+1 ∈ Z(a2 ) be a factorization with |vi+1 | = min L(a2 ) and d(vi2 , vi+1 ) ≤ C0 . The next step is to define a map ϕ : Z(H) −→ ([0, C − 1] ∪ {∞})A(H) by setting

 ϕ(z)q =

vq (z) if vq (z) < C ∞ if vq (z) ≥ C

for z ∈ Z(H) and q ∈ A(H). Then there exist α, β ∈ [0, (C + 1)|A(H)| ] with α < β N and ϕ(vα ) = ϕ(vβ ). Put N0 = 2β −2α and define ymin = (vβ vα−1 ) N0 ∈ Q(Z(H)). We claim that ymin ∈ Z(aN ) and that ymin has the required properties of the Lemma. Suppose that ϕ(vi )q = ∞ for i ∈ N0 and q ∈ A(H). Since d(vi2 , vi+1 ) ≤ C0 , we get ||vi+1 | − 2|vi || ≤ C0 and vq (vi+1 ) − vq (vi ) > vq (vi+1 ) − 2vq (vi ) + C0 ≥ −C0 + C0 = 0. Thus vq (vβ vα−1 ) > 0. Therefore, ymin ∈ Z(H) and {q ∈ A(H) | vq (y) ≥ C} ⊂ β Supp(ymin ). Since vβ is a shortest factorization of a2 , the same argument as in the proof of (19.3) shows that every σ ∈ Z(H) with Supp(σ) ⊂ {q ∈ A(H) | vq (vβ ) ≥ S}

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is a shortest factorization of πH (σ). Hence Supp(ymin ) is universally minimal. This completes the proof. We are now prepared to prove our main result. Proof of the Main Theorem. By passing to H/H × we can assume without loss of generality that H is reduced. Let N and C be as in Lemma 2.8. Let D ∈ N by Lemma 2.6 be a constant such that the inequalities | min L(cd) − min L(d)| ≤ D and | max L(cd) − max L(d)| ≤ D hold for all d ∈ H and c ∈ {πH (z) | z ∈ Z(H), vq (z) < N C for all q ∈ A(H)}. Put E = max{D + N C|A(H)|, 2M }, where M is a bound in the Structure Theorem for sets of lengths of H. Since the strong successive distance δ(H) of H is finite, there exists some B0 ∈ N such that for all h ∈ H, u ∈ Z(h) and l ∈ L(h) with ||u| − l| ≤ E there exists some factorization u ∈ Z(h) with |u | = l and d(u, u ) ≤ B0 . We claim that N and B = N (B0 + 1) satisfy the assertion of the Theorem. Thus our aim is to show that     m (19.4) L(a ) − min L(am ) ∩ [0, 2M ] = L(an ) − min L(an ) ∩ [0, 2M ] and     m L(a ) − max L(am ) ∩ [−2M, 0] = L(an ) − max L(an ) ∩ [−2M, 0]

(19.5)

hold for all a ∈ H and m, n ∈ N≥B with m ≡ n mod N . Let a ∈ H\{1} and m, n ∈ N≥B with m < n and m ≡ n mod N . We first prove equation (19.4). Let y ∈ Z(a) with |y| = min L(a) be arbitrary and let p y k ∈ Z(am ) and k ∈ [0, N − 1] with m = pN + k and n = qN + k. Put y1 = ymin q k n N y2 = ymin y ∈ Z(a ), where ymin ∈ Z(a ) satisfies the conditions in Lemma 2.8. Let I = {q ∈ A(H) | vq (y) ≥ C} and set   q vq (y) and ω = q vq (y) . ξ = q∈I

q∈A(H)\I

s s Then Supp(ξ) ⊂ Supp(ymin ), whence |ξ r ymin | = min L(πH (ξ r ymin )) for all r, s ∈ N0 . p m k p m k |−min L(am )||+ Thus we get |y1 |−min L(a ) = |ξ ymin |−min L(a )+|ω | ≤ ||ξ k ymin k k |ω | ≤ D + |ω | ≤ D + N C|A(H)| ≤ E. Similarly, we obtain |y2 | − min L(an ) ≤ E. To show (19.4) it is hence enough to prove that

|y2 | + t ∈ L(an ) ⇐⇒ |y1 | + t ∈ L(am )

(19.6)

for every t ∈ [−E, E]. Let t ∈ [−E, E] with |y2 | + t ∈ L(an ). Then there exists some y2 | = |y2 | + t and d(y2 , y˜2 ) ≤ B0 . Set y˜1 = y˜2 y1 y2−1 ∈ Q(Z(H)). y˜2 ∈ Z(an ) with |˜ We claim that y˜1 ∈ Z(H). Then y˜1 ∈ Z(am ) and |˜ y1 | = |y1 | − |y2 | + |˜ y2 | = |y1 | − |y2 | + |y2 | + t = |y1 | + t. −u y˜2 , where u = q − p. From the inequality m ≥ B we We have y˜1 = ymin get the estimate p ≥ B0 . Since d(y2 , y˜2 ) ≤ B0 , we have |w| ≤ B0 , where w =

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q−B0 y2 gcd(y2 , y˜2 )−1 . Thus ymin divides y2 w−1 = gcd(y2 , y˜2 ). Now u = q − p ≤ q − B0 implies that q−B0 u | ymin | gcd(y2 , y˜2 ) | y˜2 . ymin

The other implication in (19.6) and equation (19.5) are proved by analogous arguments.

Bibliography [1] S. Chapman and A. Geroldinger, Krull Domains and Monoids, Their Sets of Lengths, and Associated Combinatorial Problems, D.D. Anderson, editor, Factorization in Integral Domains, Marcel Dekker 189 (1997), 73–112. [2] A. Foroutan, Monotone chains of factorizations, International Journal of Commutative Rings. [3] A. Geroldinger, The Catenary Degree and Tameness of Factorizations in Weakly Krull Domains, D.D. Anderson, editor, Factorization in Integral Domains, Marcel Dekker 189 (1997), 113–153. [4] A. Geroldinger, Chains of Factorizations and Sets of Lengths, J. Algebra 188 (1997), 331–362. [5] A. Geroldinger, A structure theorem for sets of lengths, Colloq. Math. 78 (1998), 225–259. [6] A. Geroldinger and F. Halter-Koch, Congruence monoids, Acta Arith., to appear. [7] A. Geroldinger and F. Halter-Koch, Arithmetical theory of monoid homomorphisms, Semigroup Forum 48 (1994), 333–362. [8] F. Halter-Koch, On the asymptotic behaviour of the number of distinct factorizations into irreducibles, Ark. Mat. 31 (1993), 297–305. [9] F. Halter-Koch, Finitely Generated Monoids, Finitely Primary Monoids, and Factorization Properties of Integral Domains, D.D. Anderson, editor, Factorization in Integral Domains, Marcel Dekker 189 (1997), 31–72. [10] F. Halter-Koch, Ideal Systems, Marcel Dekker, 1998. [11] W. Hassler, Properties of factorizations with successive lengths in onedimensional local domains, International Journal of Commutative Rings, to appear. [12] W. Hassler, Arithmetical properties of one-dimensional, analytically ramified local domains, J. Algebra 250 (2002), 517–532. 302

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[13] W. Hassler, Factorization properties of Krull monoids with infinite class group, Colloq. Math. 92 (2002), 229–242. [14] W. Hassler, Factorization in finitely generated domains, J. Pure Appl. Algebra 186 (2004), 151–168. [15] L. R´edei, Theorie der endlich erzeugbaren kommutativen Halbgruppen, PhysikaVerlag, W¨ urzburg, 1963.

Chapter 20

UMV-Domains by

Evan Houston and Muhammad Zafrullah Abstract We call a domain R a UMV-domain if each nonzero prime ideal in the polynomial ring R[X] satisfying P ∩ R = 0 is a maximal divisorial ideal of R[X]. We characterize v-domains as integrally closed UMV-domains, and we provide characterizations of UMV-domains similar to those known for UMT-domains.

1

Introduction

Let R be an integral domain with quotient field K. If f ∈ K[X] is an irreducible polynomial, then we call the prime ideal P = f K[X]∩R[X] an upper to zero. Uppers to zero have been used by many authors to characterize ring-theoretic properties. For example, it follows from [6, Theorem 19.15] that a domain R is a Pr¨ ufer domain if and only if R is integrally closed and P  M R[X] for each upper to zero P in R[X] and each maximal ideal M of R. A corresponding result exists for Pr¨ ufer v-multiplication domains (PVMDs). A domain R is a PVMD if and only if R is integrally closed and each upper to zero in R[X] is a maximal t-ideal of R[X] [10, Proposition 2.6]. This led the authors of the present chapter to isolate this condition on uppers to zero by defining UMT-domains to be those domains in which each upper to zero is a maximal t-ideal [12]. UMT-domains were further studied in [3] and [4]. The purpose of this chapter is to study UMV-domains, domains with the property that each upper to zero is a maximal v-ideal, that is, a maximal divisorial ideal. In Section 1 we study uppers to zero. We then apply these results in Section 2 to our study of UMV-domains. We show that a domain R is a UMV-domain if and only if each upper to zero P satisfies P −1 = R[X] and c(P )v = R (where c(P ) denotes the ideal of R generated by the coefficients of the polynomials in P ). We characterize v-domains as integrally closed UMV-domains. We also point out where the similarities with UMT-domains fail to hold; to some extent the problem is that, 304

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while maximal t-ideals are plentiful (recall that for any domain R we have R =
{RM | M is a maximal t-ideal of R}), a domain may have no maximal divisorial ideals. In a brief third section, we pose several questions related to UMV-domains.

2

Uppers to Zero

We begin by reviewing terminology. Let R be a domain with quotient field K. For fractional ideals I, J of R, (I : J) = {x ∈ K | xJ ⊆ I}, and (I :R J) = {x ∈ R |  xJ ⊆ I}. We write I −1 for (R : I), and Iv for (I −1 )−1 . Finally, It = Av , where the union is taken over all finitely generated subideals A of I. Most of the facts about the v- and t-operations which we shall use can be found in [6, Section 32]. Definition 2.1. Let I = f K[X] ∩ R[X], where f is a nonconstant polynomial in K[X]. Then I is said to be almost principal if there is a nonzero element a ∈ R such that aI ⊆ f R[X]. Lemma 2.2. Let I = f K[X]∩R[X], where f is a nonconstant polynomial in K[X]. Then I −1 ∩ K[X] = (I : I). Proof. [7, Lemma 1.14]. Proposition 2.3. Let P = f K[X] ∩ R[X] with f irreducible in K[X]. Then the following statements are equivalent. (1 ) P is almost principal. (2 ) P −1  K[X]. (3 ) P −1 = (P : P ). (4 ) There is an element g ∈ R[X] \ P with gP ⊆ f R[X]. (5 ) P = (f :R[X] a) for some a ∈ R. (6 ) P = (R[X] :R[X] ψ) for some ψ ∈ K(X). Proof. The equivalence of the first four conditions is established in [7, Proposition 1.15]. Assume (1). Then we have aP ⊆ f R[X] for some 0 = a ∈ R. Thus P ⊆ (f :R[X] a). On the other hand, if h ∈ R[X] satisfies ha ∈ f R[X], then ha ∈ P ⊆ f K[X]; hence h ∈ f K[X] ∩ R[X] = P . Hence (1) ⇒ (5). (5) ⇒ (6) follows from the fact that (f :R[X] a) = (R[X] :R[X] af −1 ). Finally, assume (6). If ψ ∈ K[X], then there is some element a ∈ R with aψ ∈ R[X], whence a ∈ P , contradicting that P ∩ R = 0. Thus ψ = g/f for some g ∈ R[X] \ f K[X]. It follows that g ∈ / P , and we have gP ⊆ f R[X], proving (4). The following result is the key to our characterization of UMV-domains in Theorem 3.2 below. Lemma 2.4. Let I be an ideal of R[X], and assume that I −1 = R[X] and that I ∩ R = 0. Then c(I)v = R.

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Proof. We first observe that I −1 ⊆ K[X], since for any nonzero element a ∈ I ∩ R, we have aI −1 ⊆ R[X] ⊆ K[X]. Now let ψ ∈ I −1 \ R[X], and assume that ψ has minimal degree among elements of I −1 \ R[X]. We shall show that, in fact, ψ has degree zero. We shall then have I ⊆ (R[X] :R[X] ψ) = (R :R ψ)R[X]. It will then follow that c(I) ⊆ (R :R ψ); since (R :R ψ) is divisorial, this will imply that c(I)v = R. To see that ψ must have degree zero, write ψ = un X n + · · · + u0 . We claim that (R :R un ) ⊆ (R :R u0 ) ∩ · · · ∩ (R :R un−1 ). If not, pick b ∈ R with bun ∈ R but / R for some i < n. Then ψ  = bψ − bun X n is an element of I −1 \ R[X] of bui ∈ degree smaller than that of ψ, a contradiction. Thus the claim is true. In particular, / R. Now let g ∈ (R[X] :R[X] ψ), g = am X m + · · · + a0 . Then am un ∈ R, un ∈ whence by the claim, am ui ∈ R for each i. That is, am ∈ (R[X] :R[X] ψ). It then follows that g − am X m ∈ (R[X] :R[X] ψ). By induction on the degree of g, we obtain ai ∈ (R[X] :R[X] ψ) for each i. In particular, ai un ∈ R for each i, i.e., g ∈ (R :R un )R[X]. Hence I ⊆ (R[X] :R[X] ψ) ⊆ (R :R un )R[X], and un ∈ I −1 \ R[X]. Therefore, ψ = un has degree zero, as desired. Proposition 2.5. Let R be a domain with quotient field K, and let P = f K[X] ∩ R[X] with f irreducible in K[X]. Consider the following conditions on P . (1 ) P is maximal divisorial. (2 ) P is v-invertible. (3 ) (P : P ) = R[X]. (4 ) P −1 ∩ K = R. (5 ) c(P )v = R. (6 ) P is almost principal. (7 ) P is divisorial. Then (1) ⇒ (2) ⇔ (3) ⇒ (4) ⇔ (5), and (1) ⇒ (6) ⇒ (7). Proof. The implication (1) ⇒ (2) is proved in [5, Proposition 2.5]. That (2) ⇒ (3) is well-known and is true more generally. Indeed, let I be a v-invertible ideal of a domain R. Then (I : I) ⊆ (II −1 : II −1 ) ⊆ ((II −1 )v : (II −1 )v ) = R (by v-invertibility of I). To show that (3) ⇒ (4), let u ∈ P −1 ∩ K. By Lemma 2.2, u ∈ (P : P ) = R[X]. Hence u ∈ R[X] ∩ K = R, as desired. The equivalence (4) ⇔ (5) is straightforward. Note that (6) ⇒ (7) follows from Proposition 2.3. (3) ⇒ (2): We may assume that P −1 = R[X]. If P is not divisorial, then Pv ∩ R = 0 (since Pv  P ), and (Pv )−1 = P −1 = R[X]. By Lemma 2.4, c(P )v ⊆ c(Pv )v  R. This contradicts (5). Hence P is divisorial. Now let ψ ∈ (P P −1 )−1 , so that ψP P −1 ⊆ R[X]. Then ψP ⊆ Pv = P , and ψ ∈ (P : P ) = R[X]. Thus (P P −1 )−1 = R[X], and P is v-invertible. (1) ⇒ (6): By [5, Proposition 2.1], P = (R[X] :R[X] ψ) for some ψ ∈ K(X) \ R[X], and P is almost principal by Proposition 2.3.

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We do not know whether (4) ⇒ (3) or (7) ⇒ (6) in Proposition 2.5. However, other than these two possibilities, no implications other than those given exist. That condition (6) does not imply (5) can be illustrated with any Noetherian domain R having a divisorial prime P of height ≥ 2. This follows from the facts that P R[X] necessarily contains an upper to zero and that uppers to zero in Noetherian domains are automatically almost principal [13, Proposition 3.3]. That condition (2) does not imply (7) is illustrated by any domain R allowing an upper to zero P with P −1 = R[X]. We produce such an example at the end of this section. It is useful to observe that much of Proposition 2.5 holds for “nonprime uppers to zero”, that is without the assumption that f is irreducible. Proposition 2.6. Let R be a domain with quotient field K, and let I = f K[X] ∩ R[X], where f is a nonconstant polynomial in K[X]. Consider the following conditions on I. (2 ) I is v-invertible. (3 ) (I : I) = R[X]. (4 ) I −1 ∩ K = R. (5 ) c(I)v = R. (6 ) I is almost principal. (7 ) I is divisorial. Then (2) ⇔ (3) ⇒ (4) ⇔ (5), and (6) ⇒ (7). Proof. The only implication requiring proof is (6) ⇒ (7). Assume (6), and choose a nonzero element a ∈ R with aI ⊆ f R[X]. Then aIv ⊆ f R[X] ⊆ f K[X], whence Iv ⊆ f K[X]. Hence Iv ⊆ f K[X] ∩ R[X] = I. Let P be an upper to zero which satisfies (P : P ) = R[X] and P −1 = R[X]. Observe that the proof of (3) ⇒ (2) of Proposition 2.5 shows that P is divisorial. Also, since (P : P ) = R[X], P is v-invertible. It then follows from [5, Proposition 2.5] that P is maximal divisorial. That is, we have the following result. Proposition 2.7. With the notation of Proposition 2.5, assume that P −1 = R[X]. Then conditions (1), (2), and (3) are equivalent. Proposition 2.8. Assume that R is integrally closed. Then (a) with the notation of Proposition 2.5, conditions (1)-(5) are equivalent. (b) with the notation of Proposition 2.6, conditions (2)-(5) are equivalent (and condition (6) actually holds).

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Proof. (a) It suffices to prove (5) ⇒ (1). Suppose that B is a divisorial ideal of R[X] which properly contains P . Then B ∩ R = 0. By [17, Lemme 2], B = (B ∩ R)R[X]. Hence c(B) = B ∩ R. Now R = c(P )v ⊆ c(B)v = B ∩ R. It follows that B = R. Therefore, P is a maximal divisorial ideal. (b) It suffices to prove (5) ⇒ (2). Write I = f c(f )−1 R[X] by [17, Lemme 1]. Then c(I) = c(f )c(f )−1 . Since c(I)v = R, this shows that c(f ) is v-invertible. It follows that I is v-invertible. Also, aI ⊆ f R[X] for any a ∈ c(f ), so (6) holds. We conclude this section with the example promised above. Although we could base the proof on Lemma 2.4, we instead use the following lemma, since it may be of some independent interest. Proposition 2.9. Let P = f K[X] ∩ R[X] be an upper to zero, and let I denote the ideal generated by the constant terms of the elements of P . If I −1 = R and P −1 = R[X], then P is almost principal. Proof. Suppose that P is not almost principal. Let ψ ∈ P −1 . Assuming that I −1 = R, we shall show that ψ ∈ R[X]. Since P is not almost principal, we have ψ ∈ K[X]. Let c denote the constant term of ψ. For g ∈ P , since ψg ∈ R[X], we have ca ∈ R, where a is the constant term of g. Hence c ∈ I −1 = R. In particular, if ψ ∈ K, then ψ ∈ R. Now, since c ∈ R, the polynomial ω = (ψ − c)/X) ∈ P −1 . By induction, we have that ω ∈ R[X]. But then ψ ∈ R[X], as desired. Example 2.10. An example of an upper to zero with P −1 = R[X]. Thus P is trivially v-invertible but is not divisorial. Let k be a field, and let s, t be inn determinates over k. Set R = k[{st2 | n ≥ 0}]. This example was discussed in [7] and [11]. In particular, [11] contains a direct proof that the upper to zero P = (X − t)K[X] ∩ R[X] (where K denotes the quotient field of R) is not almost principal. It is easy to see that a monomial si tj ∈ k[s, t] lies in R iff i ≥ ϕ(j), where ϕ(j) is the number of 1’s in the binary expansion of j. To see that P −1 = R[X], it suffices to show that I −1 = R, where I is the ideal generated by the constant terms n n of P (Proposition 2.9). It is easy to see that sX 2 − st2 ∈ P for each n ≥ 1. Thus n n st2 ∈ I for each n ≥ 1. Set J = ({st2 | n ≥ 1}). Then J ⊆ I, and we need only show that J −1 = R. If u ∈ J −1 , then ust ∈ R, so that u = a/st for some a ∈ R. We shall show that a ∈ stR. For this we may assume that a is a monomial, say a = si tj . n Since a ∈ R, we have i ≥ ϕ(j). Note that for each n ≥ 1, we have (a/st)st2 ∈ R, n whence si t2 −1+j ∈ R. It follows that i ≥ ϕ(2n − 1 + j). For sufficiently large n, we have ϕ(2n − 1 + j) = 1 + ϕ(j − 1), whence i − 1 ≥ ϕ(j − 1). It follows that a = (st)(si−1 tj−1 ) ∈ stR, as desired.

3

UMV-Domains

Definition 3.1. A domain R is a UMV-domain if each upper to zero in R[X] is a maximal divisorial ideal of R[X]. Theorem 3.2. The following statements are equivalent for a domain R with quotient field K.

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(1 ) R is a UMV-domain. (2 ) Each upper to zero P in R[X] satisfies (P : P ) = R[X] = P −1 . (3 ) For each nonconstant polynomial f ∈ K[X], the ideal I = f K[X] ∩ R[X] satisfies (I : I) = R[X] = I −1 . (4 ) Each upper to zero P in R[X] satisfies P −1 = R[X] and c(P )v = R. (5 ) For each nonconstant polynomial f ∈ K[X], the ideal I = f K[X] ∩ R[X] satisfies I −1 = R[X] and c(I)v = R. Proof. (1) ⇒ (2): Let R be a UMV-domain, and let P be an upper to zero. Then P is maximal divisorial, and we have (P : P ) = R[X] by Proposition 2.5. Since P is divisorial, we also have P −1 = R[X]. (2) ⇒ (3): Let f, I be as given, and (we may) assume f ∈ R[X]. Factor f in K[X]: f = g1m1 · · · grmr , where each gi is irreducible in K[X], and set Pi = gi K[X] ∩ R[X]. Let ψ ∈ (I : I); recall from Lemma 2.2 that this implies that ψ ∈ K[X]. Then ψ(P1m1 · · · Prmr ) ⊆ ψI ⊆ I. Thus

ψ(P1m1 −1 P2m2 · · · Prmr )P1 ⊆ P1 .

Since (P1 : P1 ) = R[X], this implies that ψ(P1m1 −1 P2m2 · · · Prmr ) ⊆ R[X]. Rewrite this as

ψ(P1m1 −2 P2m2 · · · Prmr )P1 ⊆ R[X].

Then, using (2) and Lemma 2.2, we have ψ(P1m1 −2 P2m2 · · · Prmr ) ⊆ P1−1 ∩ K[X] = (P1 : P1 ) = R[X]. Repeating this as many times as necessary, we eventually obtain ψ ∈ R[X], as desired. Finally, I −1 ⊇ P1−1  R[X]. (3) ⇒ (5): Assume (3), and let I = f K[X] ∩ R[X] with f ∈ R[X]. We have I −1 = R[X] by assumption. Moreover, the condition (I : I) = R[X] implies that c(I)v = R by Proposition 2.6. (5) ⇒ (4): Trivial. (4) ⇒ (1): Proceeding contrapositively, suppose that R is not a UMV-domain, and let P be an upper to zero which is not maximal divisorial. Assume that P −1 = R[X]. Then P  J for some divisorial ideal J of R[X], and we must have J ∩R = 0. By Lemma 2.4, c(J)v = R, whence c(P )v = R. We recall that a v-domain is a domain in which each finitely generated ideal is v-invertible. Examples include all completely integrally closed domains. Theorem 3.3. The following statements are equivalent for a domain R. (1 ) R is a v-domain. (2 ) R is an integrally closed UMV-domain.

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(3 ) R is integrally closed, and every upper to zero in R[X] is v-invertible. (4 ) R is integrally closed and every upper to zero P = f K[X] ∩ R[X] with f a linear polynomial is v-invertible. Proof. (2) ⇔ (3): This follows from the fact that in the integrally closed case v-invertibility and maximal divisoriality are equivalent for uppers to zero (Proposition 2.8). (1) ⇒ (3) Suppose that R is a v-domain. It is well-known that R is integrally closed. Let P = f K[X] ∩ R[X] be an upper to zero. Then P = f c(f )−1 R[X] by [17, Lemme 1]. Since R is a v-domain, we have (c(f )c(f )−1 )v = R. Hence (P P −1 )v = (c(f )−1 c(f )v )v R[X] = R[X], and P is v-invertible. (2) ⇒ (1) Assume that R is an integrally closed domain UMV-domain, and let A be a finitely generated ideal of R. Since a principal ideal is trivially v-invertible, we may as well assume that A is not principal. Then A = c(f ) for some nonconstant polynomial f ∈ R[X]. Set I = f K[X] ∩ R[X]. Then I = f c(f )−1 R[X] by [17, Lemme 1]. By Theorem 3.2, (I : I) = R[X]. Hence by Proposition 2.6, I is vinvertible in R[X]. It follows easily that c(f ) = A is v-invertible in R. Hence R is a v-domain. (3) ⇒ (4): Trivial. (4) ⇒ (1): Assume (4). To show that R is a v-domain, it suffices by [14, Lemma 3.6] to show that every two-generated ideal of R is v-invertible. Consider a two-generated ideal A = (a, b). Let f = aX + b, and set P = f K[X] ∩ R[X]. By hypothesis P is v-invertible. Again using [17, Lemme 1], we can write P = f c(f )−1 R[X], and it is easy to see that c(f ) = A is v-invertible in R. We now compare our results on UMV-domains to those on UMT-domains. Recall that a domain R is a UMT-domain if each upper to zero in R[X] is a maximal t-ideal, and that R is a Pr¨ ufer v-multiplication domain (PVMD) if each finitely generated ideal of R is t-invertible. According to [12, Proposition 3.2], a domain is a PVMD if and only if it is an integrally closed UMT-domain. Our Theorem 3.3 above provides an exact analogue to this PVMD characterization. On the other hand, our Theorem 3.2 is much less satisfying than the corresponding situation for UMT-domains. Recall from [12, Proposition 1.1] that if R is a domain and M is a maximal t-ideal of R[X] with M ∩ R = 0, then M = (M ∩ R)R[X]. Hence, since in general a t-ideal of a domain is contained in a maximal t-ideal, we have that a domain R fails to be a UMT-domain if and only if there is an upper to zero P in R[X] and a maximal t-ideal M of R with P ⊆ M R[X]. Unfortunately, however, maximal divisorial ideals need not exist. (For example, let R be a valuation domain whose maximal ideal is not principal.) The closest that we can come to an analogue of the UMT-result is the following reworking of the statement of Theorem 3.2 (1) ⇔ (3). Proposition 3.4. Let R be a domain. Then R fails to be a UMV-domain if and only if there is an upper to zero P in R[X] for which either P −1 = R[X] or P ⊆ IR[X] for some divisorial ideal I of R.

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311

Proof. Just take I = c(P )v is the statement of Theorem 3.2 (3). In [5, Example 3.1], Gabelli and Roitman give an example of a completely integrally closed (and therefore a v-) domain R admitting an upper to zero which is a maximal divisorial ideal but not a maximal t-ideal. In view of Theorem 3.3 (and the characterization of PVMDs as integrally closed UMT-domains), any v-domain which is not a PVMD must admit such an upper to zero. The first non-PVMD v-domain was produced by Dieudonn´e in [2]; other examples were given in [15, 16], [9], and [8]. Now there is a host of such examples. For example, if R is such an example, then so is R[X] (as is mentioned in the next section below, if R is a vdomain, then R[X] is also a v-domain). Examples can also be produced using the D + XDS [X]-construction; see [18]. Thus, from our point of view, [5, Example 3.1] is another, welcome, addition to the list of v-domains which are not PVMDs. By the characterizations mentioned above, a v-domain which is not a PVMD is a UMV-domain which is not a UMT-domain. It is of interest to have nonintegrally closed examples of this phenomenon. We now give such an example. Example 3.5. An example of a nonintegrally closed UMV-domain which is not a UMT-domain. Begin with any domain D such that D is a v-domain but is not a PVMD, let k denote the quotient field, let T = k[[t2 , t3 ]] (t an indeterminate), and let R be defined by the following pullback diagram. R ⏐ ⏐ 

−−−−→

D ⏐ ⏐ 

φ

T = −−−−→ k = T /M. Note that t is integral over R, so R is not integrally closed. Also, since D is not a UMT-domain, neither is R [4, Theorem 3.7]. It remains to show that R is a UMV-domain. Accordingly, let P = f K[X] ∩ R[X] be an upper to zero in R[X], and let Q = f K[X] ∩ T [X] (K is the common quotient field of R and T ). Since T is Noetherian, Q is almost principal, and we have uQ ⊆ f T [X] for some nonzero u ∈ T . Hence t2 uP ⊆ f R[X]. Thus P is almost principal and therefore divisorial. In particular, P −1 = R[X]. We next want to show that c(P )v = R. Again, since T is Noetherian (and one dimensional), we have Q  M [X]. Therefore, since RM = T , we have P  M [X]. Now suppose c(P )v = R, and pick y ∈ c(P )−1 \ R. Then P ⊆ (R :R y)R[X], and since the ideals of R are comparable to M , we must have M  (R :R y). It is then easy to show that φ(P ) is an upper to zero in D[X] with φ(P ) ⊆ (D :D φ(y))D[X]. However, since D is a UMV-domain, this contradicts Theorem 3.2. Hence we have c(P )v = R, as desired. Another application of Theorem 3.2 completes the proof. According to [4, Theorem 1.5], a domain R is a UMT-domain if and only if RQ has Pr¨ ufer integral closure for each prime t-ideal Q of R. We have the following partial analogue.

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Theorem 3.6. If R is a UMV-domain, then RQ has Pr¨ ufer integral closure for each divisorial prime Q of R. Proof. Let R be a UMV-domain, and let Q be a divisorial prime of R. Since each upper to zero in R[X] is maximal divisorial, we have P  Q for each upper to zero P . Hence in RQ [X] the prime ideal QRQ [X] contains no uppers to zero. It follows that if T is the integral closure of RQ and N is a prime of T lying over QRQ , then N T [X] contains no uppers to zero. We then have that T is a Pr¨ ufer domain by [6, Theorem 19.15]. Corollary 3.7. If R is a v-domain, then RM is a valuation domain for each divisorial prime M of R.

4

Questions

We close with several questions and comments related to the results of the first two sections. ufer integral closure for each divisorial prime Q of R, Question 4.1. If RQ has Pr¨ is R necessarily a UMV-domain? In view of Theorem 3.6, a positive answer to this question would yield a characterization of UMV-domains comparable to the characterization of UMT-domains mentioned just before the statement of Theorem 3.6. It is known that if R is a UMT-domain (respectively, a v-domain), then so is R[X] [4, Theorem 2.4] (respectively, [1, Corollary 1.6]). This motivates our next question. Question 4.2. If R is a UMV-domain, is R[X] also a UMV-domain? Here is a related question: If R is a UMV-domain, does it follow that if P is a prime ideal of R[X, Y ] such that P has height one and satisfies P ∩ R = 0, then P is a maximal divisorial ideal of R[X, Y ]? If Question 4.1 has a negative answer, then the following question becomes interesting. ufer integral closure for each Question 4.3. If R has the property that RQ has Pr¨ divisorial prime Q of R, does R[X] have this same property? Recall that for uppers to zero t-maximality and t-invertibility are equivalent [12, Theorem 1.4]. However, in Example 2.10, we produced a domain R admitting an upper to zero P with P −1 = R[X] (so that P is trivially v-invertible but is not (maximal) divisorial). Question 4.4. Is there an example of a non-UMV-domain R with the property that every upper to zero in R[X] is v-invertible? We do not know whether Example 2.10 is such an example. At any rate no such example can be integrally closed, by Proposition 2.8. Assuming the existence of such examples, one could ask whether the property that all uppers to zero are v-invertible would extend to the polynomial ring.

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Question 4.5. If R has the property that each upper to zero P = f K[X] ∩ R[X] with f linear is maximal divisorial, is R necessarily a UMV-domain? Note that this is proved in the integrally closed case in Theorem 3.3. The corresponding question for the UMT-property has an affirmative answer. Since this has not (to our knowledge) appeared in the literature, we state it formally and sketch a proof. Proposition 4.6. Suppose that R has the property that every upper to zero P = f K[X] ∩ R[X] with f linear is a maximal t-ideal. Then R is a UMT-domain. Proof. In a manner similar to that used in the proof of Theorem 3.6, one can show that for each maximal t-ideal M of R, we have that N T [X] contains no linear uppers to zero, where T is the integral closure of RM and N is a prime of T lying over M RM . Hence for u ∈ K, we have that P = (x − u)K[X] ∩ T [X]  N [X] for each maximal ideal N of T . Thus u is the root of a polynomial not contained in N [X]. By the u, u−1 -lemma [6, Lemma 19.14], either u or u−1 lies in TN . It follows that T is a Pr¨ ufer domain. Hence R is a UMT-domain by [4, Theorem 1.5].

Bibliography [1] D.F. Anderson, E. Houston, and M. Zafrullah, Pseudo-integrality, Canad. Math. Bull. 34 (1991), 15–22. [2] J. Dieudonn´e, Sur la th´eorie de la divisibilit´e, Bull. Soc. Math. France 69 (1941), 133–144. [3] D. Dobbs, E. Houston, T. Lucas, M. Roitman, and M. Zafrullah, On t-linked overrings, Comm. Algebra 20 (1992), 1463–1488. [4] M. Fontana, S. Gabelli, and E. Houston, UMT-domains and domains with Pr¨ ufer integral closure, Comm. Algebra 26 (1998), 1017–1039. [5] S. Gabelli and M. Roitman, Maximal divisorial ideals and t-maximal ideals, manuscript. [6] R. Gilmer, Multiplicative ideal theory, Dekker, New York, 1972. [7] E. Hamaan, E. Houston, and J. Johnson, Properties of uppers to zero in R[x], Pacific J. Math. 135 (1998), 65–79. [8] W. Heinzer, An essential integral domain with a nonessential localization, Canad. J. Math. 23 (1981), 400–403. [9] W. Heinzer and J. Ohm, An essential ring which is not a v-multiplication ring, Canad. J. Math. 21 (1972), 856–861. [10] E. Houston, S. Malik, and J. Mott, Characterizations of ∗-multiplication domains, Canad. Math. Bull. 27 (1984), 48–52. [11] E. Houston, Prime t-ideals in R[X], Commutative ring theory, P.-J. Cahen, D. Costa, M. Fontana, S. Kabbaj, eds., Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 153 (1994), 163–170. [12] E. Houston and M. Zafrullah, On t-invertibility II, Comm. Algebra 17 (1989), 1955–1969. [13] J. Johnson, Three topological properties from Noetherian rings, Canad. J. Math. 34 (1982), 525–534. 314

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[14] J. Mott, B. Nashier, and M. Zafrullah, Contents of polynomials and invertibility, Comm. Algebra 18 (1990), 1569–1583. [15] M. Nagata, On Krull’s conjecture concerning valuation rings, Nagoya Math. J. 4 (1952), 29–33. [16] M. Nagata, Corrections to my paper “On Krull’s conjecture concerning valuation rings”, Nagoya Math. J. 9 (1955), 209–212. [17] J. Querr´e, Id´eaux divisoriels d’un anneau de polynˆ omes, J. Algebra 64 (1980), 270–284. [18] M. Zafrullah, The D+XDS [X]-construction from GCD-domains, J. Pure Appl. Algebra 50 (1988), 93–107.

Chapter 21

On Local Half-Factorial Orders by

Florian Kainrath

1

Introduction

Let R be an atomic domain. Then every nonunit r of R has a factorization r = u1 · . . . · un , where the ui are atoms of R. The integer n is called the length of the factorization. In general r has many such factorizations. R is called half-factorial, if for any r ∈ R all the factorizations of r have the same length. In [4] F. Halter-Koch gave a complete description of half-factorial orders in quadratic number fields. In particular, this implies the following for such an order ¯ is half-factorial, too. 2. If O: 1. Any ring between O and its integral closure O e1 en ¯ then ¯ (O : O) = p1 · . . . · pn is the prime factorization of the conductor of O in O, all exponents ei are equal to one. In view of this result one may conjecture that something similiar holds for orders in arbitrary number fields. In this chapter we study the local situation, meaning we look at local orders, i. e. local integral domains R, which are noetherian, one-dimensional, whose integral closure is a finitely generated R-module, and whose residue field is finite. We give a new criterion, when such domains are half-factorial (Theorem 3.3). As an immediate application of this criterion we obtain a result on the size of the conductor of a half-factorial order, and we show that any ring between a halffactorial local order and its quotient field is half-factorial, too. Coming back to the global situation, I will state a conjecture for local orders at the end of this chapter, which (if true) implies that any ring between an half-factorial order in some number field and its integral closure is half-factorial, too. The proof of the above mentioned criterion is mainly based on a result on the product of two subspaces in a finite extension of fields (Theorem 2.3), which is completely analogous to a result of Kneser on the sumset of two finite sets in an abelian group (see for example [6, Theorem 4.3,chap. 4.3]). 316

On Local Half-Factorial Orders

2

317

On the Product of Two Subspaces in a Finite Extension of Fields

Let R be a ring, X, Y ⊂ R and i ≥ 1. Set X ∗ Y = {xy | (x, y) ∈ X × Y } and define X ∗i inductively by X ∗1 = X and X ∗i = X ∗(i−1) ∗ X for i ≥ 2. Finally let XY be the subgroup of R generated by X ∗ Y . If X = {x} for some x ∈ R we will also write x ∗ Y , resp. xY instead of {x} ∗ Y , resp. {x}Y . Note that if R is a subring of R such that R ∗ X ⊂ X then XY is the R -submodule of R generated by X ∗ Y . In particular it is a R -submodule of R. If L | K is an extension of fields and V , W ⊂ L are K-subspaces of L, set × V = V \ {0}, T (V, W ) = {x ∈ L | xV ⊂ W } and T (V ) = T (V, V ). Note, if [L : K] < ∞, then T (V ) is the largest subfield F of L such that V is a F -subspace of L. In the following two lemmas let L | K be a finite extension of infinite fields, and V and W be two nonzero K-subspaces of L. We will also assume that L = K(x) for some x ∈ L. Note that this will be so, if and only if the set of intermediate fields of L | K is finite. Lemma 2.1. There exist nonzero v ∈ V , w ∈ W having the following property: whenever U is a K-subspace of V W containing vw, we have T (U ) ⊂ T (V W ). Proof. First note that for (v, w) ∈ V × × W × the condition stated in the lemma is equivalent to: F vw ⊂ V W for all intermediate fields F of L | K, such that F ⊂ T (V W ). Let F be the set of all intermediate fields F of L | K, such that F ⊂ T (V W ). By our assumption on L | K this is a finite set. For F ∈ F set XF = {(v, w) ∈ V × × W × | F vw ⊂ V W } . Since F ⊂ T (V W ) and since V × ∗ W × generates V W we have XF = V × × W × . Next let µ : V × × W × → L be the multiplication map, µ(v, w) = vw. Then clearly XF = µ−1 (T (F, V W )) . Now endow V × W and L with the K-Zariski topologies. Note that V × × W × is an open subset of V × W , and hence is irreducible (K being infinite). Since µ is the restriction of a bilinear map to V × × W × it is continous. Further T (F, V W ) is a K-subspace of L, and hence is closed. The above equation implies now that XF is closed in V × × W × , too. So the family X = (XF )F ∈F consists of proper, closed subsets of V × × W × . The irreducibility of V × × W × implies now that the union of X does not equal V × × W × . Now it is clear that any (v, w) ∈ V × × W × not belonging to this union satisfies the condition stated at the beginning of the proof. Lemma 2.2. Suppose that dimK (V W ) − dimK (V ) > 0. Then there exist nonzero K-subspaces V  and W  of L, such that:

318

Kainrath

1. V  W  ⊂ V W and T (V  W  ) ⊂ T (V W ). 2. dimK (V  W  ) − dimK (V  ) < dimK (V W ) − dimK (V ). 3. dimK (V  ) + dimK (W  ) = dimK (V ) + dimK (W ). Proof. Note that dimK (V W ) − dimK (V ) > 0 implies, that V w = V W for any nonzero w ∈ W . Choose (v, w) ∈ V × × W × as in Lemma 2.1. Since V w = V W and V × ∗ W × generates V W , there exist nonzero v  ∈ V and w ∈ W , such that / V w. v  w ∈ Define now V  and W  by V  = V + v  w−1 W

and

W  = W ∩ v

−1

wV

.

Then v ∈ V  and w ∈ W  , so that in particular both are nonzero. We now show that V  and W  have the properties 1-3 of the lemma. 1. Obviously vw ∈ V  W  ⊂ V W . T (V  W  ) ⊂ T (V W ), too.

By our choice of v and w this implies

2. We have v  w w−1 ∈ V  \ V and hence dimK (V  ) > dimK (V ), which implies dimK (V  W  ) − dimK (V  ) < dimK (V W ) − dimK (V ) by 1. 3. dimK (V  ) + dimK (W  ) = dimK (V + v  w−1 W ) + dimK (W ∩ v  

= dimK (V + v w

−1

−1



W ) + dimK (V ∩ (v w 

= dimK (V ) + dimK (v w = dimK (V ) + dimK (W )

−1

wV ) =

−1

W )) =

W) =

.

Theorem 2.3. Let L | K be a finite extension of fields such that L = K(x) for some x ∈ L, and let V , W be two K-subspaces of L. Then dimK (V W ) ≥ dimK (V ) + dimK (W ) − [T (V W ) : K]. Proof. Let K  be any infinite field containing K, such that K  and L are linearly disjoint over K. Then the K  -algebra L = K  ⊗K L is a field. Moreover inside L we have L = K  L. Set now V  = K  V = K  ⊗K V , W  = K  W = K  ⊗K W . Then we have V  W  = K  (V W ) = K  ⊗K V W . Moreover we have T (V  W  ) = K  T (V W ) = K  ⊗K T (V W ) (see for example [3, Prop. 12, Chap. I, S2.10]). Since dimK (U ) = dimK
(K  ⊗K U ) for any K-subspace U of L, we see that the assertion of the theorem holds for (L, K, V, W ), if and only if it does so for (L , K  , V  , W  ). We may therefore suppose that K is infinite.

On Local Half-Factorial Orders

319

If V = 0 or W = 0, then V W = 0 and T (V W ) = L, so the assertion of the theorem is obvious. From now on we suppose that V and W are nonzero. Since wV ⊂ V W for any nonzero w ∈ W we have dimK (V W ) − dimK (V ) ≥ 0. We use induction on dimK (V W ) − dimK (V ). Suppose that dimK (V W ) − dimK (V ) = 0. Choose a nonzero w ∈ W . Then V W = wV . Thus W w−1 ⊂ T (V ) = T (wV ) = T (V W ), so that dimK (W ) − [T (V W ) : K] ≤ 0, and hence dimK (V W ) = dimK (V ) ≥ dimK (V ) + dimK (W ) − [T (V W ) : K]. Let now dimK (V W )−dimK (V ) > 0. Choose V  and W  as in Lemma 2.2. Then dimK (V  W  ) − dimK (V  ) < dimK (V W ) − dimK (V ), so that by induction dimK (V  W  ) ≥ dimK (V  ) + dimK (W  ) − [T (V  W  ) : K] . Using 1. and 3. of Lemma 2.2 we obtain dimK (V W ) ≥ dimK (V  W  ) ≥ dimK (V  ) + dimK (W  ) − [T (V  W  ) : K] ≥ ≥ dimK (V  ) + dimK (W  ) − [T (V W ) : K] = = dimK (V ) + dimK (W ) − [T (V W ) : K] .

Remark 2.4. This theorem is completely analogous to a theorem of Kneser about the cardinality of the sumset of two finite sets in an abelian group (see for example [6, Theorem 4.3,chap. 4.3]). We now assume that L | K is an extension of finite fields. For any finite set X we let #X be its cardinality and set q = #K. We will need the following lemma which is due to G. Lettl. Lemma 2.5. Let V be a nonzero K-subspace of L and set n = dimK (V ). Then #V ∗ V ≤ q 2n−1 − q n−1 + 1

.

In particular if n > 1, then #V ∗ V < q 2n−1 . Proof. The second assertion follows immediately from the first one. To proof the first one, let A be the subset of V × × V × containing all pairs which are linearly independent and let B be its complement in V × ×V × . Note that #A = (q n −1)(q n − q) and #B = (q n − 1)(q − 1). Finally let µ : V × × V × → L be the multiplication map. For any (v, w) ∈ A the set A∩µ−1 (vw) contains the set {(vx, x−1 w), (wx, x−1 v) | x ∈ K × }. Since v, w are linearly independent, this set contains exactly 2(q − 1) elements. Hence #µ(A) ≤

(q n − 1)(q n − q) #A = 2(q − 1) 2(q − 1)

.

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Kainrath

Similarly, for any (v, w) ∈ B the set B ∩ µ−1 (vw) contains the set {(vx, x−1 w) | x ∈ K × } which has cardinality q − 1. Therefore #µ(B) ≤

#B = qn − 1 q−1

.

Using these two inequalities and the inequality q ≤ 2(q − 1) (since q ≥ 2) we obtain #V ∗ V = #(µ(A) ∪ µ(B)) + 1 ≤ #µ(A) + #µ(B) + 1 ≤ (q n − 1)(q n − q) + qn − 1 + 1 = 2(q − 1)   q n n−1 = (q − 1) (q +1 +1≤ − 1) 2(q − 1)

≤

≤ (q n − 1)(q n−1 − 1 + 1) + 1 = = q 2n−1 − q n−1 + 1 .

Theorem 2.6. Let L | K be an extension of finite fields. Suppose that V is a nonzero K-subspace of L, such that 1 ∈ V ∗ V and V ∗ V = V V . Then V ∗ V = V V is a subfield of L. Proof. Set K  = T (V V ) and V  = K  V . Then clearly V V = V  V  = V  ∗ V  . Hence, by replacing K by K  and V by V  , we may assume that T (V V ) = K. Then Theorem 2.3 implies dimK (V V ) ≥ 2 dimK (V ) − 1. Comparing this with Lemma 2.5 we obtain dimK (V ) = 1. Then dimK (V V ) = 1, too. Since 1 ∈ V V , V V = K is a subfield of L. Remark 2.7. 1. The content of the last theorem may be informally stated as: suppose that V is nonzero K-subspace of L such that V ∗ V contains 1 and is closed under taking sums. Then it is also closed under taking products. 2. As the following example shows, Theorem 2.6 is not true for infinite fields: Let K be the maximal 2-extension of Q. Then K is closed under taking square roots. Let x be any algebraic number of degree 5 over Q and set L = K(x). Then [L : K] = 5. Define V = K + Kx. Then V V = K + Kx + Kx2 is not a subfield of L. On the other hand, since K contains all square roots of its elements, every polynomial over K of degree ≤ 2 is reducible. Clearly this implies V V = V ∗ V .

3

Half-Factorial Orders

Recall from the introduction that in this chapter a local order R is an one-dimensional, ¯ is a finitely generated noetherian, local integral domain, whose integral closure R R-module, and whose residue field is finite. It is well known that such a ring is half-factorial if and only if it has the following property: ¯ is local, and each atom of R is a prime of R ¯ R

(21.1)

On Local Half-Factorial Orders

321

(see [1, Theorem 2.4 and Theorem 6.3], or see[2, Theorem 2.12]). ¯ is local, too. In the following we fix a local order R, whose integral closure R ¯ Then R is a DVR and R is a Cohen-Kaplansky domain, i. e. R has up to units only finitely many atoms (see [1, Theorem 2.4]). We will use the following notations: m (resp. m ¯ ) ¯ k (resp. k)

... ...

¯ the maximal ideal of R (resp. R) ¯ the residue field of R (resp. R)

¯ → k¯ π: R

...

the natural map .

We will identify k with π(R). ¯ and an integer i ≥ 1 set For a prime p of R ¯ × | rpi ∈ R} and Vi,p (R) = π(Ui,p (R)) ∪ {0} . Ui,p (R) = {r ∈ R ¯ Since R ¯ is a finitely generated RThen obviously Vi,p (R) is a k-subspace of k. e × ¯ module, m ¯ ⊂ R, for some e ≥ 1. Hence Ui,p (R) = R and Vi,p (R) = k¯ for all large ¯ × then enough i. If r ∈ R Ui,rp (R) = r−i ∗ Ui,p (R) and Vi,rp (R) = π(r)−i ∗ Vi,p

.

(21.2)

For any integers i, j ≥ 1 we have Ui,p (R) ∗ Uj,p (R) ⊂ Ui+j,p (R) and Vi,p (R) ∗ Vj,p (R) ⊂ Vi+j,p (R). Finally, if p ∈ R, then (Ui,p (R))i≥1 and (Vi,p (R))i≥1 form ascending chains. Lemma 3.1. Let R be as above. Then R is half-factorial if and only if for all ¯ we have (some) prime p of R Ui,p (R) = U1,p (R)∗i

.

(21.3)

Proof. This is more or less obvious from (21.1). A detailed proof is given in [7, Proposition 16]. ¯ = m. ¯ and any integer Lemma 3.2. Suppose that mR ¯ Then for any prime p of R × ¯ ¯ if and only if Vi,p (R) = k. i ≥ 1 we have Ui,p (R) = R ¯ if Ui,p (R) = R ¯ The ¯ × . Now suppose that Vi,p (R) = k. Proof. Clearly Vi,p (R) = k, i i+1 i i+1 ¯ prime p defines a natural isomorphism m ¯ /m ¯ → k, given by ap + m ¯ → π(a), ¯ From our definitions it follows immediately, that Vi,p (R) is the image where a ∈ R. ¯ i+1 )/m ¯ i+1 under this isomorphism. Hence m ¯i ∩ R + m ¯ i+1 = m ¯ i. of (m ¯i ∩ R + m ¯=m ¯i = m ¯ i . By Nakayama’s Lemma we Our assumption mR ¯ implies m ¯ i ∩ R + mm i × ¯ obtain m ¯ ⊂ R and hence Ui,p (R) = R . ¯ Then using Theorem 3.3. Let R be a local order with a local integral closure R. ¯ the above notations the following are equivalent for any (some) prime p of R: 1. R is half-factorial. ¯×. 2. U1,p (R) ∗ U1,p (R) = R

322

Kainrath

¯ 3. V1,p (R) ∗ V1,p (R) = k. ¯ then by (21.2) Proof. Note first that if one of 2. or 3. holds for some prime p of R, ¯ it holds for any prime of R. ¯ so that 1. ⇒ 2.: Let p be any atom of R. Then by (21.1) p is a prime of R, ∗i ¯ mR = m. ¯ Further Ui,p (R) = U1,p (R) holds for any i ≥ 1 by Lemma 3.1. This ∗i (R) = Vi,p (R) for all i ≥ 1. Using this equation for i = 2 we see that implies that Vi,p V1,p (R) ∗ V1,p (R) is a k-subspace of k¯ and hence V1,p (R) ∗ V1,p (R) = V1,p (R)V1,p (R). Since p ∈ R we have 1 ∈ V1,p (R) ⊂ V1,p (R) ∗ V1,p (R). Using Theorem 3.3, we see that V2,p (R) = V1,p (R) ∗ V1,p (R) is a subfield of k¯ containing V1,p (R). For large i ¯ Lemma 3.2 gives us we have k¯ = Vi,p (R) = V1,p (R)∗i and therefore V2,p (R) = k. × ¯ now R = U2,p (R) = U1,p (R) ∗ U1,p (R). 2. ⇒ 3.: clear. 2. ⇒ 1.: clear by Lemma 3.1. ¯ In particular ¯ such that V1,p (R)∗V1,p (R) = k. 3. ⇒ 2.: Let p be some prime of R this implies that V1,p (R) = 0, and hence U1,p (R) = ∅. Therefore there exists a prime ¯ contained in R. We may therefore assume that p ∈ R. In particular mR ¯ = m. of R ¯ ¯ ¯ From V1,p (R) ∗ V1,p (R) = k we deduce that Vi,p (R) = k for all i ≥ 2. Lemma 3.2 ¯ × for all i ≥ 2. Thus m ¯ 2 ⊂ R. implies now Ui,p (R) = R ¯ By assumption there exist u1 , u2 ∈ U1,p (R) such Let now r be any unit of R. ¯ that π(r) = π(u1 )π(u2 ), so that r = u1 u2 + ap = u1 (u2 + au−1 p) for some a ∈ R. Since m ¯ 2 ⊂ R we have (u2 + au−1 1 p) ∈ U1,p (R).

1

Remark 3.4. In a special case the equivalence of 1. and 2. in the last theorem has already been observed in [7, Proposition 16]. Corollary 3.5. Let R be a local half-factorial order. Then m ¯ 2 ⊂ R. Proof. This has already been observed in the above proof of 3. ⇒ 2. Corollary 3.6. Let R be a local half-factorial order. Then any ring between R and its quotient field is half-factorial, too. Proof. Let S be such a ring. We may assume that S is not the quotient field K of ¯ ⊂ S¯ = K. Since R ¯ is a DVR, it is a maximal subring of K, so that R. Then R ¯ = S, ¯ i. e. R ⊂ S ⊂ R. ¯ In particular S is a local order with local integral closure R ¯ Let p be any prime of R. ¯ Then clearly U1,p (R) ⊂ U1,p (S). The claim of the R. Corollary follows now from Theorem 3.3. ¯ We can state this result in another way: Let v be the normalized valuation on R and let a(R) be the maximum of all v(a), where a runs through the set of atoms of R. Then clearly, R is half-factorial if and only if a(R) = 1. Then the last corollary ¯ if R is half-factorial. I shows that a(S) ≤ a(R) for any ring S between R and R, conjecture that this holds generally: ¯ Then Conjecture 3.7. Let R be any local order with local integral closure R. ¯ a(S) ≤ a(R) for any S between R and R.

On Local Half-Factorial Orders

323

Finally I mention (without proof) the following fact: If this conjecture is true (or only if a(R) ≤ 2) then any ring between a half-factorial order in some number field and its integral closure is half-factorial, too. Note added in proof. After submitting this chapter I recognized that Thereom 2.3 has already been proved in [5].

Bibliography [1] D. D. Anderson and J. L. Mott, Cohen-Kaplansky domains: Integral domains with a finite number of irreducible elements, J. Algebra 148 (1992), 17–42. [2] D.D. Anderson and D.F. Anderson, Elasticity of factorizations in integral domains, J. Pure Appl. Algebra 80 (1992), 217–235. [3] N. Bourbaki, Commutative algebra, chapters 1-7, Elements of Mathematics, Springer, 1989. [4] F. Halter-Koch, Factorization of algebraic numbers, Ber. Math. Stat. Sektion i.Forschungszentrum Graz 191 (1983), 1–24. [5] Xiang-Dong Hou, Ka Hin Leung, and Qing Xiang, A generalization of an additition theorem of Kneser, J. Number Theory 97 (2002), 1–9. [6] M. B. Nathanson, Additive number theory, inverse problems and the geometry of sumsets, Springer, 1996. [7] M. Picavet-L’Hermitte, Factorization in some orders with a pid as integral closure, Algebraic Number Theory and Diophantine Analysis (F. Halter-Koch and R. F. Tichy, eds.), de Gruyter, 2000, pp. 365–390.

324

Chapter 22

On Factorization in Krull Domains with Divisor Class Group Z2k by

Karl M. Kattchee Abstract Let Υ(G) denote the set of elasticities of all Krull domains with divisor class group G, where G is a finite abelian group. Equivalently, Υ(G) may be viewed as the set of elasticities of all block monoids of the form B(G, S), where S generates the group G. Write Υj (G) for the subset of Υ(G) which arises from the block monoids B(G, S) where S consists of at most j elements. We show that Υ(Z2k ) = Υ2 (Z2k ), for k = 1, 2, 3, 4, 5, 6.

1

Introduction

A common way of measuring the extent to which a Krull domain R fails to be a unique factorization domain is by means of the elasticity ρ(R), which is defined by m ρ(R) = sup{ : a1 · · · am = b1 · · · bn , for some irreducibles ai , bj ∈ R}. n The divisor class group Cl(R) is central to the computation of ρ(R). If Cl(R) is trivial, then R is a unique factorization domain (UFD) [7, Proposition 6.1], and we obviously have ρ(R) = 1. It is possible, however, to have ρ(R) = 1 without R being a UFD, and in that case, R is called a half-factorial domain (HFD). In the article [12], Zaks defined HFDs and showed that every Krull domain with divisor class group Z2 is an HFD. This chapter extends that result, in a sense. On the other hand, if Cl(R) is nontrivial, then the value of ρ(R) depends on the set S(R) of (nontrivial) divisor classes which contain height-one primes: One 325

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Kattchee

forms the so-called block monoid B(Cl(R), S(R)) (defined below) and uses the fact [3, Lemma 3.2] that ρ(R) = ρ(B(Cl(R), S(R))), where the elasticity of the block monoid is defined in the obvious way. There are algorithms for computing the elasticities of block monoids (see [2] or [10]), and the computer implementations of these algorithms are efficient provided Cl(R) and S(R) are not too large. In this chapter, we shall concern ourselves with the problem, given a finite abelian group G, of enumerating the set of all elasticities ρ(R) which arise as R ranges among all Krull domains with divisor class group G. We denote this set by Υ(G), as in [1]. Recall that the pair (G, S), where G is an abelian group and S ⊆ G\{0}, is called realizable if there is a Dedekind domain R such that Cl(R) = G and S = {g ∈ G : g contains a nonprincipal prime ideal of R}. This concept extends naturally to Krull domains, where the set S represents those nonzero divisor classes which contain height-one prime ideals. Since it is known [8, Corollary 1.5] that (G, S) is a realizable pair if and only if S generates G as a group, we may write Υ(G) = {ρ(B(G, S)) : S ⊆ G \ {0} and S generates G}. The set Υ(G) may thus be studied entirely in the context of block monoids. The case where G is a cyclic group has been dealt with in the articles [1] and [6], where the emphasis is on G = Zp , and in the article [10], where the set Υ(Z2p ) is considered. In the above articles, the general program for studying Υ(G) is to acquire a relatively small set of values which is known to contain Υ(G), and then, by means of explicit computations and ad hoc reasoning, to determine which of the values are actually in Υ(G). Let us focus on the set Υ(Z2k ). The aforementioned result of Zaks settles the case k = 1, that is, Υ(Z2 ) = {1}.

(22.1)

In principle, to get Υ(Z2k ), one simply lists all generating subsets of Z2k , and, for each S in the list, computes ρ(B(Z2k , S)). As the value of k grows, this becomes an unwieldy–even insurmountable–task. To address this problem, let j be any positive integer, G be any finite abelian group, and define the set Υj (G) as follows: Υj (G) := {ρ(R) : Cl(R) = G and #S(R) ≤ j}. Note that Υj (G) ⊆ Υ(G), for all j. Our main result is Theorem 1.1. Υ2 (Z2k ) = Υ(Z2k ), for k = 1, 2, 3, 4, 5, 6.

On Factorization in Krull Domains with Divisor Class Group Z2k

327

It would be interesting to know the smallest value of k for which the theorem fails, or else to prove that it holds for all k. The reader will quickly realize that if the theorem does hold in general, more conceptual techniques will be necessary in order to prove it. It is also natural to wonder, given a finite abelian group G, what is the smallest value of j such that Υj (G) = Υ(G), though that problem will not be addressed in this chapter.

2

Block Monoids and k = 1, 2, 3, 4

First we make the block monoid concept precise (see [9] for more details): If G is a finite abelian group and g1 , . . . , gt are (not necessarily distinct) nonzero t elements of G such that i=1 gi = 0, then we refer to the system g1 g2 g3 · · · gt as a zero-system (or block) of G. We do not distinguish between two blocks if one can be obtained from the other by a permutation. The collection B(G) of all zero-systems of G is a commutative monoid under the operation of concatenation, and if S ⊆ G \ {0}, then we denote by B(G, S) the submonoid consisting of those blocks which involve only elements of S. The trivial (empty) block is the identity of B(G, S) and is denoted by 1. A nonzero block σ is called irreducible if no proper nontrivial subsystem of σ is also a block, that is, σ = αβ ⇒ α = 1 or β = 1, for any blocks α, β. If S = {a1 , . . . , ak }, then we usually write elements of B(G, S) in the form σ = ae11 ae22 · · · aekk , where ei denotes the number of repetitions of ai in the zerosystem σ. To each block σ = ae11 ae22 · · · aekk in B(G, S) we associate the cross number k(σ), which is defined by k ei , k(σ) = o(a i) i=1 where o(ai ) denotes the order of ai in G. The elasticity of a block monoid H = B(G, S) is defined in the obvious way: ρ(H) = sup{

m : σ1 σ2 · · · σm = τ1 τ2 · · · τn , for some irreducibles σi , τj ∈ H}. n

The following fact, whose proof is based on ideas in [11] and [5], considerably simplifies the computation of the elasticity when G = Zpk . Lemma 2.1. [1, Lemma 5(a)] For any S ⊆ Zpk \ {0}, we have ρ(B(Zpk , S)) = max{k(σ)−1 : σ is an irreducible block in B(Zpk , S)}. Whenever S is finite, the block monoid B(G, S) has a finite number of irreducible elements (by a Dickson’s Lemma argument), so Lemma 2.1 makes sense, and the elasticity of a block monoid of the form B(Zpk , S) is easily obtained, once the set of irreducible elements is known.

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Kattchee

For future reference, we also mention that if G is any finite abelian group and S ⊆ S  ⊆ G \ {0G }, then ρ(B(G, S  )) ≤ ρ(B(G, S)),

(22.2)

and if φ is an automorphism of the group G, then B(G, S) ∼ = B(G, φ(S)).

(22.3)

The proofs are left as easy exercises. The following lemma assists in the computation of Υ(Zpk ) by providing a relatively small set of values which is known to contain Υ(Zpk ). Lemma 2.2. [1, Corollary 10] (a) Let p be an odd prime integer. Then Υ(Zpk ) ⊆ {

pk pk pk , , · · · , , 1}, 2 3 ξ

k

where ξ = # 2p3 $. (b) Let p = 2. Then Υ(Z2k ) ⊆ {2k−1 ,

2k−1 2k−1 2k−1 , ,··· , , 1}, 2 3 ξ

k

where ξ = # 23 $. The next proposition settles three cases of our main theorem. Proposition 2.3.

(a) Υ2 (Z2 ) = Υ(Z2 ) = {1}

(b) Υ2 (Z4 ) = Υ(Z4 ) = {2, 1} (c) Υ2 (Z8 ) = Υ(Z8 ) = {4, 2, 1} Proof. The only block monoid B(Z2 , S) with #S ≤ 2 is B(Z2 , {1}), which clearly has elasticity 1, so Υ2 (Z2 ) = {1}. This fact, along with Eq (22.1), proves part (a). To prove (b) and (c), note that Υ2 (Z4 ) ⊆ Υ(Z4 ) ⊆ {2, 1},

(22.4)

Υ2 (Z8 ) ⊆ Υ(Z8 ) ⊆ {4, 2, 1},

(22.5)

and where the right-hand containments follow from Lemma 2.2. Now, using the methods of the article [6] for computing elasticities of block monoids of the form B(Zn , {1, a}), we have  1 if a = 2 ρ(B(Z4 , {1, a})) = 2 if a = 3,

On Factorization in Krull Domains with Divisor Class Group Z2k and

⎧ ⎪ ⎨1 ρ(B(Z8 , {1, a})) = 2 ⎪ ⎩ 4

329

if a = 2, 4 if a = 3, 5, 6 if a = 7.

It follows immediately that the containments in Eqs (22.4) and (22.5) are all equalities. Before proceeding further, the following lemma is necessary. Lemma 2.4. Let S ⊆ Z2k \{0} such that S generates Z2k . Then either ρ(B(Z2k , S)) = 1 or ρ(B(Z2k , S)) ≥ 2. Proof. First, we prove the lemma in the specific case where S = {1, a} by induction on k: By Proposition 2.3, the result holds for k = 1, 2 and 3. Now suppose that k > 3. If a is even, then it is not hard to see that B(Z2k , {1, a}) and B(Z2k−1 , {1, a2 }) are isomorphic (see [4, Lemma 1.1] for details on the underlying “normalization” process). Thus a ρ(B(Z2k , {1, a})) = ρ(B(Z2k−1 , {1, })), 2 and the inductive hypothesis may be applied. If a is odd, then consider the block k k σ = 12 −ξa aξ , where ξ = # 2a $. Note that σ is an irreducible block, and k(σ) =

ξ 1 2k − ξa + k ≤ , 2k 2 2

whence ρ(B(Z2k , {1, a})) ≥ 2, by Lemma 2.1. To prove the general case, first note that since S generates Z2k , we may assume that 1 ∈ S. If S = {1}, then obviously ρ(B(Z2k , S)) = 1. Otherwise, pick a ∈ S \{1} and observe that ρ(B(Z2k , S)) ≥ ρ(B(Z2k , {1, a}) ≥ 2.

Proposition 2.5. Υ2 (Z16 ) = Υ(Z16 ) Proof. By Lemma 2.2, we know that Υ2 (Z16 ) ⊆ Υ(Z16 ) ⊆ {8/1, 8/2, 8/3, 8/4, 8/5, 1}. The methods of [6] give ⎧ ⎪ ⎪ ⎪1 ⎪ ⎪ ⎪ ⎨8/4 ρ(B(Z16 , {1, a})) = 8/3 ⎪ ⎪ ⎪8/2 ⎪ ⎪ ⎪ ⎩8/1

if if if if if

a = 2, 4, 8 a = 6, 9, 10, 12 a = 3, 11 a = 5, 7, 13, 14 a = 15,

(22.6)

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Kattchee

so the left-hand containment in Eq (22.6) is equality provided 8/5 ∈ / Υ(Z16 ). But Lemma 2.4 ensures that fact.

Proposition 2.5 provides a simple illustration of the process which will be employed to find Υ(Z32 ) and Υ(Z64 ) in the next section, where the following lemma will be put to use. Lemma 2.6. If {1, a} ⊆ S ⊆ Zpk \ {0}, 1 = a, and ρ(B(Zpk , {1, a})) = 1, then ρ(B(Zpk , S \ {a})) = ρ(B(Zpk , S)). Proof. For convenience, set q = pk . Since ρ(B(Zq , {1, a})) = 1, it follows from Lemma 2.1 that each irreducible block in B(Zq , {1, a}) has cross number equal to one. Note that 1q−a a1 and 1q are irreducible blocks. We may thus replace a, wherever it appears in a block, with 1a , and obtain a new block with the same cross number. Now let σ be any irreducible block in B(Zq , S) such that ρ(B(Zq , S)) = k(σ)−1 . Note that k(σ) ≤ 1. If σ does not involve a, then also σ ∈ B(Zq , S \ {a}), whence ρ(B(Zq , S \{a})) ≥ ρ(B(Zq , S)). The reverse inequality is obvious, so equality holds, and we are done. If σ does involve a, then repeat the process outlined in the last paragraph to obtain a block τ which has the same cross number as σ but does not involve a. Note that τ is irreducible; otherwise there would be an irreducible block α in B(Zq , S) such that k(α)−1 is greater than ρ(B(Zq , S)), which is impossible, in view of Lemma 2.1. Now, to complete the proof of the lemma, apply to τ the same argument as to σ, when σ did not involve a.

3

The Sets Υ(Z32) and Υ(Z64)

In the last section, Theorem 1.1 was verified for the values k = 1, 2, 3, 4. The cases k = 5 and k = 6 are approached in a similar manner. We appeal once more to the methods of [6] to compute ⎧ 1 ⎪ ⎪ ⎪ ⎪ ⎪ 16/8 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨16/6 ρ(B(Z32 , {1, a})) = 16/4 ⎪ ⎪ ⎪ 16/3 ⎪ ⎪ ⎪ ⎪ ⎪ 16/2 ⎪ ⎪ ⎩ 16

if if if if if if if

a = 2, 4, 8, 16 a = 12, 17, 18, 20, 24 a = 3, 6, 11, 22 a = 5, 7, 9, 10, 13, 14, 23, 25, 26, 28 a = 19, 27 a = 15, 21, 29, 30 a = 31,

(22.7)

On Factorization in Krull Domains with Divisor Class Group Z2k

331

and

⎧ 1 ⎪ ⎪ ⎪ ⎪ ⎪ 32/16 ⎪ ⎪ ⎪ ⎪ ⎪ 32/12 ⎪ ⎪ ⎪ ⎪ 32/11 ⎪ ⎪ ⎪ ⎪ ⎪ 32/8 ⎪ ⎪ ⎪ ⎨32/7 ρ(B(Z64 , {1, a})) = ⎪ 32/6 ⎪ ⎪ ⎪ ⎪ ⎪ 32/5 ⎪ ⎪ ⎪ ⎪ 32/4 ⎪ ⎪ ⎪ ⎪ ⎪ 32/3 ⎪ ⎪ ⎪ ⎪ ⎪32/2 ⎪ ⎪ ⎩ 32

if if if if if if if if if if if if

a = 2, 4, 8, 16, 32 a = 24, 33, 34, 36, 40, 48 a = 6, 12, 22, 44 a = 3, 43 a = 5, 10, 13, 14, 17, 18, 20, 26, 28, 46, 49, 50, 52, 56 a = 11, 23, 35, 39 a = 37, 38, 45, 54 a = 7, 19, 27, 55 a = 9, 15, 25, 29, 30, 41, 42, 47, 53, 57, 58, 60 a = 51, 59 a = 21, 31, 61, 62 a = 63. (22.8)

Therefore, Υ2 (Z32 ) = {16,

16 16 16 16 16 , , , , , 1}, 2 3 4 6 8

(22.9)

and

32 32 32 32 32 32 32 32 32 32 , , , , , , , , , , 1}. 2 3 4 5 6 7 8 11 12 16 Applying Lemmas 2.2 and 2.4 as in Proposition 2.5 above, we have Υ2 (Z64 ) = {32,

Υ2 (Z32 ) ⊆ Υ(Z32 ) ⊆ {16,

16 16 16 16 16 16 16 , , , , , , , 1}, 2 3 4 5 6 7 8

(22.10)

(22.11)

and

32 32 32 32 , , , . . . , , 1}. (22.12) 2 3 4 16 Our aim is to show that the left-hand containments in Eqs (22.11) and (22.12) are in fact equalities. Verification of the following two lemmas, which contain the cross numbers of some useful irreducible zero-systems, is left to the reader. Υ2 (Z64 ) ⊆ Υ(Z64 ) ⊆ {32,

Lemma 3.1. Below is a chart of cross numbers of selected irreducible blocks in B(Z32 ). k(σ) σ (i) 13 171 121 1/4 (ii) 171 35 3/16 3/16 (iii) 175 111 (iv) 37 111 1/4 (v) 13 171 62 1/4 (vi) 13 171 222 1/4 (vii) 12 34 181 1/4 1/4 (viii) 34 201 (ix) 14 32 221 1/4 (x) 14 112 61 1/4 1/4 (xi) 114 201 (xii) 12 114 181 1/4

332 Lemma 3.2. Below is B(Z64 ). σ (i) 17 331 241 (ii) 335 39 (iii) 339 435 (iv) 37 431 (v) 11 331 65 (vi) 17 331 122 (vii) 19 331 225 (viii) 17 331 442 (ix) 310 341 (x) 11 39 361 (xi) 38 401 (xii) 14 32 221 (xiii) 12 36 441 (xiv) 432 345 (xv) 15 435 361 (xvi) 438 401 (xvii) 432 67 9 (xviii) 1 431 121

Kattchee a chart of cross numbers of selected irreducible blocks in k(σ) 1/4 32/7 32/7 1/8 3/16 1/4 3/16 1/4 3/16 7/32 1/4 1/4 3/16 3/16 7/32 1/4 1/4 7/32

The correctness of the next lemma is guaranteed by the correctness of existing algorithms for computing the irreducible elements of a block monoid and by the correctness of Lemma 2.1; it has been verified by the author on a computer. Lemma 3.3. The following are elasticities of some useful block monoids. (a) ρ(B(Z32 , {1, 17, 18, 20, 24})) = 2. (b) ρ(B(Z32 , {1, 3, 12, 24, 6})) = 8/3. (c) ρ(B(Z32 , {1, 11, 12, 24, 22})) = 8/3. (d ) ρ(B(Z64 , {1, 33, 34, 36, 40, 48})) = 2. (e) ρ(B(Z64 , {1, 3, 6, 12, 24, 48})) = 32/11. (f ) ρ(B(Z64 , {1, 43, 22, 44, 24, 48})) = 32/11. We now present the final two ingredients in the proof of Theorem 1.1. Proposition 3.4. Υ2 (Z32 ) = Υ(Z32 ). Proof. In view of Eqs (22.9) and (22.11), it is sufficient to show that 16/7 and 16/5 are not elements of Υ(Z32 ). To see that 16/7 ∈ / Υ(Z32 ), suppose for a contradiction that S is such that ρ(B(Z32 , S)) = 16/7. Since S must generate Z32 , there is an automorphism φ of Z32 such that 1 ∈ φ(S), so by Eq (22.3) we may assume without loss that 1 ∈ S. In view of Eqs (22.2) and (22.7), we have S ⊆ {1, 2, 4, 8, 16, 12, 17, 18, 20, 24},

On Factorization in Krull Domains with Divisor Class Group Z2k

333

and, by Lemma 2.6, we may assume further that S ⊆ {1, 12, 17, 18, 20, 24}. We are already assuming that 1 ∈ S, but we must also have 17 ∈ S for the following reason: If 17 ∈ / S and we write S = {1, s2 , s3 , . . . , st }, then the elements of S other than 1 are all even, and the “normalization” process of [4, Lemma 1.1] may be employed to obtain B(Z32 , S) ∼ = B(Z16 , {1,

st s2 , . . . , }). 2 2

The elasticity of the block monoid on the right is an element of the set Υ(Z16 ), which does not contain 16/7 (see Proposition 2.5). Therefore, 17 ∈ S. Since k(13 171 121 ) = 1/4 (by Lemma 3.1(i)), we know that ρ(B(Z32 , {1, 12, 17})) ≥ 4. By Eq (22.2), then, we know 12 ∈ / S. Therefore, S ⊆ {1, 17, 18, 20, 24}, but since ρ(B(Z32 , {1, 17, 18, 20, 24})) = 2 (by Lemma 3.3(a)), we may apply Eq (22.2) again to see that ρ(B(Z32 , S)) ≤ 2, for the contradiction. To see that 16/5 ∈ / Υ(Z32 ), suppose that S is such that ρ(B(Z32 , S)) = 16/5. In view of Eqs (22.2), (22.7), and Lemma 2.6, we may assume that S ⊆ {1, 12, 17, 18, 20, 24, 3, 6, 11, 22}. We are still assuming that 1 ∈ S, so by another application of the “normalization” argument of [4], the set S must contain at least one of 17, 3 or 11. But Lemma 3.1(ii), (iii), and (iv) say that k(171 35 ) = k(175 111 ) = 3/16 and k(37 111 ) = 1/4, so, by Lemma 2.1 and Eq (22.2), S can contain at most one of 17, 3 or 11. If 17 ∈ S, then by Lemma 3.1(i), (v), and (vi), none of 12, 6, 22 are in S. Thus S ⊆ {1, 17, 18, 20, 24}, and Eq (22.2) and Lemma 3.3(a) imply that ρ(B(Z32 , S) ≤ ρ(B(Z32 , {1, 17, 18, 20, 24}) = 2, which is a contradiction. If 3 ∈ S, then Lemma 3.1(vii), (viii), and (ix) imply that S ⊆ {1, 12, 24, 3, 6}, but then ρ(B(Z32 , S)) ≤ ρ(B(Z32 , {1, 12, 24, 3, 6})) = 8/3 (by Eq (22.2) and Lemma 3.3(b)), which is a contradiction. If 11 ∈ S, then Lemma 3.1(x), (xi), and (xii) imply that S ⊆ {1, 12, 24, 11, 22}, but then ρ(B(Z32 , S)) ≤ ρ(B(Z32 , {1, 12, 24, 11, 22})) = 8/3 (by Eq (22.2) and Lemma 3.3(c)), which is a contradiction. Therefore, the assumption that S is such that ρ(B(Z32 , S)) = 16/5 is absurd, and the proof is complete. The logic in the next proof is similar to Proposition 3.4. Proposition 3.5. Υ2 (Z64 ) = Υ(Z64 ). Proof. In view of Eqs (22.10) and (22.12), it is sufficient to prove that 32/d ∈ / Υ(Z64 ), for d = 9, 10, 13, 14, and 15. Suppose that S is such that ρ(B(Z64 , S)) = 32/d, for d = 13, 14, or 15. Assuming, as before without loss, that 1 ∈ S, Eqs (22.2) and (22.8) imply that S ⊆ {1, 2, 4, 8, 16, 32, 24, 33, 34, 36, 40, 48}.

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By Lemma 2.6, we have S ⊆ {1, 24, 33, 34, 36, 40, 48}, and we can argue that 33 ∈ S using the same “normalization” argument as in the proof of Proposition 3.4, whence Eq (22.2) and Lemma 3.2(i) imply that S ⊆ {1, 33, 34, 36, 40, 48}. But then Eq (22.2) and Lemma 3.3(d) imply that ρ(B(Z64 , S)) ≤ ρ(B(Z64 , {1, 33, 34, 36, 40, 48})) = 2, which is a contradiction. Now suppose that S is such that ρ(B(Z64 , S)) = 32/d, for d = 9 or 10. Again, we assume without loss that 1 ∈ S, and after applying Eqs (22.2), (22.8), and Lemma 2.6, we have S ⊆ {1, 24, 33, 34, 36, 40, 48, 6, 12, 22, 44, 3, 43}. Recycling the “normalization” argument one more time, together with Eq (22.2) and Lemma 3.2(ii), (iii), and (iv), we must have exactly one of 33 ∈ S, 3 ∈ S or 43 ∈ S. If 33 ∈ S, then Lemma 3.2(i), (v), (vi), (vii), and (viii) imply that S ⊆ {1, 33, 34, 36, 40, 48}, but then, by Lemma 3.3(d), ρ(B(Z64 , S)) ≤ ρ(B(Z64 , {1, 33, 34, 36, 40, 48})) = 2, which is a contradiction. If 3 ∈ S, then Lemma 3.2, parts (ix) through (xiii), implies that S ⊆ {1, 24, 48, 6, 12, 3}, but then, according to Lemma 3.3(e), we have ρ(B(Z64 , S)) ≤ ρ(B(Z64 , {1, 24, 48, 6, 12, 3})) = 32/11, which is a contradiction. Finally, if 43 ∈ S, then Lemma 3.2, parts (xiv) through (xviii), implies that S ⊆ {1, 24, 48, 22, 44, 43}, and it follows by Lemma 3.3(f) that ρ(B(Z64 , S)) ≤ ρ(B(Z64 , {1, 24, 48, 22, 44, 43})) = 32/11, resulting in another contradiction. This completes the proof. Theorem 1.1 is now established. The curious reader may wonder, as does the author, whether the theorem holds for all k. That is an open question. It is known, however, that Υ2 (Z19 ) = Υ(Z19 ), since 19/6 ∈ Υ(Z19 ) \ Υ2 (Z19 ) [1, p 2552], so if 2k is replaced with 19k in Theorem 1.1, then the statement is false for k = 1. Acknowledgements: The author wishes to thank Scott Chapman and Bill Smith for the numerous conversations in which many of the ideas contained in this chapter originated. The contents of this chapter are based partly on the results of some computer experimentation; the author thanks Pedro Garc´ıa-S´anchez and Bill Smith for providing two of the programs which were used. Finally, the author thanks the referee for the helpful remarks.

Bibliography [1] D.F. Anderson, S.T. Chapman, On the Elasticities of Krull Domains with Finite Cyclic Divisor Class Group, Communications in Algebra 28(5) (2000), 2543–2553. [2] S. Chapman, J. Garc´ıa-Garc´ıa, P. Garc´ıa-S´anchez, J. Rosales, Computing the Elasticity of a Krull Monoid, Linear Algebra Appl. 336 (2001), 191–200. [3] S. Chapman, A. Geroldinger, Krull monoids, their sets of lengths and associated combinatorial problems, Factorization in Integral Domains (Anderson, D.D., Ed.), Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 189 (1997), 73–112. [4] S. Chapman, U. Krause, E. Oeljeklaus, Monoids Determined by a Homogenous Linear Diophantine Equation and the Half-factorial Property, J. Pure Appl. Algebra 151 (2000), 107–133. [5] S.T. Chapman, W.W. Smith, An Analysis Using the Zaks-Skula Constant of Element Factorizations in Dedekind Domains, J. Algebra 159 (1993), 176– 190. [6] S.T. Chapman, W.W. Smith, On Factorization in Block Monoids Formed by {1, a} in Zn , Proc. Edinburgh Math. Soc. 46 (2003), 257–267. [7] R. Fossum, The Divisor Class Group of a Krull Domain, Springer-Verlag, New York, 1973. [8] A.P. Grams, The distribution of prime ideals of a Dedekind domain, Bull. Austral. Math. Soc. 11 (1974), 429–441. [9] F. Halter-Koch, Finitely generated monoids, finitely primary monoids, Factorization in Integral Domains (Anderson, D.D., Ed.), Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 189 (1996), 31–72. [10] K. Kattchee, Elasticities of Krull Domains with Finite Divisor Class Group, Linear Algebra Appl. 384C, 171–185. [11] U. Krause, A Characterization of Algebraic Number Fields with Cyclic Class Group of Prime Power Order, Math Z. 186 (1984), 143–148. 335

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[12] A. Zaks, Half-factorial domains, Bull. Amer. Math. Soc. 82 (1976), 721–24.

Chapter 23

Integral Morphisms by

Jack Maney Abstract A half-factorial domain (HFD) R is an atomic domain where, given any collection of irreducibles {α1 , α2 , · · · , αm , β1 , β2 , · · · , βn } with α1 α2 · · · αm = β1 β2 · · · βn , then n = m. In a paper by J. Coykendall [3], a generalization of the length function of Zaks [9], called the boundary map, was introduced. In this chapter, we look at generalizations of the boundary map, called integral morphisms. Using integral morphisms, we generalize many of the author’s results in [6].

1

Introduction

We recall that a domain R is called atomic if every nonzero nonunit can be factored into irreducibles (or atoms). An atomic domain R is called a half-factorial domain, or HFD, if whenever we have a collection of irreducibles {α1 , α2 , · · · , αm , β1 , β2 , · · · , βn } with α1 · · · αm = β1 · · · βn then n = m. HFDs were originally studied by Carlitz in [2], although Zaks, in [8], was the first to coin the terminology of “HFD”. HFDs are a generalization of unique factorization, and they have been extensively studied as of late. In [3], the following tool was introduced. Definition 1.1. Let R be an HFD with quotient field K. If R = K, we define the π1 π2 · · · πt with boundary map ∂R : K \ {0} −→ Z by ∂R (α) = t − k, where α = δ1 δ2 · · · δk πi , δj irreducible. If R = K, we define ∂R (α) = 0 for all nonzero α. We remark that the boundary map is a homomorphism of abelian groups, and it is well-defined precisely when R is an HFD. Also, for all x ∈ R \ {0}, ∂R counts the number of irreducibles in any irreducible factorization of x (and, of course, ∂R (x) = 0 ⇔ x ∈ U (R)). 337

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The boundary map was used by Coykendall in [3] to prove that if we have an order R of a ring of algebraic integers R, and if R is an HFD, then so is R. In [6], the author used the boundary map to study overrings of a general HFD, and in [5], the author defined and studied a class of HFDs called boundary valuation domains by using the boundary map. However, there are many examples of homomorphisms of abelian groups from the nonzero elements of the quotient field of a domain to the integers which share many of the nice properties of the boundary map. Example 1.2. Let R be a Dedekind domain with quotient field K. Let φ : K \ {0} −→ Z be given by φ( ab ) = n−m where aR = P1 P2 · · · Pn and bR = Q1 Q2 · · · Qm (Pi , Qj prime). This is a well-defined homomorphism of abelian groups. What is more, for all x ∈ R \ {0}, φ(x) ≥ 0 and φ(x) = 0 if and only if x ∈ U (R). √ Note that in Example 1.2, R need not be an HFD. For example, Z[ −14] is well-known to be a Dedekind domain that is not an HFD. With this in mind, we wish to extend the boundary map to a more general notion of maps from K ∗ to Z. By N, Z, and Q, we mean the natural numbers, integers, and complex numbers respectively. If S is a subset of a ring, then by S ∗ we mean S \ {0}. Most of the following definitions are due to Gonzalez and Pellerin ([4]). Definition 1.3 ([4]). Let R be a domain with quotient field K. We say that φ is an integral morphism on R if φ : K ∗ −→ Z is a homomorphism of abelian groups. We denote the set of all integral morphisms on R by M(R). Note that an arbitrary map in M(R) can send units of R to positive or negative values. Since R may have a lot of units, resulting in “chaotic” integral morphisms, we distinguish a special class of integral morphisms. Definition 1.4. Let R be a domain with quotient field K. We say that φ ∈ M(R) is a faithful integral morphism (over R) if φ(u) = 0 for all u ∈ U (R). We denote the set of all faithful integral morphisms on R by M0 (R). Definition 1.5 ([4]). Let R be a domain with quotient field K. We say that φ ∈ M(R) is a positive integral morphism (or that φ is positive) if φ(x) ≥ 0 for all nonzero x ∈ R. We say that a positive integral morphism φ on R is a strictly positive integral morphism (or we say φ is strictly positive) if φ(x) > 0 for all nonzero nonunit x ∈ R. We denote the set of all positive integral morphisms on R and strictly positive integral morphisms on R by P(R) and P+ (R), respectively. We will denote the zero map from K ∗ to Z by 0, trusting that the chance of confusing it with the zero element of R will be minimal. We will concern ourselves mostly with positive and strictly positive integral morphisms. However, in the spirit of Definition 1.5, we also have the following definition.

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Definition 1.6 ([4]). Let R be a domain with quotient field K. We say that φ ∈ M(R) is negative (strictly negative) if, given any nonzero x ∈ R, φ(x) ≤ 0 (φ is negative and given any nonzero nonunit x ∈ R, φ(x) < 0). We denote the set of all negative integral morphisms (strictly negative integral morphisms) by −P(R) (−P+ (R)). Let φ ∈ M(R). We make a few observations: 1. K ∗ /ker(φ) ∼ = im(φ) ∼ = nZ for some nonnegative integer n. Also, im(φ) ∼ =Z if and only if φ = 0. 2. If φ ∈ P(R), then U (R) = Ker(φ) ∩ R if and only if φ ∈ P+ (R). 3. We only need to know what φ does to the nonzero elements of R to know what it does on K ∗ . Or, put more formally, any homomorphism of monoids from R∗ to Z has a unique extension to an integral morphism on K ∗ . And conversely, given an integral morphism, we may restrict it down to R∗ to get a homomorphism of monoids from R∗ to Z. This is why we refer to φ ∈ M(R) as an integral morphism on R, rather than on K ∗ . 4. Finally, there is a one to one correspondence between positive integral morphisms on R and negative integral morphisms on R. For if φ ∈ P(R), then −φ ∈ −P(R), and likewise if ψ ∈ P(R), then −ψ ∈ −P(R) (where, of course, by −φ, we mean the function that takes α ∈ K ∗ to −1 · φ(α) for all α ∈ K ∗ ). By the exact same reasoning, there is a one to one correspondence between + P (R) and −P+ (R). It is for this primary reason that we will not concern ourselves with negative integral morphisms.

2

Preliminary Results

Theorem 2.1. For a domain R with quotient field K, the following hold: 1. Pointwise addition of integral morphisms is a well-defined binary operation on M(R). 2. Under addition, M(R) is an abelian group with identity 0. 3. If M(R) is nontrivial, then M(R) is torsion-free. 4. Under addition, M0 (R) is a subgroup of M(R). 5. If φ ∈ P(R) and ψ ∈ P+ (R), then φ + ψ ∈ P+ (R). 6. P(R) is a submonoid of M(R). 7. P+ (R)–if it is nonempty–is a subsemigroup of M(R). 8. M(R) is partially ordered by: φ ≤ ψ if and only if φ(x) ≤ ψ(x) for all x ∈ R∗ . What is more, M0 (R) inherits this ordering from M(R), and φ ∈ P(R) if and only if φ ≥ 0. Proof: Obvious. 

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Definition 2.2. We call the ordering on M(R) given in part 8 of Theorem 2.1 the R-ordering on M(R). This raises the following questions: since M(R) is a partially ordered Abelian group, when is M(R) the group of divisibility for some domain T ? Also, if R is an HFD, then clearly ∂R is the smallest element (under the Rordering) of P+ (R). If R is a domain such that φ ∈ P+ (R) is the smallest element, then is R an HFD? Finally, given a domain R with quotient field K, how are the factorization properties of R related to the factorization properties of the monoid domains R[X; P(R)] and K[X; P(R)]? Proposition 2.3. Let R be a domain, and let φ ∈ P(R). Then, for all u ∈ U (R), φ(u) = 0. In other words, < P(R) >⊆ M0 (R), where < P(R) > is the subgroup of M(R) generated by P(R). Proof: Since φ ∈ P(R), we must have φ(u) = 0 or φ(u) > 0. But, if φ(u) > 0, / P(R), a contradiction. Therefore then φ(u−1 ) = −φ(u) < 0 implying that φ ∈ φ(u) = 0. The last statement clearly follows.  This next theorem essentially says that faithful integral morphisms have a “high degree of freedom” on nonzero principal primes in atomic domains. Theorem 2.4. Let R be an atomic domain with nonzero prime elements. Denote the collection of nonassociate nonzero primes of R by Γ = {pα }α∈Λ . For each α ∈ Λ, choose an integer nα , and let φ ∈ M0 (R). Then there exists ψ ∈ M0 (R) such that φ(pα ) = nα for each α ∈ Λ, and ψ(x) = φ(x) for each x ∈ R∗ that is a not divisible by any pα . If φ ∈ P(R) and each nα ≥ 0, then ψ ∈ P(R). If φ ∈ P+ (R) and each nα > 0, then ψ ∈ P+ (R). Proof: For each α ∈ Λ, define ψ linearly on the irreducibles of R by  nα if xR = pα R, for some α ∈ Λ, ψ(x) = φ(x) if xR = pα R for all α ∈ Λ. Since R is an atomic domain, we may factor any nonzero nonunit into irreducibles. However, in any irreducible factorization of a fixed nonzero nonunit element, pα appears a unique number of times for each α ∈ Λ. Thus ψ is well-defined, and it is clear that ψ gives a homomorphism of abelian monoids from R∗ to Z, and thus has a unique extension to a faithful integral morphism. The last two statements clearly follow.  Corollary 2.5. Let R be an Then φp is a positive integral  n φp (x) = 0

atomic domain, and let p be a nonzero prime of R. morphism, where, for x ∈ R∗ , if p divides x in R exactly n times, if p does not divide x in R.

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Proof: Let φ = 0 and k = 1 in Theorem 2.4.  Corollary 2.6. Suppose R is atomic and that M0 (R) is linearly ordered. Then there can exist at most (up to associates in R) one nonzero prime in R. Proof: Suppose not. Then there are two nonzero nonassociate primes p1 , p2 in R. Choose any φ ∈ P(R) such that φ(p2 ) ≥ 1 (we may do this by Theorem 2.4). Define ψ on R∗ by: ⎧ ⎪ ⎨φ(p1 ) + 1 if αR = p1 R, ψ(α) = φ(p2 ) − 1 if αR = p2 R, ⎪ ⎩ φ(α) otherwise. Note that since φ(p2 ) ≥ 1, we have ψ ∈ P(R). However we do not have φ ≤ ψ, since ψ(p2 ) < φ(p2 ), and we also do not have ψ ≤ φ, since φ(p1 ) < ψ(p1 ). This contradicts our assumption that M0 (R) is linearly ordered.  When is M(R) = M0 (R)? When is M0 (R) =< P(R) >? Proposition 2.7. Given φ ∈ M(R), φ ∈< P(R) > if and only if it can be written in the form φ1 − φ2 for φ1 , φ2 ∈ P(R) Proof: An arbitrary nonzero element of < P(R) > looks like ξ = a1 ψ1 + a2 ψ2 + · · · + an ψn with each ψi ∈ P(R) and ai ∈ Z. Collecting the positive and negative coefficients allows us to write ξ in the required form. The other implication is obvious.  Recall that an atomic domain R is a bounded factorization domain, or BFD, if for every nonzero nonunit x ∈ R, there exists N (x) ∈ N such that any irreducible factorization of x has at most N (x) irreducibles. Theorem 2.8. Let R be a domain with quotient field K. Consider the following conditions: 1. 2. 3. 4. 5.

P+ (R) = ∅. R is an HFD. R is a BFD. R is atomic. R is a Dedekind domain.

Then 2 ⇒ 1, 5 ⇒ 1, and 1 ⇒ 3 ⇒ 4. What is more, 1  2, and 3  1. Proof: (2 ⇒ 1): If R is an HFD, then ∂R ∈ P+ (R). (1 ⇒ 3 ⇒ 4): Suppose P+ (R) = ∅. The proof of why R is atomic is almost identical to that of part 1 of Proposition 3.11 in [6]. Suppose that there exists some nonzero nonunit x ∈ R such that x cannot be factored into irreducibles. Let φ ∈ P+ (R), and fix φ(x) = k > 0.

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Since x cannot be factored into irreducibles, we may write x = α1 β1 with α1 , β1 nonunits of R, and β1 nonfactorable into atoms. So, write β1 = α2 β2 with α2 , β2 nonunits, and β2 nonfactorable into atoms. Continue this process. At the k th step, we have: x = α1 α2 · · · αk βk ⇒ k = φ(x) =

k 

 φ(αi ) + φ(βk ) ≥ k + 1,

i=1

a contradiction. Therefore there exists no such x, and R is atomic. It is clear that given any nonzero nonunit x ∈ R, x can be factored into at most φ(x) irreducibles, whence R is a BFD. (5 ⇒ 1): See Example 1.2. (3  1): See Example 3.1. (1  2): It is well-known that there exist non-HFD Dedekind domains. 

3

Examples

One might ask if the atomicity of a domain R implies that P+ (R) is nonempty. We not only give an example of an atomic domain where P+ (R) is empty, but the example is a BFD. Example 3.1. Let V = F2 [{xα | α ∈ Q, α ≥ 0}]M , where M is the ideal generated by {xα | α ∈ Q, α > 0}. Let I = xV , and let R = (F2 + xV )xV . All of the nonunits of R are of the form xb (xa1 + xa2 + · · · + xan ) with 1 ≤ a1 < a2 < · · · < an , 1 ≤ a1 < 2 and b = 0 or b ≥ 1. If we let f = xa1 + xa2 + · · · + xan , then f is irreducible. To see why, suppose f = (xb1 + xb2 + · · · + xbk )(xc1 + xc2 + · · · + xcm ) with 1 ≤ b1 < b2 · · · < bk and 1 ≤ c1 < c2 · · · < cm . It is then clear that the smallest possible power of x in the product of the right hand side is b1 + c1 . But then we must have b1 + c1 ≥ 2, whence x|f and a1 ≥ 2, a contradiction. Therefore f is irreducible. So, given an arbitrary nonzero nonunit xb (xa1 + xa2 + · · · + xan ) ∈ R with b, ai as above, it is clear that xb can be factored into at most q irreducibles, where b = st with t, s ∈ N and t = sq + r (with r = 0 or 0 < r < s). Therefore R is a BFD. We now show that not only is P+ (R) empty, but P(R) = {0}. Suppose there exists some φ ∈ P(R) \ {0}. r+2 1 x r+1 r+1 is an element of the quotient field of R. If φ(x) = r > 0, then x = x 1 Therefore r = φ(x) = (r + 1)φ(x r+1 ), a contradiction, whence φ(x) = 0. So, let α = xb (xa1 + · · · + xan ) be an arbitrary nonzero nonunit of R with b, ai as above. Then φ(α) = φ(xa1 + · · · + xan ) = φ(1 + xa2 −a1 + · · · + xan −a1 ). Now, there exists j ∈ N such that 1 ≤ 2j (a2 − a1 ). Since char(R) = 2, this gives us j j j that 2j φ(α) = φ(1 + x2 (a2 −a1 ) + · · · + x2 (an −a1 ) ). However, 1 + x2 (a2 −a1 ) + · · · + j x2 (an −a1 ) ∈ U (R), so φ(α) = 0, and P(R) = {0}.

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We also might ask if R is not atomic, then is P(R) trivial? We give an example to show that this need not be the case. Example 3.2. Let F be any field, and let R = F [x, xy , xy2 , · · · ]M where M is the ideal of F [x, xy , xy2 , · · · ] generated by {x, xy , xy2 , · · · }. ym Any nonzero element of R can be written in the form n u(x, y) with u(x, y) ∈ x U (R), m ≥ 0, and n ≤ 0 if m = 0. Consequently, in K (the quotient field of R), ym every element can be written in the form n u(x, y), where again u ∈ U (R), but x this time m, n ∈ Z. Note that if φ ∈ P(R), we can figure out what φ does on K ∗ simply by defining φ on x and y, and extending linearly. y Note also that we must have φ(x) = 0, for if φ(x) > 0, then n ∈ R for all x y n implies that φ( n ) = φ(y) − nφ(x). Since this holds for all n, there exists some x large enough n with φ( xyn ) < 0, whence φ ∈ / P(R), a contradiction. However, φ may send y to any nonnegative integer that we wish. Therefore P(R) = {0}. Also, note that the only elements in R that can be mapped by a positive integral morphism to a positive integer are elements that cannot be written as products of irreducibles in R. This is also an example of a domain that does not satisfy the conclusion of Theorem 2.4, thus showing that the assumption of atomicitiy for this theorem is needed. Example 3.3. Let F be any finite field. Consider the polynomial extension R = F [x]. It is easy to see that any unit of R is a root of unity. So, let φ ∈ M(R) be given, along with some u ∈ U (R). Since u is a root of unity, un = 1 for some n ≥ 1. Thus φ(un ) = nφ(u) = φ(1) = 0, whence φ(u) = 0 implying that φ ∈ M0 (R) and M(R) = M0 (R). Example 3.4. Consider R = Z(2) = { ab ∈ Q | b ≡ 1 mod(2)}. Then ∂Z ∈ M(R) \ M0 (R), since 13 ∈ U (R), but ∂Z ( 13 ) = −1. Thus M0 (R)  M(R). This next example shows that given any domain R and an indeterminate x, we never have P(R[x]) nor P(R[[x]]) nontrivial. Example 3.5. Let R be a domain. For R[x], let φ(f (x)) = deg(f (x)) for all f (x) ∈ R[x]∗ . It is clear that φ ∈ P(R[x]). For R[[x]] and f (x) ∈ R[[x]]∗ , let φ(f (x)) = n where n is the largest possible number of factors of x that can be factored out of f (i.e., f (x) = xn g(x) with g(0) = 0).

4

Similarities to the HFD Case: φ-Positivity and φ-Completeness

In [6], the author introduced the following two definitions.

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Definition 4.1. Let R be an HFD with quotient field K, and let T be an overring of R. We call T a boundary positive overring of R if for all nonzero x ∈ T , ∂R (x) ≥ 0.

Other equivalent terminology to that expressed in Definition 4.1 would be that R ⊆ T is a boundary positive extension or T is boundary positive over R. Definition 4.2. Let R be an HFD with quotient field K, and let T be an overring of R. We say that T is a boundary complete overring with respect to R if x ∈ T with ∂R (x) = 0 implies x ∈ U (T ). In other words, T is boundary complete over R if no nonunit of T has zero boundary (over R, of course). Other equivalent terminology to that expressed in Definition 4.2 would be that R ⊆ T is boundary complete, or T is boundary complete over R. We will now generalize the notions of boundary positivity and boundary completeness to integral morphisms. Many of these results are generalizations of the results in [6]. In this section, unless otherwise stated, R will denote a domain with quotient field K, and T will denote an overring of R. Definition 4.3. Let φ ∈ P(R). We say that T is φ-positive (or T is a φ-positive overring of R) if for all x ∈ T ∗ , φ(x) ≥ 0. We say that T is positive (or T is a positive overring of R) if T is φ-positive for all φ ∈ P(R). Definition 4.4. Let φ ∈ P(R). We say that T is φ-complete (or T is a φ-complete overring of R) if x ∈ T ∗ with φ(x) = 0 implies that x ∈ U (T )–i.e. if T has no nonunits that are sent to zero by φ. We say that T is complete (or T is a complete overring of R) if T is φ-complete for all φ ∈ P+ (R) Proposition 4.5. Suppose that T is φ-positive for some φ ∈ P+ (R). Then no nonzero nonunit element of R becomes a unit of T . Proof: Let x be a nonzero nonunit element of R, so that φ(x) > 0. If x−1 ∈ T , then φ(x−1 ) < 0, contradicting the φ-positivity of T .  The following example shows that we cannot replace “φ ∈ P+ (R)” with “φ ∈ P(R) \ {0}” in the statement of Proposition 4.5. Example 4.6. Let R = Z[x, y], and let φ ∈ P(R) be the x-adic valuation on R. Let T = Z[x, y, y −1 ]. Then it is clear that T is a φ-positive overring of R. However, y is a nonunit of R that becomes a unit of T . Proposition 4.7. Let T be any almost integral overring of R. Then T is a positive overring of R.

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Proof: Let φ ∈ P(R) be arbitrary, and let x ∈ T be given. Then there exists r ∈ R∗ such that rxn ∈ R for all n ≥ 0. Since rxn ∈ R, we must have φ(rxn ) ≥ 0. Also, φ(rxn ) = φ(r) + nφ(x) ≥ 0. Now, since r ∈ R∗ , we have φ(r) ≥ 0. So, if the above inequality is to hold for all n, we must have φ(x) ≥ 0.  If R is a domain and if φ ∈ P(R), then is there any φ-positive overring of R that is not contained in R ? Lemma 4.8. Let φ ∈ P(R) \ {0}, with im(φ) = mZ. Suppose there exists some x ∈ T ∗ with φ(x) < 0. Then, there exists z ∈ T ∗ such that φ(zx) = 0. Proof: Let y ∈ T ∗ be such that φ(y) is minimal positive. Suppose im(φ) = mZ, and denote φ(x) = −jm, φ(y) = km for k, j > 0. There are three cases. If k|j, say kw = j for some w ∈ R, then φ(y w x) = 0, and we’re done (by a similar argument, we’re done if j|k). If k > j with j  k, then there exist q, r ∈ N such that k = qj + r with 0 < r < j < k. Then, k − qj = r, whence km + q(−jm) = rm, and φ(yxq ) = rm. However, since yxq ∈ T ∗ , and since 0 < φ(yxq ) = rm < jm < km, this contradicts the minimality of φ(y). So, assume k < j, with k  j. Again there exist q, r ∈ N such that j = qk + r with 0 < r < k < j. So, jm − q(km) = rm, which gives φ( xy1 q ) = rm, and φ(xy q ) = −rm, whence φ(xy q+1 ) = (k − r)m. However, xy q+1 ∈ T ∗ , and 0 < φ(xy q+1 ) = (k − r)m < km, again violating the minimality of φ(y).  Corollary 4.9. Let φ ∈ P(R) \ {0}. Assume T is φ-complete. Then x ∈ T ∗ with φ(x) < 0 implies x ∈ U (T ). Proof: Let x ∈ T ∗ with φ(x) < 0. By Lemma 4.8, there exists z ∈ T ∗ such that φ(zx) = 0. But T is φ-complete. Thus zx ∈ U (T ), and x ∈ U (T ).  Theorem 4.10. Let φ ∈ P(R) \ {0}. The following are equivalent: 1. T = K. 2. T is φ-complete and not φ-positive. Proof: Clearly, if T = K, then T is φ-complete and not φ-positive. So, assume T is φ-complete and not φ-positive. Then, there exists some u ∈ T ∗ with φ(u) < 0. By Corollary 4.9, u ∈ U (T ). Let x ∈ R∗ \ U (R) be arbitrary. We have φ(x) ≥ 0. Clearly, there exists a sufficiently large n ∈ N such that φ(un x) < 0. However, φ(un x) < 0 implies that un x ∈ U (T ), again by Corollary 4.9. Thus un x ∈ U (T ) gives us that x ∈ U (T ), and since every nonzero nonunit element of R becomes a unit in T , we see that T = K. 

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Proposition 4.11. Let φ ∈ P(R). Then, there exists an overring Tφ of T that is φ-complete with respect to R. Furthermore, if T is a φ-positive overring of R, then Tφ is also a φ-positive overring of R. Proof: Let S = {x ∈ T ∗ | φ(x) = 0}. It is easy to see that S is a multiplicatively closed set in T . So, consider TS = Tφ . U (Tφ ) = { st | t ∈ S}. So, let α ∈ Tφ with φ(α) = 0. Write α = st . Then 0 = φ( st ) = φ(t) − φ(s) = φ(t) implies that φ(t) = 0, whence t ∈ S. Therefore α ∈ U (Tφ ), and Tφ is φ-complete. Suppose T is also φ-positive. Choose st ∈ Tφ . Then φ( st ) = φ(t) − φ(s) = φ(t) ≥ 0, since T was assumed to be φ-positive.  Definition 4.12. We call Tφ from Proposition 4.11 the φ-completion of T . Corollary 4.13. Let φ ∈ P(R) \ {0}. Then, the following are equivalent: 1. T is not φ-positive. 2. Tφ = K. Proof: Suppose T is not φ-positive. Then Tφ is an overring of R which is not φ-positive (since T is not φ-positive) and φ-complete. Thus, by Theorem 4.10, Tφ = K. On the other hand, suppose Tφ = K. We cannot have T φ-positive without contradicting Proposition 4.11.  Proposition 4.14. Let φ ∈ P(R), and let T be any overring of R. Then there exists an overring S of R that is contained in T and is maximal amongst all φ-positive overrings contained in T . In particular, there exists a maximal φ-positive overring of R. Proof: Consider the collection Γ of overrings of R contained in T that are φpositive. Since R ∈ Γ, Γ = ∅. Partially order Γ by set inclusion. It is also easy to see that each chain in Γ has an upper bound in Γ. Therefore, by Zorn’s Lemma, Γ has a maximal element. The last statement follows by letting T = K.  Proposition 4.14 raises some questions: when, for a fixed φ ∈ P(R), does R have a unique maximal φ-positive overring? Also, if φ, ψ ∈ P+ (R) with T1 and T2 maximal φ-positive and ψ-positive overrings of R (respectively), then does T1 = T2 ? Finally, if φ, ψ ∈ P+ (R) with φ and ψ distinct, then is an overring T of R φ-positive if and only if it is ψ-positive? Proposition 4.15. Let R be a domain, and suppose there exists some fixed φ ∈ P(R) \ {0}. Let T be a φ-positive and φ-complete overring of R. Then T is a BFD (and hence atomic). Proof: Suppose T is not atomic. Then, there exists a nonzero nonunit t ∈ T that cannot be factored as a product of irreducibles. Let φ(t) = k > 0.

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In particular, t is not irreducible, so we can write t = α1 β1 where α1 , β1 are nonzero nonunits in T and one of α1 or β1 cannot be factored into irreducibles, say β1 . Well, β1 is not irreducible, so we may write β1 = α2 β2 where α2 , β2 are nonzero nonunits in T and one of α2 or β2 –say β2 –cannot be factored into irreducibles. Continue this process, and note that φ(αi ), φ(βj ) > 0 since each αi , βj is a nonunit of T (and T is φ-positive and φ-complete). At the k th step, we have t = α1 α2 · · · αk βk ⇒ φ(t) =

k 

 φ(αi ) + φ(βk ) ≥ k + 1

i=1

which is a contradiction. Therefore T is atomic. Given a fixed nonunit x ∈ T , any irreducible factorization of x contains at most φ(x) < ∞ irreducibles. Therefore T is a BFD.  Note that an analogous Proposition in [6] says that if R is an HFD with a boundary positive, boundary complete overring T , then every irreducible of R is irreducible in T . We give an example to show that this property does not generalize– even when φ ∈ P+ (R). Example 4.16. Let K be any field, x an indeterminate over K, and let T = K[x], R = K[x2 , x3 ] (i.e. R is the subring of T that consists of all polynomials over K with no linear terms). Let φ(f ) be the degree of f for all f ∈ R∗ . It is clear that φ ∈ P+ (R) and that T is φ-positive and φ-complete over R. However, x2 is irreducible in R, but not in T . Theorem 4.17. Let R be a domain with quotient field K and overring T . 1. P(T ) ⊆ P(R). 2. T is a positive overring of R if and only if P(R) = P(T ). 3. If P+ (R) = ∅, and T is positive and complete, then P+ (R) = P+ (T ). Proof: For 1, let φ ∈ P(T ). Since R ⊆ T , φ is nonnegative on R∗ , whence φ ∈ P(R). For 2, suppose T is a positive overring of R. Let φ ∈ P(R). Since T is positive, φ is nonnegative on all nonzero elements of T . Therefore φ ∈ P(T ), and P(R) ⊆ P(T ). This, along with the other inclusion provided in 1, gives us the equality. Now, suppose that P(R) = P(T ). Let φ ∈ P(R) be arbitrary. If φ is not positive on T , then φ ∈ / P(T ), violating our assumption of equality. Therefore T is φ-positive for all φ ∈ P(R), whence T is a positive overring of R. For 3, let P+ (R) = ∅ and suppose T is φ-positive and φ-complete for all φ ∈ P+ (R). Then, for φ ∈ P+ (R), we get that φ ∈ P(T ) by 2. However, since φ is strictly positive on all nonzero nonunit elements of T , φ ∈ P+ (T ). Let ψ ∈ P+ (T ). Since T is φ-positive for some φ ∈ P+ (R), we see that all nonunits of R remain nonunits of T . So, given any nonzero nonunit x ∈ R, ψ(x) > 0, and R ⊆ T implies that ψ(y) ≥ 0 for any y ∈ R∗ . Therefore ψ ∈ P+ (R). 

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Example 4.18. Let F be any field, let {xi }∞ i=1 be a countable family of indeterminates over F , and let R = F [x1 , x2 , · · · ]. Let Tn = F (x1 , · · · , xn )[xn+1 , · · · ]. Let K = F (x1 , x2 , · · · ) be the quotient field of R. Then, we have the following infinite tower of rings: R  T1  T2  · · ·  K. Now, R is a UFD, and thus has a boundary map. Since Ti is not boundary positive for each i ≥ 2 (∂R ( x1i ) = −1), we see that each Ti is not a positive overring. In fact, by the exact same argument, Tj is not a positive overring of Ti for i < j. So, this gives us the following infinite descending tower: P(R) P(T1 ) P(T2 ) · · · P(K) = {0}. Recall that if R is an atomic domain, we define the elasticity of R (denoted by ρ(R)) to be: ρ(R) = sup{

m | α1 · · · αm = β1 · · · βn ; αi , βj irreducible in R}. n

Elasticity was introduced by Valenza in [7], and an excellent discussion of elasticity can be found in [1]. We now produce a relationship between elasticity and overrings that are complete with respect to a strictly positive integral morphism on R, generalizing a result in [6]. Theorem 4.19. Let R be a domain with φ ∈ P(R) \ {0}, and T = K a φ-complete overring of R. Suppose further that there exists some fixed N ∈ N such that given any irreducible x ∈ T , φ(x) ≤ N . Then ρ(T ) ≤ N . Proof: First note that T is atomic (and is, in fact, a BFD), by Proposition 4.15. So, ρ(T ) makes sense. Note also that T φ-complete and T = K implies that T is also φ-positive (Proposition 4.10). Suppose ρ(T ) > N . Since ρ(T ) is the least upper bound of {

m |α1 α2 · · · αm = β1 β2 · · · βn , αi , βj irreducible in T } n

and since ρ(T ) > N , we must have some m, n such that

m n

> N and

α1 · · · αm = β1 · · · βn for some irreducibles αi , βj in T . So, φ(α1 · · · αm ) = φ(β1 · · · βn ). Note that m n > N implies that m > nN , m > n, and m > N . Now, since T is φ-positive and boundary complete, we must have 1 ≤ φ(αi ) ≤ N for each 1 ≤ i ≤ m. So, this gives us: N < m ≤ φ(α1 · · · αm ).

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Also, since φ(βj ) ≤ N for each j, we must have φ(β1 · · · βn ) ≤ nN . However, φ(β1 · · · βn ) = φ(α1 · · · αm ), so this gives: N < m ≤ φ(α1 · · · αm ) ≤ nN < m. This gives our desired contradiction.  Corollary 4.20. Let R be a domain with φ ∈ P+ (R), and suppose that there exists some fixed N ∈ N such that for each irreducible x in R, φ(x) ≤ N . Then ρ(R) ≤ N . Proof: Note that R is atomic by Theorem 2.8, since R is a BFD. So, letting T = R in Theorem 4.19 gives us the result.  Theorem 4.21. Let R be a domain with quotient field K and φ ∈ P(R)\ {0}. Then the following are equivalent: 1. 2. 3. 4.

φ is a valuation (after defining φ(0) = ∞). There exists a rank 1 DVR overring V of R such that V is φ-positive. For all α ∈ K ∗ with φ(α) ≥ 0, R[α] is φ-positive. For all α ∈ K ∗ with φ(α) > 0, R[α] is φ-positive.

Proof: (1 ⇒ 2):, (3 ⇒ 4): Obvious. (2 ⇒ 1): Let v be the associated valuation with V . Since V is φ-positive and a rank 1 DVR, it is clear that V is then also φ-complete. Let the maximal ideal of V be zV , and let φ(z) = m. It is then clear that φ = mv, whence φ is a valuation. (2 ⇒ 3): Choose α ∈ K ∗ with φ(α) > 0. Then we must have α ∈ V , whence R ⊆ R[α] ⊆ V and R[α] is φ-positive. (4 ⇒ 2): Let S = {x ∈ K ∗ | φ(x) > 0}. It is then clear that R[S] is φ-positive. We show that (R[S]) , the complete integral closure of R[S], is a rank 1 DVR (it is clear that (R[S]) is φ-positive by Proposition 4.7). Choose α ∈ K ∗ . If φ(α) > 0, then α ∈ R[S] ⊆ (R[S]) . If φ(α) < 0, then α−1 ∈ R[S] ⊆ (R[S]) . If φ(α) = 0, then choose any r ∈ R∗ with φ(r) > 0. Then for any k ≥ 0, φ(rαk ) = φ(r) > 0, whence r ∈ (R[S]) . Thus (R[S]) is a valuation domain, and it is atomic since it is φ-complete. Therefore (R[S]) is the rank 1 DVR that we seek. 

5

Connections with the Group of Divisibility

In this section, R will denote a domain with quotient field K and group of divisibility G = G(R). Theorem 5.1. Given any φ ∈ M0 (R), we may define φ on G by φ(αU (R)) = φ(α). What is more, if φ ∈ P(R), and if αU (R) ≤ βU (R) in G, then φ(α) ≤ φ(β) in Z. Proof: To show the first claim, we only show that φ : G(R) −→ Z is welldefined, for it is easy to see that φ is a homomorphism of abelian groups. So, let αU (R) = γU (R) in G. Then there exists some u ∈ U (R) with αu = γ. Thus, φ(αu) = φ(α) + φ(u) = φ(α) = φ(γ)

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and φ is well-defined on G. β ∈ R. For the second claim, let φ ∈ P(R), and suppose αU (R) ≤ βU (R). Then α β So, since φ ∈ P(R), we see that φ( α ) ≥ 0, which gives φ(β) − φ(α) ≥ 0, and φ(β) ≥ φ(α).  So, given φ ∈ M0 (R), and given α ∈ K ∗ , we will identify φ(α) with φ(αU (R)). We recall the following definition: Definition 5.2. We say that R is a rank-n discrete valuation ring (rank-n DVR), n 0 for n ≥ 1 if G(R) ∼ Z under the dictionary order. Specifically, (m1 , · · · , mn ) ≤ = i=1

(k1 , · · · kn ) if and only if either the first nonzero entry in (k1 − m1 , k2 − m2 , · · · , kn − mn ) is positive, or all of the entries of (k1 − m1 , k2 − m2 , · · · , kn − mn ) are zero, in which case (m1 , · · · , mn ) = (k1 , · · · , kn ). We now use the extension of faithful integral morphisms to the group of divisibility to characterize M0 (R) when R is a rank-n DVR. Theorem 5.3. Let R be a rank-n DVR. Denote by xi the element corresponding to the n-vector in G with a 1 in the ith coordinate and zeroes elsewhere. Then, without regard to order, M0 (R) ∼ = G(R). M0 (R) is order isomorphic to G(R) under the following partial order on M0 (R)(*): φ ≤ ψ if and only if either the first nonzero entry of (ψ(x1 ) − φ(x1 ), ψ(x2 ) − φ(x2 ), · · · , ψ(xn ) − φ(xn )) is positive or all entries of (ψ(x1 ) − φ(x1 ), ψ(x2 ) − φ(x2 ), · · · , ψ(xn ) − φ(xn )) are zero (in which case φ = ψ). Proof: Consider the map f : M0 (R) −→ G given by f (φ) = (φ(x1 ), φ(x2 ), · · · , φ(xn )). Clearly, f is well-defined, by virtue of φ being well-defined. It is also clear that f is a homomorphism. Also, if f (φ) = (0, 0, · · · , 0), then φ(xi ) = 0 for all i, and it follows that φ = 0. Thus Ker(f ) is trivial. Now, let (m1 , · · · , mn ) ∈ G be arbitrary. It is clear that we may then define an integral morphism φ with φ(xi ) = mi for all i. Therefore f is onto. Now, order M0 (R) with ordering (*). It is then obvious that this ordering gives an order-preserving isomorphism between M0 (R) and G.  Corollary 5.4. Let R, G, and xi be as in Theorem 5.3. Then the following conditions are equivalent: 1. 2. 3. 4.

M0 (R) is order isomorphic to G under the R-ordering. P+ (R) = ∅. M0 (R) =< P(R) >=< P+ (R) ∪ {0} >. n = 1.

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Proof: Note that if n = 1, then R is a rank-1 DVR. Thus R has a unique (up to a unit in R) nonzero prime–call it z. Thus, to choose a map in M0 (R), we only need to choose the image for z. Thus M0 (R) ∼ = Z, and it is easy to see that this isomorphism is order-preserving under the R-ordering of M0 (R). Also, since each choice of the image of z under a faithful gives either a strictly positive or strictly negative map, it is clear that M0 (R) =< P(R) >=< P+ (R) ∪ {0} >, and that P+ (R) = ∅. Thus 4 ⇒ 1, 2, 3. (1 ⇒ 4): Suppose 1. holds and n > 1. Then, there exist φ, ψ ∈ M0 (R) with φ ←→ (1, 0, 0, · · · , 0) and ψ ←→ (2, −1, 0, 0, · · · , 0). Clearly, under the dictionary ordering of G, (1, 0, · · · , 0) ≤ (2, −1, 0, · · · , 0). However ψ(x2 ) = −1 < φ(x2 ) = 0. This is a contradiction. Therefore n = 1. (2 ⇒ 4): Let φ ∈ P+ (R), and suppose n > 1. Then, in particular, we must have φ(x2 ) > 0. However, xxk1 ∈ R for any k ∈ N, since xxk1 ←→ (1, −k, 0, · · · , 0) > 2

2

(0, 0, · · · , 0). However, for a big enough k, we must have kφ(x2 ) = φ(xk2 ) > φ(x1 ), whence φ( xxk1 ) < 0, a contradiction. Thus n = 1. 2 (3 ⇒ 4): Suppose again that n > 1. By the calculations in the previous paragraph, any positive integral morphism must be zero on x2 . However, there exists a φ ∈ M0 (R) such that φ(x2 ) = −1. Thus we have φ ∈ M0 \ < P(R) >, a contradiction.  Under what conditions on R are we guaranteed that–without paying attention to order–M0(R) ∼ = G(R)? When are we guaranteed a partial ordering on M0 such that M0 and G(R) are order isomorphic?

6

Integral Valuation Domains

In [5], the author studied the following class of HFDs. Definition 6.1. Let R be an HFD with quotient field K. We say that R is a boundary valuation domain, or BVD, if for every α ∈ K ∗ with ∂R (α) = 0, then α or α−1 is an element of R. Furthering our generalizations, we wish to extend the idea of BVDs to integral morphisms on domains. Definition 6.2. Let R be a domain with quotient field K, and let φ ∈ M(R). We say that R is an integral valuation domain with respect to φ if for every α ∈ K ∗ with φ(α) = 0, we have α or α−1 an element of R. Clearly, if R is a valuation domain, then R is an integral valuation domain with respect to any integral morphism. Also, if R is a BVD, then R is an integral valuation domain with respect to ∂R . We will mostly concern ourselves with integral valuation domains with respect to positive integral morphisms. Our aim is to show that for atomic domains, the concepts of integral valuation domain with respect to a positive integral morphism and BVD coincide.

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Theorem 6.3. Let R be a domain with quotient field K, and let φ ∈ P(R). Then the following are equivalent: 1. R is an integral valuation domain with respect to φ. 2. For all α ∈ K ∗ with φ(α) > 0, we have α ∈ R. 3. If x, y ∈ R∗ with 0 ≤ φ(x) < φ(y), then x|y in R. Proof: (1 ⇒ 2): Let R be an integral valuation domain with respect to φ, and / R, then α−1 ∈ R. But φ(α−1 ) < 0, and suppose that α ∈ K ∗ with φ(α) > 0. If α ∈ this contradicts the positivity of φ. (2 ⇒ 3): Let x, y ∈ R with 0 ≤ φ(x) < φ(y). Then φ( xy ) > 0. So, by 2, xy ∈ R. (3 ⇒ 1): Let α ∈ K ∗ be given with φ(α) = 0. Write α = ab with a, b ∈ R. Suppose φ(α) > 0. Then 0 ≤ φ(b) < φ(a) and by 3, ab = α ∈ R. Similarly, if φ(α) < 0, then α−1 ∈ R. Thus R is an integral valuation domain with respect to φ.  Theorem 6.4. Let R be a domain with quotient field K and complete integral closure R , and let φ ∈ P(R) \ {0}. If R is an integral valuation domain with respect to φ, then R is a rank-1 DVR. What is more, R is the unique maximal φ-positive overring of R, and if R is atomic, then U (R ) ∩ R = U (R). Proof: We first show that R is a valuation domain. Let α ∈ K ∗ be given. If φ(α) > 0, then α ∈ R ⊆ R . If φ(α) < 0, then α−1 ∈ R ⊆ R . If φ(α) = 0, then take any x ∈ R∗ with φ(x) > 0. Then, for any n ≥ 0, φ(xαn ) = φ(x) > 0, implying that xαn ∈ R, whence α ∈ R . Thus R is a valuation domain. Since R is φ-positive and φ-complete, we see, by Proposition 4.15, that R is atomic. Thus R is a rank-1 DVR. The fact that R is the unique maximal φ-positive overring of R follows from the fact that R is a rank-1 DVR. Clearly, U (R) ⊆ U (R ) ∩ R. Suppose R is atomic, and let x ∈ (U (R ) ∩ R) \ U (R). Then, since R is atomic, we may find an irreducible π ∈ R with π|x. Then φ(π) = φ(x) = 0. Let y be any irreducible of R such that φ(y) > 0. Then, φ(π −1 y) = φ(y) > 0 ⇒ π −1 y ∈ R. But then y = π(π −1 y), a contradiction, since y was irreducible in R. Therefore U (R) = U (R ) ∩ R, as claimed.  Proposition 6.5. Let R be a domain with complete integral closure R . Suppose U (R ) ∩ R = U (R) and that R is a rank-1 DVR with zR the maximal ideal of R . Then P(R) \ {0} = P+ (R). In particular, if R is an atomic integral valuation domain with respect to some φ ∈ P(R) \ {0}, then P(R) \ {0} = P+ (R). Proof: Every nonzero nonunit of R can be written as uz n where n ≥ 0 and u ∈ U (R ). Since φ is not the zero map, we must have φ(z) > 0. Suppose that r = uz n ∈ R with φ(r) = 0. Then n = 0, whence r = u ∈ U (R ) ∩ R = U (R). Thus R is φ-complete, and φ ∈ P+ (R). The last statement follows from Theorem 6.4.  Theorem 6.6. Let R be an atomic domain with quotient field K. Then the following are equivalent:

Integral Morphisms 1. 2. 3. 4. 5.

R R R R R

is is is is is

353

an integral an integral an integral an integral a BVD.

valuation valuation valuation valuation

domain domain domain domain

with with with with

respect respect respect respect

to to to to

φ for some some φ for

all φ ∈ P+ (R). φ ∈ P+ (R). φ ∈ P(R) \ {0}. all φ ∈ P(R) \ {0}.

What is more, if one (hence all) of conditions 1-5 hold, then P(R) \ {0} = P+ (R) = {m∂R }∞ m=1 . Proof: Clearly 1 ⇒ 2 ⇒ 3. (3 ⇔ 4): Clearly 4 ⇒ 3, so suppose R is an integral valuation domain with respect to some fixed φ ∈ P(R) \ {0}, and let ψ ∈ P(R) \ {0} be arbitrary. By Proposition 6.4, R is a rank-1 DVR and U (R) = U (R ) ∩ R. By Corollary 6.5, φ, ψ ∈ P+ (R). So, let φ(z) = k (where, again, z is the unique nonzero prime of R ). Let α ∈ K ∗ with ψ(α) > 0. Then we may write α = uz n with n > 0. We then see that φ(α) = nk > 0 ⇒ α ∈ R (Theorem 6.3). Thus R is an integral valuation domain with respect to ψ. (3, 4 ⇒ 5): Suppose R is an integral valuation domain with respect to φ for some φ ∈ P(R) \ {0}. By Proposition 6.5, φ ∈ P+ (R). Let α1 α2 · · · αm = β1 β2 · · · βn where αi , βj are irreducible in R (such irreducible factorizations exist, since R is atomic). Suppose n > m. Now, there must exist some i, 1 ≤ i ≤ m such that φ(αi ) > φ(βi ). If not, then for all i, φ(αi ) ≤ φ(βi ), and we have φ(α1 · · · αm ) ≤ φ(β1 · · · βm ) < φ(β1 · · · βn ), a contradiction. So, without loss of generality, φ(α1 ) > φ(β1 ). Thus α1 , β1 are irreducible in R, we may cancel them and obtain

α1 β1

∈ R. Since

uα2 · · · αm = β2 · · · βn where u ∈ U (R). Continuing by induction, we arrive at a contradiction. Thus R must be an HFD. So, to see why R is a BVD, let x, y be nonunits of R, and consider xy . Let us fix two irreducible factorizations of x and y, say x = π1 π2 · · · πt+k , y = δ1 δ2 · · · δt where k > 0 and each πi , δj is irreducible in R. Without loss of generality, we may assume that no πi is an associate to any δj when we form xy . Suppose φ(x) = φ(y). Then, by the exact same argument used above we may (without loss of generality) say that φ(δ1 ) > φ(π1 ). Thus, since R is an integral valuation domain with respect δ1 δ1 to φ, we see that φ( ) > 0, implying that ∈ R. Since δ1 , π1 are irreducibles of π1 π1 R, it is then apparent that π1 and δ1 are associates in R, a contradiction.

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y y Therefore φ(x) = φ(y). If φ(y) > φ(x), then ∈ R. But ∂R ( ) < 0, a contrax x x diction. Therefore φ(x) > φ(y) and ∈ R, whence R is a BVD. y (5 ⇒ 1): Let R be a BVD, and choose any φ ∈ P+ (R). We know that R , the complete integral closure of R, is a rank-1 DVR. So, let the unique prime of R be z. Then any irreducible of R is of the form uz for u ∈ U (R ). Since R is a positive overring of R, we see that φ(U (R )) = {0}. Let φ(z) = m > 0. Then it is easy to see that φ = m∂R , whence R is an integral valuation domain with respect to φ.  The following example shows that we need the condition of atomicity in Theorem 6.6. Example 6.7. Let F be any field, and let R = F [x, xy , xy2 , · · · ]M , where M = (x, xy , xy2 , · · · ). Let φ ∈ P(R) be the map linearly defined by φ(y) = 1, φ(x) = 0. Since R is a valuation domain, it is clear that R is an integral valuation domain with respect to φ. However, R is not a BVD, since R is not atomic.

Acknowledgements: The author wishes to thank both the editor of this volume and the referee.

Bibliography [1] D.F. Anderson, Elasticity of factorizations in integral domains: a survey, Factorization in integral domains (Iowa City, IA, 1996), Lecture Notes in Pure and Appl. Math., Marcel Dekker 189 (1997), 1–29. [2] L. Carlitz, A characterization of algebraic number fields with class number two, Proc. Amer. Math. Soc. 11 (1960), 391–392. [3] J. Coykendall, The half-factorial property in integral extensions, Comm. Algebra 27 (1999), 3153–3159. [4] N. Gonzalez and S. Pellerin, Boundary map and overrings of half-factorial domains, preprint. [5] J. Maney, Boundary valuation domains, to appear in J. Algebra. [6] J. Maney, On the boundary map and overrings, submitted. [7] R.J. Valenza, Elasticity of factorization in number fields, J. Number Theory 36 (1990), no. 2, 212–218. [8] A. Zaks, Half-factorial domains, Bull. Amer. Math. Soc. 82 (1976), 721–723. [9] A. Zaks, Half-factorial domains, Israel J. Math. 37 (1980), 281–302.

355

Chapter 24

A Special Type of Invertible Ideal by

1

Stephen McAdam and Richard G. Swan

Introduction

In this work, I will always be a nonzero ideal (possibly I = R) in a commutative domain R. We are concerned with the following concept. Definition 1.1. I is called an S-ideal if there exists an ideal J with I + J = R and with IJ principal. S-ideals are crucial in proving the main result of [MS], which concerns an analogue to unique factorization domains. The present chapter focuses on S-ideals. The definition shows S-ideals are invertible, and Lemma 1.4 shows they are also 2-generated. Corollary 1.7 shows an arbitrary 2-generated invertible ideal is isomorphic to an S-ideal, but need not be one itself. Furthermore, we show that if I is an S-ideal then there are elements a and c such that for any m ≥ n ≥ 1, I n = (am , cn ) = (an , cm ) (Proposition 1.11). We also show (Proposition 1.14) that if a1 a2 · · · an = c1 c2 · · · cm = 0, with the ai pairwise comaximal and the cj arbitrary, and if for 1 ≤ i ≤ n and 1 ≤ j ≤ m we let Iij = (ai , cj ), then the Iij are n m S-ideals, and for a fixed i, j=1 Iij = (ai ), while for a fixed j, i=1 Iij = (cj ). We further show that if each cj is a product of prime elements, then each Iij is actually principal (Corollary 1.17). We begin with two results which follow easily from the definition of S-ideals. Proposition 1.2. Nonzero principal ideals are S-ideals. Proof. If I = (b), let J = (1 − b). Lemma 1.3. [MS, Lemma 1.2] Let a and c be in R. The following are equivalent. i) (a, c) = (a2 , c). 356

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ii) There is an element b with (a, b) = R and with c | ab. Proof. Suppose (i) holds, and write a = a2 r + cd. Let b = 1 − ar, and note ab = cd. Conversely, if (ii) holds, then a ∈ aR = a(a, b) = (a2 , ab) ⊆ (a2 , c). Thus (a, c) = (a2 , c). Lemma 1.4. If J is an ideal with I + J = R and with IJ = (c), then I = (a, c) = (a2 , c) for any a ∈ I with (a) + J = R. Proof. Write ar + j = 1. For x ∈ I, we have x = xar + xj ∈ (a) + IJ ⊆ (a, c). Thus I = (a, c). Now since aj ∈ IJ = (c) we have c divides aj. We also have (a, j) = R. By Lemma 1.3, I = (a, c) = (a2 , c). We now give two characterizations of S-ideals. (The statement of Proposition 1.5(b) was used as the definition of S-ideals in [MS].) Proposition 1.5. The following are equivalent. a) I is an S-ideal. b) There are elements a and c in I such that I = (a, c) = (a2 , c). c) I is finitely generated and I/I 2 is principal. Proof. a) ⇒ b): This is immediate from Lemma 1.4. b) ⇒ a): Suppose I = (a, c) = (a2 , c). By Lemma 1.3, there is a b ∈ R with (a, b) = R and with c | ab. Let J = (b, c). Clearly I + J = R, and (c) ⊆ I ∩ J = IJ ⊆ (c), so that IJ = (c). b) ⇒ c): Suppose I = (a, c) = (a2 , c). Then clearly I is finitely generated and I/I 2 is generated by the image of c. c) ⇒ b): Suppose I is finitely generated and I/I 2 is principal, generated by the image of c ∈ I. Let K = I/(c) and note that K is finitely generated and K 2 = K. By [K, Theorem 76], there is an element α ∈ K with (1 + α)K = 0. Thus −α is an idempotent which generates K. If −α is the image of a ∈ I, then I = (a, c) = (a2 , c). We thank the referee for calling our attention to [GH] which has some results related to the present ones. For instance, let a ∈ I with I invertible, and suppose there are only finitely many maximal ideals containing a. Then [GH, Theorem 3] shows there is a c ∈ I with I = (a, c). We now strengthen that. Proposition 1.6. If I is an invertible ideal contained in only finitely many maximal ideals, then I is an S-ideal. Furthermore, if a ∈ I is such that only finitely many maximal ideals contain a, then there exists c ∈ I with I = (a, c) = (a2 , c). Proof. A slight variation on the proof of [K, Theorem 60] (the well-known fact that an invertible ideal in a domain with only finitely many maximal ideals is principal) shows that if I is invertible and M1 , M2 , . . . , Mn is a finite set of maximal ideals, then there exists c ∈ I such that none of Mi , 1 ≤ i ≤ n, contain cI −1 . We apply

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that to either the finitely many maximal ideals containing I, or, if there exists an a ∈ I as hypothesized, to the possibly larger finite set of maximal ideals containing a. If we let J equal the resulting cI −1 , we get I + J = R and IJ = (c), which shows I is an S-ideal. If such an a exists, we also have (a) + J = R, so that Lemma 1.4 shows I = (a, c) = (a2 , c). We use Proposition 1.5 to show that a 2-generated invertible ideal might not be an S-ideal, but must be isomorphic to an S-ideal. Corollary 1.7. i) Every 2-generated invertible ideal is isomorphic to an S-ideal. ii) If J is a nonprincipal 2-generated invertible ideal in a domain D, and if X is an indeterminate, then X(JD[X]) is a 2-generated invertible D[X]-ideal which is not an S-ideal. Proof. i) Let H = (x, y) be invertible. Then there exist χ, ψ ∈ H −1 with χx + ψy = 1. We may assume χ = 0. As χx = (χx)2 + ψx(χy), we see that χH = (χx, χy) = ((χx)2 , χy). By Proposition 1.5(b) ⇒ (a), χH is an S-ideal isomorphic to H. ii) Let I = X(JD[X]). Clearly I is a 2-generated invertible ideal. There is a Dhomomorphism I/I 2 → J which, for Xj(X) ∈ X(JD[X]) = I, sends Xj(X) + I 2 to j(0) ∈ J. Clearly this map is onto. Suppose I/I 2 is principal, generated by Xf (X) + I 2 , and suppose Xj(X) + I 2 = (d(X) + I 2 )(Xf (X) + I 2 ). Then j(0) = d(0)f (0). Therefore, our map shows f (0) generates J as a D-ideal. As that violates the hypothesis, we see I/I 2 is not principal. By Proposition 1.5(a) ⇒ (c), I is not an S-ideal. S-ideals can be used to prove the following somewhat curious fact about nonprincipal 2-generated invertible ideals. Proposition 1.8. Let I be a nonprincipal 2-generated invertible ideal. Then there is an infinite list x1 , x2 , x3 , . . . of distinct elements of I, such that if n ≥ 1 and if y and z are monomials of degree n consisting of products of powers of some of these xi such that no xi appears in both y and z, then I n = (y, z). Proof. We first treat the case that I is an S-ideal and n = 1. Using Proposition 1.5(a) ⇒ (b), let I = (a, c) = (a2 , c), so that I = (am , c) for all m ≥ 1. Let x1 = c, x2 = a, and for i ≥ 2, let xi+1 = x2 x3 x4 · · · xi + c. Since for j > 1, xj is congruent modulo c to a positive power of a, we see that for j > 1, I = (c, xj ) = (x1 , xj ). Now if 1 < j < i, then since xi ≡ c mod xj , we have (xi , xj ) = (c, xj ) = I, and so I = (xi , xj ) whenever i = j. In particular, since I is not principal, we see our list consists of distinct elements. This does the case n = 1 when I is an S-ideal. However, by Corollary 1.7(i), we immediately see that the case n = 1 holds even after dropping the restriction that I be an S-ideal. Now let n ≥ 1. We already have x1 , x2 , x3 , . . . such that any two of these generate I. For ease of notation, we will assume this list has been ordered such that er+1 er+2 xr+2 · · · xess , with e1 + e2 + · · · + er = n = er+1 + y = xe11 xe22 · · · xerr and z = xr+1 er+2 +· · ·+es . Pick any j and k with 1 ≤ j ≤ r and r +1 ≤ k ≤ s. We already know I = (xj , xk ). Therefore xj I −1 +xk I −1 = R, so that xj I −1 and xk I −1 are comaximal

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for any such choice of j and k. It follows that (x1 I −1 )e1 (x2 I −1 )e2 · · · (xr I −1 )er and (xr+1 I −1 )er+1 (xr+2 I −1 )er+2 · · · (xs I −1 )es are comaximal. Thus yI −n + zI −n = er+1 er+2 xr+2 · · · xess )I −n = R, from which we see I n = (y, z). (xe11 xe22 · · · xerr )I −n + (xr+1 The next lemma discusses a certain circumstance in which a product of S-ideals is again an S-ideal. (However, Example 1.18 shows that fails to hold in general.) In particular, the next lemma shows a power of an S-ideal is an S-ideal (although that also follows easily from the definition). Lemma 1.9. Let a, f1 , f2 , . . . , fn be in R. Suppose (a, fj ) = (a2 , fj ) for all j. Then   (a, fj ) = (am , fj ) for all m ≥ n. (Example 1.19 shows m ≥ 1 is not enough.) If each (a, fi ) is nonzero (and so an S-ideal by Proposition 1.5), then their product is an S-ideal. Proof. Since (a, fi ) = (am , fi ) for all m ≥ 1, we see that for m ≥ n we have     (am , fi ) ⊆ (a, fi ) = (am , fi ) ⊆ (am , fi ). The final sentence follows by letting m first be n and then be 2n. Lemma 1.10. [MS, Lemma 1.4] Suppose I = (e, c) = (e2 , c). Then there are elements a, b, and d such that I = (a, c) = (a2 , c), (a, b) = R = (c, d), and ab = cd. Proof. Let a = e + cz with z to be determined, and note that I = (a, c) = (a2 , c). Write e = e2 r + cs, and let b = 1 − ar. Clearly (a, b) = R, and ab = a − a2 r = cd where d = s + z − 2erz − rcz 2 . We will now select z so as to make (c, d) = R. Note that d ≡ s + z(1 − 2er) mod c. As er = (er)2 + csr ≡ (er)2 mod c, we see (1 − 2er)2 ≡ 1 mod c. Specifying z = (1 − s)(1 − 2er), we see d ≡ 1 mod c, and it follows that (c, d) = R. Proposition 1.11. Let I be an S-ideal. Then there are elements a and c such that for any m ≥ n ≥ 1, I n = (am , cn ) = (an , cm ). Proof. By Proposition 1.5 and Lemma 1.10, there are elements a, b, c, and d with I = (a, c), with (a, b) = R = (c, d), and with ab = cd. By Lemma 1.3, I = (a, c) = (a2 , c) and I = (c, a) = (c2 , a). Applying Lemma 1.9 to both representations of I shows I n = (am , cn ) and I n = (cm , an ) for all m ≥ n ≥ 1. Lemma 1.12. Let a, f1 , f2 , . . . , fn be in R. The following are equivalent.   a) (a, fj ) = (a2 , fj ).   b) (a, fj ) = (a, fj ) and (a, fj ) = (a2 , fj ) for all j.   Therefore, if I = (a, fi ) = (a2 , fi ) is an S-ideal, then I can be factored into  a product (a, fi ) in which each factor is an S-ideal. Proof. The final sentence follows from a) ⇒ b) and Proposition 1.5.  a) ⇒ b): By (a) and Lemma 1.3, there is a b with (a, b) = R such that fj divides ab. As each fj divides ab, Lemma 1.3 shows (a, fj ) = (a2 , fj ) for all j.   Now let g = f2 f3 · · · fn . We have (a, f1 )(a, g) ⊆ (a, f1 g) = (a, fj ) = (a2 , fj ) ⊆

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 (a, f1 )(a, g). Thus (a, fj ) = (a, f1 )(a, g). Since g divides ab, Lemma 1.3 also gives (a, g) = (a2 , g), and so induction applied to (a, g) = (a, f2 f3 · · · fn ) yields the result.   b) ⇒ a): Assuming (b), Lemma 1.9 shows (a, fj ) = (am , fj ) for all m ≥ n.     However, by (b) we also have (a, fj ) = (a, fj ). Thus (a, fj ) = (am , fj ) for all m ≥ n, and so it also holds for m = 2. Lemma 1.13. Suppose (x, y) = R. Then for any c ∈ R, (xy, c) = (x, c)(y, c). Proof. Clearly (x, c)(y, c) ⊆ (xy, c). Since (x, c) and (y, c) are comaximal, c ∈ (x, c) ∩ (y, c) = (x, c)(y, c), and the lemma follows. We now present an interesting result involving S-ideals and comaximal factorizations. Proposition 1.14. Let a1 a2 · · · an = c1 c2 · · · cm = 0, where the ai are pairwise comaximal and the cj are arbitrary. For 1 ≤ i ≤ n and 1 ≤ j ≤ m, let Iij = (ai , cj ). m Then the Iij are S-ideals. Furthermore, for a fixed i, j=1 Iij = (ai ), and for a n fixed j, i=1 Iij = (cj ). Proof. As the ai are pairwise comaximal, Lemma 1.3 shows Iij = (ai , cj ) = (a2i , cj ), so that Proposition 1.5 shows Iij is an S-ideal. Now fix j. Lemma 1.13 shows    n ai , cj ) = (cj ). Next, fix i. Lemma 1.3 shows (ai , cj ) = (a2i , cj ). i=1 Iij = (  m Thus Lemma 1.12 shows (ai ) = (ai , cj ) = j=1 Iij . If to the hypothesis of Proposition 1.14 we add the condition that each cj is a product of prime elements, then the ideals Iij turn out to be principal. However, to get a little more mileage, we recall the following definition from [MS]. Definition 1.15. Let p be a nonzero nonunit element of R. We say p is a pseudoprime element if for every pair of elements a and b in R, if p divides ab and if (a, b) = R, then either p divides a or p divides b. (Note that prime elements are pseudo-prime.) Lemma 1.16. i) Suppose (a, p) = (a2 , p) and p is pseudo-prime. Then either (a, p) = (p) or (a, p) = R. ii) Suppose I = (a, c) = (a2 , c) and c is a product of pseudo-prime elements. Then I is principal. Proof. i) By Lemma 1.3, there is a b with (a, b) = R and with p | ab. As p is pseudo-prime, either p | a, in which case (a, p) = (p), or p | b, in which case R = (a, b) ⊆ (a, p). ii) This follows from Lemma 1.12(a) ⇒ (b) and part (i). Lemma 1.16(ii) (and Proposition 1.5) show that if every nonzero nonunit in R is a product of pseudo-primes, then every S-ideal is principal. For more in a similar vein, see [MS, Section 1].

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Corollary 1.17. Let a1 a2 · · · an = c1 c2 · · · cm = 0, where the ai are pairwise comaximal and each cj is either a unit or a product of pseudo-prime elements. Then there are elements fij , 1 ≤ i ≤ n, 1 ≤ j ≤ m such that for 1 ≤ i ≤ n, ai = fi1 fi2 · · · fim and for 1 ≤ j ≤ m, cj = f1j f2j · · · fnj . Proof. Define Iij as in Proposition 1.14. We claim each Iij is principal. If cj is a unit, this is clear. Otherwise cj is a product of pseudo-primes. Since Iij = (ai , cj ) = (a2i , cj ), Lemma 1.16 proves the claim.  Now let Iij = (hij ). Proposition 1.14 shows that for fixed i, hij = (ai ) and  for fixed j, hij = (cj ). By multiplying each hi1 by an appropriate unit, we may  in fact assume that for fixed i, hij = ai . For each fixed j, there is a unit uj such     that uj hij = cj . As ai = cj , we see uj = 1. Now let f1j = uj h1j , and for 1 < i ≤ n, let fij = hij . Example 1.18. We show the product of two S-ideals might not be an S-ideal, even if one factor is principal. Let K be a 2-generated invertible ideal which is not an S-ideal (see Corollary 1.7(ii)). By Corollary 1.7(i), there is an S-ideal I isomorphic to K, and so there exist nonzero elements b and c with cI = bK. We claim cI is not an S-ideal. Otherwise, bK is an S-ideal, and so there is a J with bK + J = R and (bK)J principal. Now K + J = R and KJ is principal, contradicting that K is not an S-ideal. Example 1.19. We give an example showing that in Lemma 1.9, the condition “for all m ≥ n” cannot be replaced with the stronger condition “for all m ≥ 1”. Let I be a nonprincipal proper S-ideal (which exist by Corollary 1.7(i)). By Proposition 1.5, we may write I = (x, y) = (x2 , y), with x = 0. Let a = x2 and f = xy. We have (a, f ) = x(x, y) = x(x3 , y) = (a2 , f ). Therefore, Lemma 1.9 shows (a, f )2 = (a2 , f 2 ). We will show this does not equal (a, f 2 ) by showing a is not in (a, f )2 . Suppose it were. Then using a | f 2 we would have (a) ⊆ (a, f )2 ⊆ (a), so that (x)2 = (a) = (a, f )2 = (xI)2 . Cancelling x2 contradicts that I is proper.

Bibliography [GH] R. Gilmer and W. Heinzer, On the number of generators of an invertible ideal, J. Algebra 14 (1970), 139–151. [K] I. Kaplansky, Commutative Rings, Chicago: University of Chicago Press, 1974. [MS] S. McAdam and R. G. Swan, Unique comaximal factorization, J. Algebra 276 (2004), 180–192.

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Chapter 25

Factorization into Radical Ideals by

Bruce Olberding Abstract We characterize the domains R for which every proper ideal of R is a product of radical ideals. We show that given a Boolean space X there exists a domain R such that Max(R) is homeomorphic to X and every proper ideal of R is a product of radical ideals. Domains for which every proper ideal is a product of radical ideals are almost Dedekind; so by varying the Boolean space X we obtain by this construction examples of almost Dedekind domains whose finitely generated maximal ideals correspond to the isolated points of X.

1

Introduction

In a paper of 1978 N. Vaughan and R. Yeagy prove that if a domain R has the property that every proper ideal is a product of radical ideals, then R is an almost Dedekind domain; that is, RM is a Dedekind domain for each maximal ideal M of R [19, Theorem 2.4]. Following Vaughan and Yeagy, we say that a ring R for which every proper ideal is a product of radical ideals (i.e. “semi-prime” ideals) is an SP -ring. Thus SP-domains are almost Dedekind. In [19] it is shown that if R is a certain non-Noetherian almost Dedekind domain constructed by W. Heinzer and J. Ohm in [13], then R is an SP-domain (see Example 4.1 of the present chapter). Highlighting another class of such examples, Yeagy shows in [20] that if R is an almost Dedekind domain that is a union of a tower of Dedekind domains, then R is an SP-domain if and only if R has no critical maximal ideals. Here, a maximal ideal M is critical if every finite subset of M is contained in a square of a maximal ideal of R. It is noted in this same article that one of the first examples of a non-Noetherian almost Dedekind domain – Nakano’s example of the rings of integers in the number field obtained by adjoining all pth roots of 363

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unity, p a prime [18] – has no critical maximal ideals. Hence Nakano’s example is an SP-domain. H. Butts and Yeagy give in [2] an example of a non-Noetherian almost Dedekind domain that is a tower of Dedekind domains but which has a critical maximal. Thus the class of SP-domains is properly contained in the class of almost Dedekind domains. In Section 2 we generalize the results of Vaughan and Yeagy and prove that a domain R is an SP-domain if and only if R is a Pr¨ ufer domain of Krull dimension 1 having no critical maximal ideals, and we give in Theorem 2.1 several other characterizations of this class of rings. Since an almost Dedekind domain R has Krull dimension 1 and is integrally closed, R is Dedekind if and only if every maximal ideal of R is finitely generated. We turn in Sections 3 and 4 to the question of the existence of SP-domains with respect to the sparsity or density of the collection of finitely generated maximal ideals in the maximal spectrum. As Nakano’s example suggests, one encounters non-Noetherian almost Dedekind domains when considering certain infinite integral extensions of almost Dedekind domains. For more on constructing almost Dedekind domains in this manner, see [11]. In [10], R. Gilmer mentions two other techniques for constructing almost Dedekind domains: the Nagata function ring and the Jaffard-Kaplansky-Ohm Theorem. The latter theorem will be used in the present chapter, so we discuss it later. The former method relies on the fact that for a domain R, the Nagata function ring R[X]S , where S is the collection of polynomials whose coefficients generate R, is an almost Dedekind B´ezout domain that is not a Dedekind domain if and only if R is an almost Dedekind non-Dedekind domain. A fourth construction, motivated by the theory of integer-valued polynomials, was introduced recently by K. A. Loper. It is necessary that R be an almost Dedekind domain with all residue fields finite in order for the ring Int(R) of integer-valued polynomials of R to be a Pr¨ ufer domain [1, Proposition VI.1.5]. The search for almost Dedekind domains having finite residue fields motivates a number of interesting examples. See [17] for a survey of the methods used to construct such examples. The key elements of the construction given in the present note are the JaffardKaplansky-Ohm and Stone Representation Theorems, as well as Example 2.2 in [13] and Examples III.4.1 and III.4.6 in [7]. The basic idea behind the construction is present in these examples, but the construction we give here does not seem to have been carried out in the literature. Our work here consists mainly in combining the above theorems and examples in the right sequence. Unlike the approaches for constructing almost Dedekind domains discussed in [17], in which it is desired to have finite residue fields, since our method relies on the Jaffard-Kaplansky-Ohm Theorem it does not afford much control over the residue fields. The finitely generated maximal ideals of an almost Dedekind domain R with nonzero Jacobson radical correspond to the isolated points in the maximal spectrum Max(R) of R (see Lemma 3.1). We show in Section 3 that given any Boolean topological space X, there exists an SP-domain R with nonzero Jacobson radical such that Max(R) is homeomorphic to X. By specifying Boolean spaces with interesting collections of isolated points, we obtain then examples of SP-domains whose

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finitely generated maximal ideals are distributed in various ways. Since their introduction by R. Gilmer in his 1964 paper [9], the class of almost Dedekind domains has arisen frequently in multiplicative ideal theory. Over 50 instances now appear in Mathematical Reviews. An incomplete sampling of what is known for this class of rings can be found in [1, 8, 9, 11, 12, 14, 16]. Acknowledgement. This chapter grew out of a recent series of articles with Laszlo Fuchs and Bill Heinzer that also consider various ideal decompositions [3, 4, 5, 6, 14]. I thank both of them for very helpful comments and suggestions on this chapter.

2

SP-Domains

In this section we characterize SP-domains in several different ways. First we review the following notation and terminology. (a) If F is the quotient field of R and X and Y are R-submodules of F , then [Y : X] = {q ∈ F : qX ⊆ Y }. Thus an ideal A of R is invertible if and only if [R : A]A = R. (b) A Pr¨ ufer domain R is an integral domain for which every finitely generated ideal is invertible; equivalently, RM is a valuation domain for all maximal ideals M of R. Hence an almost Dedekind domain is Pr¨ ufer. (c) In the theorem below we require two facts about Pr¨ ufer domains: (1) If A and B are finitely generated ideals, then A ∩ B is finitely generated, and (2) if P is
∞ a prime ideal, then i=1 P i is a prime ideal [11]. (d) Recall from the introduction that a maximal ideal M of a domain R is critical if every finite subset A of M is contained in the square of some maximal ideal of R. Equivalently, M is critical if and only if every finitely generated ideal of R is contained in the square of some maximal ideal of R. (e) If R is an almost Dedekind domain and 0 = a ∈ R, then we define a mapping γa : Max(R) → Z by γa (M ) = vM (a), where vM is the rank one discrete valuation corresponding to the valuation ring RM . This mapping is upper semi-continuous if for all n ∈ Z, the set γa−1 ([n, ∞)) is closed. As noted in the introduction, an SP-domain is almost Dedekind. Thus one may draw characterizations of SP-domains from the following theorem. Theorem 2.1. The following statements are equivalent for an almost Dedekind domain R. (i) R is an SP-domain. (ii) R has no critical maximal ideals. (iii) If A is a proper finitely generated ideal of R, then ideal of R.

√ A is a finitely generated

(iv ) Every proper principal ideal of R is a product of radical ideals.

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(v ) For every nonzero a ∈ R, the function γa : Max(R) → Z is upper semicontinuous and has finite image. (vi) For every proper ideal A of R, there exist radical ideals J1 ⊆ J2 ⊆ · · · ⊆ Jn such that A = J1 J2 · · · Jn . (vii) Every proper nonzero ideal A of R can be represented uniquely as a product A = J1 J2 · · · Jn of radical ideals Ji such that J1 ⊆ J2 ⊆ · · · ⊆ Jn . Proof. (i) ⇒ (ii) Let M be a maximal ideal of R, and let A be a nonzero principal ideal contained in M . By (i) A = J1 · · · Jn for some radical ideals J1 , . . . , Jn of R. Thus some Ji ⊆ M . Since A is principal, Ji is invertible, hence finitely generated. If Ji ⊆ N for some maximal ideal N , then since RN is one-dimensional, Ji RN = N RN , so Ji is not contained in N 2 . It follows that the finitely generated ideal Ji is contained in M but not in a square of a maximal ideal of R. Hence M is not critical. (ii) ⇒ (iii) First we claim that for each maximal ideal M and finitely generated ideal A ⊆ M , there exists a finitely generated radical ideal J of R such that A ⊆ J ⊆ M . Let A be a proper finitely generated ideal and M be a maximal ideal with A ⊆ M . By (ii) there is a finitely generated ideal B contained in M but not in the square of any maximal ideal. Let J = A + B. If N is a maximal ideal of R containing J, then JRN ⊆ N RN . By assumption BRN ⊆ N 2 RN , since otherwise B ⊆ N 2 RN ∩ R = N 2 , a contradiction. Thus BRN = N RN and this forces JRN = N RN for all maximal ideals N containing J. Hence J is the intersection of all maximal ideals that contain it. Moreover, J is finitely generated and √A ⊆ J ⊆ M. Now let A be a proper finitely generated ideal of R and set J = A. We claim that J is finitely generated. If A = 0, the claim is clear, so suppose A = 0. To prove the claim it suffices to show that [R : J]RM = [RM : JRM ] for all maximal ideals M of R. For once this is established, it follows that since JRM is principal for all maximal ideals M of R, [R : J]J = R, proving that J is invertible, hence finitely generated. Let M be a maximal ideal of R and q be an element of the quotient field of R such that qJ ⊆ RM . We exhibit an element b ∈ R \ M such that bqJ ⊆ R. As we have seen, there exists a finitely generated radical ideal J1 of R such that A ⊆ J1 ⊆ M . Since A and J1 are invertible, it is the case that A = J1 B1 for some finitely generated ideal B1 of R. If B1 ⊆ M , then we may repeat this argument to obtain a finitely generated radical ideal J2 ⊆ M and finitely generated ideal B2 such that B1 = J2 B2 . Continuing in this manner, we have either that A = J1 · · · Jn Bn for some Bn ⊆ M or there are radical ideals J1 , J2 , . . . ⊆ M such that for all k > 0, A ⊆ J1 · · · Jk .
∞ In this latter case, A ⊆ k=1 M k . However, M = M 2 and R has Krull dimension 1, so as noted in statement (c) before the theorem this forces A = 0. Thus there Jn Bn exist finitely generated radical ideals J1 , J2 , . . . , Jn such √ that A = J1 J2 · · · √ for some finitely generated ideal Bn ⊆ M . Now J = A = J1 ∩ · · · ∩ Jn ∩ Bn , √ so since qJRM ⊆ RM and Bn ⊆ M , we have q(J1 ∩ · · · ∩ Jn ) ⊆ RM . Since J1 ∩ · · · ∩ Jn is an intersection of finitely generated ideals this intersection is finitely

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generated (see statement (c) above). It follows that there exists b ∈ R \ M such that bq(J1 ∩ · · · ∩ Jn ) ⊆ R. Hence bqJ ⊆ R, as claimed. Thus q ∈ [R : J]RM , and it follows that [RM : J] = [R : J]RM for all maximal ideals M containing A. (iii) ⇒ (iv) Let A be a proper principal ideal √ of R. If A = 0, there is nothing to prove, so suppose that A = 0, and set J1 = A. Since A and J1 are invertible, A = J1 B1 for some invertible ideal B1 of R. If B1 = R, then the claim is proved. √ Otherwise, set J2 = B1 . Then A = J1 J2 B2 for some√finitely generated ideal B2 of R. Moreover, since J1 is finitely generated and J1 = A, there exists n > 0 such that J1n ⊆ A ⊆ J2 . Since J2 is a radical ideal, this implies J1 ⊆ J2 . Repeating this argument we obtain either that A is a product of radical ideals or there is a chain of radical ideals J1 ⊆ J2 ⊆ · · · ⊆ Jn ⊆ · · · such that for each k > 0, ∞ A ⊆ J1 J2 · · · Jk . In the latter case let M be a maximal ideal containing k=1 Jk .
∞ Then A ⊆ i=1 M i . Since M = M 2 and R is a one-dimensional domain, this implies A = 0 (see statement (c) above), contrary to assumption. Hence it must be that A is a product of radical ideals of R. (iv) ⇒ (v) Let a be a nonzero element of R and n be a positive integer. We claim first that γa has a finite image. By (iv), there are distinct radical ideals J1 , . . . , Jk and positive integers e1 , . . . , ek such that aR = J1e1 · · · Jkek . Let M be a maximal ideal of R containing a. Define X = {i ∈ {1, 2, . . . , k} : Ji ⊆ M }. For each i ∈ X, since Ji is a radical ideal and RM is a DVR, it follows that  Ji RM = M RM . Now Ji RM = RM for all i ∈ X, so J1e1 · · · Jkek RM = i∈X M ei RM .  Hence γa (M ) = vM (a) = i∈X ei . It follows that for any maximal ideal M , γa (M ) = 0 or γa (M ) is a sum of some of the ei ’s. Therefore, γa has a finite image. Let n > 0. We show next that V := γa−1 ([n, ∞)) = {M ∈ Max(R) : a ∈ M n } is closed in Max(R). Let M ∈ V . As above, define X = {i ∈ {1, 2, . . . , k} : Ji ⊆  the set F := {X ⊆ M }. Then, as noted above, i∈X ei = γa (M ) ≥
n. Thus   {1, 2, . . . , k} : i∈X ei ≥ n} is nonempty. Set A = X∈F ( i∈X Ji ). We show that V is closed by verifying that V = {M ∈ Max(R) : M ⊇ A}. If M ∈ V , then as  we have established above, there is subset X of {1, 2, . . . , k} such that i∈X ei ≥ n  and M ⊇ i∈X Ji ⊇ A. Hence V ⊆ {M ∈ Max(R) : M ⊇ A}. Conversely, let M ∈ Max(R) with M ⊇ A. Then since A is a finite intersection, X ∈ F . Hence Ji ⊆ M for all i ∈ X, and it follows that i∈X Ji ⊆ M for some   aR = J1e1 · · · Jkek ⊆ i∈X Jiei ⊆ i∈X M ei ⊆ M n . Hence a ∈ M n , so M ∈ V . 

(v) ⇒ (vi) Let A be a proper nonzero ideal of R. For a maximal ideal M of R denote by vM (A) the smallest element in {vM (a) : a ∈ A}. Let X = {vM (A) : M ∈ Max(R) and A ⊆ M }. We claim first that X is finite. Let a be a nonzero element of A. Then by (v) γa has finite image, so {vM (a) : M ∈ Max(R)} is finite. Since vM (A) ≤ vM (a) for all M ∈ Max(R), it follows that X is finite. Now write X = {f1 , f2 , . . . , fn } for some positive integers f1 < f2 < · · · < fn . For each i = 1, 2, . . . , n, let Vi = {M ∈ Max(R) : A ⊆ M fi }. We claim that each Vi

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is a closed subset of Max(R). For each i, we have: Vi

= {M ∈ Max(R) : a ∈ M fi for all a ∈ A} = {M ∈ Max(R) : M ∈ γa−1 ([fi , ∞)) for all a ∈ A}  = γa−1 ([fi , ∞)) a∈A

By (v), each γa−1 ([fi , ∞)) is a closed subset of Max(R), so since Vi is an intersection of closed subsets, Vi is closed.
For each i, define Ji = M∈Vi M , and note that since Vn ⊆ Vn−1 ⊆ · · · ⊆ V1 , we have A ⊆ J1 ⊆ J2 ⊆ · · · ⊆ Jn . Set B = J1f1 J2f2 −f1 · · · Jnfn −fn−1 . We claim that A = B. It suffices to show that ARM = BRM for all maximal ideals M of R. If M is a maximal ideal of R that contains B, let k be the largest integer f −f ≤ n such that J1 ⊆ · · · ⊆ Jk ⊆ M . Then BRM = J1f1 · · · Jk k k−1 RM = M fk RM . Moreover, since each Vi is closed, Vk is the smallest member of the chain Vn ⊆ · · · ⊆ V1 such that M ∈ Vk . Since vM (A) ∈ {f1 , f2 , . . . , fn }, this forces vM (A) = fk . Hence ARM = M fk RM = BRM . On the other hand, suppose that M is a maximal ideal of R that contains A. Then vM (A) = fk for some k ≤ n, so ARM = M fk RM . Thus M ∈ Vk but not in any Vm where k < m ≤ n. Since each Vi is closed this implies that J1 ⊆ · · · ⊆ Jk ⊆ M f −f but Jm ⊆ M for any m with k < m ≤ n. Thus BRM = J1f1 · · · Jk k k−1 RM = fk M RM = ARM . This proves that A = B. (vi) ⇒ (vii) Suppose that A = J1 · · · Jn = K1 · · · Km with 0 = J1 ⊆ · · · ⊆ Jn and K1 ⊆ · · · ⊆ Km . Observe that a maximal ideal M of R contains J1 if and only if M √ contains A; if and only if M contains K1 . Thus since R is one-dimensional, J1 = A = K1 . In [8] it is shown that a domain is almost Dedekind if and only if its ideals have the cancellation property. Thus J2 · · · Jn = K2 · · · Km and an induction argument completes the proof. (vii) ⇒ (i) This is clear. Corollary 2.2. A domain R is an SP-domain if and only if R is Pr¨ ufer domain of Krull dimension 1 having no critical ideals. Proof. If R is an SP-domain, then R is an almost Dedekind domain by the result of Vaughan and Yeagy. Thus R has Krull dimension 1, and by Theorem 2.1 R has no critical maximal ideals. Conversely, assume that R is Pr¨ ufer, of dimension 1 and R has no critical ideals. Then M = M 2 for all maximal ideals M of R. Since RM is a valuation domain, this implies M RM is a principal ideal of RM , and since R has dimension 1, it is the case that R is almost Dedekind. Hence by Theorem 2.1 R is an SP-domain. Since we have considered here only domains, the following question remains open.

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Question 2.3. Which rings have the property that every proper ideal is a product of radical ideals?

3

Constructing SP-Domains from Boolean Spaces

Before stating the main theorem of this section we review the connection between some topological and ideal-theoretic notions. A topological space X is totally disconnected if for all x, y ∈ X, there exists a clopen set containing x but not y. A totally disconnected space is necessarily Hausdorff. A Boolean space is a totally disconnected compact space. Since the maximal spectrum of a ring R is quasicompact, it follows that Max(R) is a Boolean space if and only if Max(R) is totally disconnected. Thus Max(R) is a Boolean space if and only if for every pair of distinct maximal ideals M and N of Max(R), R modulo its Jacobson radical is a product of two reduced rings, one containing the image of M , the other containing the image of N . Lemma 3.1. If M is a maximal ideal of an almost Dedekind domain R, then {M } is an isolated point of Max(R) if and only if the Jacobson radical of R is nonzero and M is a finitely generated ideal. Proof. Let J denote the Jacobson radical of R, and let M be a maximal ideal of R. Set X = Max(R) \ {M }, and suppose that M is an isolated point of Max(R).

Then R = ( N ∈X N ) + M , so there exists b ∈ N ∈X N and c ∈ M such that 1 = b + c. Note that 0 = bc ∈ J, so J is nonzero. If m ∈ M , then m = mc + mb implies m + J = mc + J. Hence M = Rc + J. Also, since R is almost Dedekind there exists d ∈ M such that M RM = dRM . Set A = (c, d)R. We claim that A = M . Indeed, ARM = dRM = M RM , and if N is any maximal ideal distinct from M , then ARN = RN , since c ∈ N implies 1 − b = c ∈ N , a contradiction to the assumption that b ∈ N . Hence A = M since this equality holds locally. Conversely, suppose that the Jacobson radical J is nonzero and M is a finitely generated ideal of R. Then M RM = JRM , so M ⊆ JRM . Now M is finitely generated, so there exists b ∈ R \ M such that bM ⊆ J. Thus b is contained in every maximal ideal of R distinct from M but not in M itself, so the set X = Max(R) − {M } is a closed subset of Max(R). Hence {M } is open. Our main goal in this section is to establish: Theorem 3.2. If X is a Boolean space, then there exists an SP-domain R such that Max(R) is homeomorphic to X. This domain R is necessarily almost Dedekind and the isolated points of X correspond to the maximal ideals of R that are finitely generated. If R is a domain and Max(R) is a Boolean space, then it follows from the discussion at the beginning of this section that R has nonzero Jacobson radical. Thus if R is an almost Dedekind domain and Max(R) is a Boolean space, then by Lemma 3.1 the isolated points of Max(R) correspond to the maximal ideals of R that are finitely generated.

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Thus to prove Theorem 3.2 it remains to show that for any Boolean space X there exists an SP-domain R such that Max(R) is homeomorphic to X. The proof of this occupies the rest of the section. We emphasize first several aspects of the construction. Filters of a lattice. If L is a lattice with respect to meet ∧ and join ∨, then a filter F on L is a nonempty collection of elements of L such that for all a, b ∈ F , a ∧ b ∈ F and for all c ∈ L with a ≤ c, c ∈ F . A filter F is principal if there exists a ∈ F such that for all b ∈ F , a ≤ b. The lattice of clopen sets. If X is a topological space, then the collection L(X) of clopen subsets of X is closed under complements, finite intersections and finite unions. Hence L(X) is a Boolean algebra with respect to meet ∩ and join ∪. The space of ultrafilters of L(X). An ultrafilter on the Boolean algebra L(X) is a proper filter U on L(X) such that if a∨b ∈ U for a, b ∈ L(X), then a ∈ U or b ∈ U. The space of ultrafilters of L(X) is the collection of its ultrafilters topologized by a basis of open sets of the form {U : U is an ultrafilter not containing F }, where F is some filter of L(X). Stone Representation Theorem. Let X be a Boolean space, and let L(X) be the Boolean algebra consisting of the clopen sets of X. Then X is homeomorphic to the space of ultrafilters of L(X) [15, Theorem 7.10, page 100]. The space of prime filters in a lattice-ordered group. If G is a lattice ordered group, denote by G+ the nonnegative elements of G. A filter P of the lattice G+ is a prime filter if for all f, g ∈ G+ , f + g ∈ P implies f ∈ P or g ∈ P . The space of prime filters of G+ is the set of all prime filters of G+ topologized by a basis of open sets of the form {P : P is a prime filter not containing F }, where F is a filter of G+ . The group of divisibility of a B´ezout domain. A B´ezout domain is an integral domain for which every finitely generated ideal is principal. Let R be a B´ezout domain with quotient field Q, and let R∗ and Q∗ denote the units in R and Q, respectively. Then Q∗ /R∗ is an abelian group; indeed it is a lattice ordered group with respect to the order ≤ defined by xR∗ ≤ yR∗ if and only if yx−1 ∈ R [7, Proposition III.4.5]. The correspondence A → {aR∗ : a ∈ A} and F → {a ∈ R : aR∗ ∈ F } is an order-preserving bijection between ideals A of R and filters F in the positive cone of Q∗ /R∗ . This bijection is such that prime ideals correspond to prime filters and principal ideals correspond to principal filters [7, Proposition III.4.6]. The Jaffard-Kaplansky-Ohm Theorem. If G is a lattice-ordered abelian group, then there exists a B´ezout domain R such that if Q is the quotient field of R, then Q∗ /R∗ and G are isomorphic as lattice-ordered groups [7, Theorem III.5.3]. Hence, by our remarks above regarding the group of divisibility, Spec(R) is homeomorphic to the space of prime filters of the lattice G+ , where G+ denotes the set of positive elements of G. Turning now to the proof of Theorem 3.2, fix once and for all a Boolean space X. By the Stone Representation Theorem, X is homeomorphic to the space of

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ultrafilters of the Boolean algebra L(X) of clopen sets of X. We construct now a lattice-ordered abelian group G such that the space of proper prime filters of G+ is homeomorphic to the space of ultrafilters of L(X). Once such a G is constructed, we will use the Jaffard-Kaplansky-Ohm Theorem to obtain a B´ezout domain R such that Max(R) is homeomorphic to X. Let C be the set of all functions from X to Z. Then C is an abelian group with respect to pointwise addition; that is, for all f, g ∈ C, (f + g)(x) = f (x) + g(x) for all x ∈ X. For any A ⊆ X, let χA be the characteristic function defined by χA (x) = 1 for all x ∈ A and χA (x) = 0 for all x ∈ X \ A. Following Example III.4.1 in [7], define a lattice-ordered subgroup G of C by ZχA . G= A∈L(X)

The group G is lattice-ordered with respect to the ordering ≤ defined by f ≤ g if and only if f (x) ≤ g(x) for all x ∈ X. Join and meet are defined by (f ∨ g)(x) = max{f (x), g(x)} for all x ∈ X and (f ∧ g)(x) = min{f (x), g(x)} for all x ∈ X. An argument such as that in Lemma 3.3 (i) below shows that G is the set of all bounded continuous functions in C.1 Lemma 3.3. Let f be a member of a proper prime filter P of G+ . (i) There exist a1 , a2 , . . . , an ∈ N and nonempty disjoint sets A1 , A2 , . . . , An ∈ n n L(X) such that f = i=1 ai χAi and f −1 (N) = i=1 Ai . (ii) There exists B ∈ L(X) such that B ⊆ f −1 (N), χB ≤ f and χB ∈ P . (iii) If g > 0 and f −1 (N) ⊆ g −1 (N), then g ∈ P . m Proof. (i) Let A = f −1 (N), and write f = i=1 bi χBi , where b1 , . . . , bm are nonzero integers and B1 , . . . , Bm are nonempty members of L(X). Let Y be the collection of all nonempty subsets of A that can be obtained as an intersection of m distinct sets C1 , C2 , . . . , Cm , such that for each i = 1, 2, . . . , m, either Ci = Bi or Ci = A \ Bi . Since L(X) is a Boolean algebra, each member of Y is in L(X). The members of Y form a partition for A, and for each i = 1, 2, . . . , m, there is a subset of members in Y that partitions Bi . Write Y = {A1 , A2 , . . . , An }. Then there are integers n a1 , a2 , . . . , an such that f = i=1 ai χAi . After renumbering if necessary we may assume that no ai is zero. Moreover since the Ai ’s are disjoint and f > 0, it must  be that each ai > 0. Thus f −1 (N) = ni=1 Ai . n (ii) Write f = i=1 ai χAi as in (i). Since P is a prime filter, there exists i such that χAi ∈ P and Ai ⊆ f −1 (N). Since each ai > 0 it follows that χAi ≤ f . (iii) By (ii) there exists A ∈ L(X) such that χA ∈ P , χA ≤ f and A ⊆ f −1 (N). Thus A ⊆ g −1 (N) and since 0 < g it follows that χA ≤ g. Since χA ∈ P and P is a filter, this forces g ∈ P . 1 This construction also generalizes that of Example 2.2 in [13]. The group G is there defined both as the set of all functions on X = N that are eventually constant and as the set of all functions f such that f (X) is bounded, but it is clear from the context that this second definition is a misstatement.

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Lemma 3.4. If U is an ultrafilter of L(X), then PU := {f ∈ G+ : f −1 (N) ∈ U} is a proper prime filter of G+ . Similarly, if P is a proper prime filter of G+ , then UP := {f −1 (N) : f ∈ P } is an ultrafilter of L(X). Proof. We show first that PU is a proper prime filter. Since the characteristic function of X is in PU , PU is nonempty. Suppose that f, g ∈ PU . Then f −1 (N), g −1 (N) ∈ U and since U is a filter of L(X), f −1 (N) ∩ g −1 (N) ∈ U. Now x ∈ f −1 (N)∩g −1 (N) if and only if f (x) > 0 and g(x) > 0; if and only if (f ∧g)(x) > 0. Hence (f ∧ g)−1 (N) = f −1 (N) ∩ g −1 (N) ∈ U, so f ∧ g ∈ PU . Next suppose that f ∈ PU , h ∈ G+ and f ≤ h. Then f −1 (N) ⊆ h−1 (N), so since f −1 (N) ∈ U and U is a filter, h−1 (N) ∈ U. Thus h ∈ PU , and this shows that PU is a filter. Since the empty set is not contained in U it follows that 0 ∈ PU ; hence PU is a proper filter of G+ . Finally, suppose that f, g ∈ G+ and f + g ∈ PU . If x ∈ X, then (f + g)(x) > 0 if and only if f (x) > 0 or g(x) > 0. Thus f −1 (N) ∪ g −1 (N) = (f + g)−1 (N) ∈ U, and since U is an ultrafilter, f −1 (N) ∈ U or g −1 (N) ∈ U. Thus f or g is in PU , and PU is a prime filter. We show next that UP is an ultrafilter. Since X ∈ UP , UP is nonempty. Let A, B ∈ UP . Then A = f −1 (N) and B = g −1 (N) for some f, g ∈ P . As above, A ∩ B = (f ∧ g)−1 (N), so since P is a filter, f ∧ g ∈ P and A ∩ B ∈ UP . Suppose now that C ∈ L(X) with A ⊆ C. As above, (f + χC )−1 (N) = A ∪ C = C. Moreover f ≤ f + χC , so since P is a filter, f + χC ∈ P . Hence C ∈ UP . This proves UP is a filter. Since 0 ∈ P , the empty set is not contained in UP . Thus UP is a proper filter of L(X). It remains to show that UP is an ultrafilter. Let A, B ∈ L(X) and suppose that A ∪ B ∈ UP ; that is, A ∪ B = f −1 (N) for some f ∈ P . Let n be the maximum value in f (X). Then f ≤ nχA + nχB , so since P is a prime filter, χA ∈ P or χB ∈ P . Thus A ∈ UP or B ∈ UP . Lemma 3.5. The mapping U → PU is a homeomorphism from the space of ultrafilters U of L(X) to the space of proper prime filters of G+ . Proof. Let α denote the mapping U → PU . We claim first that α is a bijection. Suppose that PU = PV for ultrafilters U and V of L(X). If U = V, then without loss of generality we may assume that there exists A ∈ U \ V. Thus χA ∈ PU \ PV , since χ−1 A (N) = A ∈ U \ V. Hence α is injective. To see that α is surjective, suppose that P is a proper prime filter of G+ . We claim that α(UP ) = P ; that is, PUP = P . Let f ∈ PUP . Then f −1 (N) ∈ UP , so that f −1 (N) = g −1 (N) for some g ∈ P . By Lemma 3.3 (iii) this implies f ∈ P . Thus PUP ⊆ P . Conversely, if g ∈ P , then (by definition) g −1 (N) ∈ UP , so g ∈ PUP . This proves α(UP ) = P , and we conclude that α is bijective. We show finally that α is continuous. Let A be a closed set of the space of proper prime filters of G+ . Then α−1 (A) = {UP : P ∈ A}. We claim that α−1 (A) = {U :

P ∈A UP ⊆ U}. Suppose that U is an ultrafilter of L(X) and P ∈A UP ⊆ U. Then there exists a proper prime filter Q on G+ such that U = UQ . We claim that Q ∈ A.

Since A is closed, it suffices to show that P ∈A P ⊆ Q. Let f ∈ P ∈A P . Then

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f −1 (N) ∈ P ∈A UP ⊆ UQ . Thus there exists g ∈ Q such that g −1 (N) = f −1 (N). By
Lemma 3.3 (iii) f ∈ Q. This proves P ∈A P ⊆ Q, so Q ∈ A and U = UQ ∈ α−1 (A). It follows that α−1 (A) is closed and α is a continuous mapping. Applying the Jaffard-Kaplansky-Ohm Theorem, we have now a B´ezout domain R with quotient field Q and an isomorphism λ : Q∗ /R∗ → G of lattice ordered groups such that Spec(R) is homeomorphic to the space of prime filters of G+ . By Lemma 3.5 the space of proper prime filters of G+ is homeomorphic to the space of ultrafilters of L(X), and by the Stone Representation Theorem, X is homeomorphic to the space of ultrafilters of L(X). Since L(X) is a Boolean algebra, the ultrafilters of L(X) are precisely the filters of L(X) that are maximal among proper filters of L(X). Thus the nonzero prime ideals of R are maximal ideals and X is homeomorphic to Max(R). This proves that R is a one-dimensional B`ezout domain such that Max(R) is homeomorphic to X. It remains to show that R is an SP-domain. Since R is one-dimensional and Pr¨ ufer it suffices by Corollary 2.2 to show that R has no critical maximal ideals. To this end, define ν : R − {0} → Q∗ /R∗ by ν(x) = xR∗ for all x ∈ R − {0}. Let λ : Q∗ /R∗ → G be the homomorphism of lattice-ordered groups given by the Jaffard-Kaplansky-Ohm Theorem, and define µ = λ ◦ ν. Then the maximal ideals of R correspond via µ to the proper prime filters of G+ ; similarly, nonzero principal ideals of R correspond to proper principal filters of G+ . Let M be a maximal ideal of R. We claim that M is not critical. Define P = µ(M ). Then P is a proper prime filter of G+ . By Lemma 3.3 (ii) there exists a clopen set A of X such that χA ∈ P . Let F = {g ∈ G+ : χA ≤ g}. We claim that F is the intersection of all prime filters containing χA . Let g be an element of this intersection and set B = g −1 (N). If χA ≤ g, then A ⊆ B. Thus A ∩ (X \ B) is nonempty. Since B is clopen, so is X \ B. Thus A ∩ (X \ B) ∈ L(X) and there exists an ultrafilter U of L(X) containing A ∩ (X \ B). Hence U contains A but not B. Consequently, χA ∈ PU but g ∈ PU , a contradiction. Thus F is the intersection of all the prime filters containing χA . Now F is the principal filter of G+ generated by χA , so I := µ−1 (F ) is a principal ideal of R. Also, if L is a prime filter of G+ containing χA , then µ−1 (L) is a prime ideal of R containing I. Since F is the intersection of all the prime filters of G+ containing F , then I is the intersection of all prime ideals of R containing I. Hence I is a principal radical ideal contained in M . Since R is a one-dimensional domain, IRN = N RN for all maximal ideals N of R containing I. Thus I is contained in M but not in the square of any maximal ideal of R. This completes the proof of Theorem 3.2.

4

Some Examples

In this section we give some examples to illustrate how Theorem 3.2 can be applied. The first example was constructed in [13] using a direct method in a nontopological setting. That the ring is an SP-domain was first established in [19]. It follows in

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our context from Theorem 3.2. Example 4.1. There exists an SP-domain R such that Max(R) is infinite and R has exactly one maximal ideal that is not finitely generated. The one point compactification X of N is a Boolean space whose isolated points correspond to the members of N. Hence by Theorem 3.2 there exists an SP-domain R with Max(R) homeomorphic to X. This ring R has exactly one maximal ideal that is not finitely generated; it corresponds to the single nonisolated point of X. Example 4.2. There exists an SP-domain R having countably many principal maximal ideals and 2c maximal ideals that are not finitely generated, where c is the cardinality of the continuum. Let βN denote the Stone-Cech compactification of the discrete space N. Then βN is a Boolean space whose isolated points are the images of the elements of N. By Theorem 3.2 there exists an SP-domain R such that Max(R) is homeomorphic to βN . Example 4.3. There exists an SP-domain having no finitely generated maximal ideals. Let X be the classical Cantor space. Then X is a Boolean space and every point of X is an accumulation point of X (indeed, this is true for any Cantor space); hence X has no isolated points. Applying Theorem 3.2 to the space X, one obtains the desired example. In the above example R has uncountably many maximal ideals. This is necessary when Max(R) is Hausdorff, since if X is a compact (Hausdorff) space having no isolated points, then X must be uncountable. (I thank Pat Morandi for pointing this out to me.) Since the space Max(R) is quasi-compact for any commutative ring R, if R is an almost Dedekind domain such that Max(R) is Hausdorff and R has only countably many maximal ideals, then the Jacobson radical of R is nonzero and by Lemma 3.1 some maximal ideal of R must be finitely generated. Example 4.4. There exists an SP-domain R with infinitely many maximal ideals and nonzero Jacobson radical such that R has the property that for every proper ideal A of R, every nonunit of R/A is contained in a principal maximal ideal of R/A. By Theorem 3.2 it is required to construct a Boolean “scattered” space X; that is, X must be a Boolean space such that every subspace of X contains an isolated point. (A scattered space is necessarily totally disconnected, so any quasi-compact scattered space is a Boolean space.) There are a number of examples of scattered spaces in the literature (see for example Chapter 6, Section 17 of [15]). We give one simple construction here that yields interesting examples. The justification for the example is contained in Example 17.3, page 272, of [15]. Let (X, ≤) be a well ordered set. Then X is a scattered Boolean space with respect to the order topology, and the isolated points of X are precisely the smallest element of X and the immediate successors of elements in X. By suitable choices of X one may construct various examples of scattered Boolean spaces having infinitely many nonisolated points. Taking another point of view, a Boolean algebra A is superatomic if every homomorphic image B of A is atomic, that is, for each 0 = b ∈ B, there exists c ∈ B such

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that 0 < c ≤ b and no other member d of B satisfies 0 < d < c. A Boolean algebra A is superatomic if and only if the space of ultrafilters of A is a scattered (Boolean) space (see Chapter 6, Section 17 of [15]). Thus superatomic Boolean algebras give rise to almost Dedekind domains with “many” finitely and nonfinitely generated maximal ideals. In [14] it is shown that an integral domain R with nonzero Jacobson radical has the property that every proper nonzero ideal of R is an irredundant intersection of powers of maximal ideals if and only if R is an almost Dedekind domain such that Max(R) is a scattered space. Example 4.5. An R-module M is superdecomposable if every nonzero summand of M is the direct sum of two nonzero submodules. The group of divisiblity arising as in Section 3 from a Boolean space is used in Examples 6.1 and 6.2 of Chapter VII, Section 6, of [7] to construct B´ezout domains R such that Q/R is superdecomposable. In these cases the underlying lattice of clopen sets is an atomless Boolean algebra. In view of Theorem 3.2 we can add to these examples that the domain R constructed therein is an SP-domain. Also, since the lattice is atomless, the space of ultrafilters contains no isolated points; hence these SP-domains have no finitely generated maximal ideals.

Bibliography [1] P. J. Cahen and J. L. Chabert, Integer-valued polynomials, Math. Surveys 48, 1997. [2] H. S. Butts and R. W. Yeagy, Finite bases for integral closures, J. Reine Angew. Math. 282 (1976), 114–125. [3] L. Fuchs, W. Heinzer and B. Olberding, Commutative ideal theory without finiteness conditions: primal ideals, Trans. Amer. Math. Soc., to appear. [4] L. Fuchs, W. Heinzer and B. Olberding, Maximal prime divisors in arithmetical rings, Venezia 2002 Proceedings, Marcel Dekker, to appear. [5] L. Fuchs, W. Heinzer and B. Olberding, Commutative ideal theory without finiteness conditions: completely irreducible ideals, submitted. [6] L. Fuchs, W. Heinzer and B. Olberding, Commutative ideal theory without finiteness conditions: irreducibility in the quotient field, in preparation. [7] L. Fuchs and L. Salce, Modules over non-Noetherian domains, Math. Surveys 84 (2001). [8] R. Gilmer, Jr., The cancellation law for ideals in a commutative ring, Canad. J. Math. 17 (1965), 281–287. [9] R. Gilmer, Jr., Integral domains which are almost Dedekind, Proc. Amer. Math. Soc. 15 (1964), 813–818. [10] R. Gilmer, Jr., Pr¨ ufer domains and rings of integer-valued polynomials, J. Algebra 129 (1990), 502–517. [11] R. Gilmer, Jr., Muliplicative ideal theory, Queen’s University Press, 1992. [12] A. Grams, Atomic rings and the ascending chain condition for principal ideals, Proc. Camb. Phil. Soc. 75 (1974), 321–329. [13] W. Heinzer and J. Ohm, Locally Noetherian commutative rings, Trans. Amer. Math. Soc. 158 (1971), 273–284. [14] W. Heinzer and B. Olberding, Unique irredundant intersection of completely irreducible ideals, preprint. 376

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[15] S. Koppelberg, Handbook of Boolean algebras, vol. 1, North Holland, 1989. [16] M. Larsen and P. McCarthy, Multiplicative theory of ideals, Academic Press, 1971. [17] K. A. Loper, Constructing examples of integral domains by intersecting valuation domains, Non-Noetherian Commutatitive Ring Theory, Kluwer (2000), 325–340. [18] N. Nakano, Idealtheorie in einem speziellen unendlichen algebraischen Zahlk¨ orper, J. Sci. Hiroshima Univ. Ser. A. 16 (1953), 425–439. [19] N. H. Vaughan and R. W. Yeagy, Factoring ideals in semiprime ideals, Canad. J. Math. 30 (1978), 1313–1318. [20] R. W. Yeagy, Semiprime factorizations in unions of Dedekind domains, J. Reine Angew. Math. 310 (1979), 182–186.

Chapter 26

Strongly Primary Ideals by

Gyu Whan Chang, Hoyoung Nam, and Jeanam Park Abstract Let R be an integral domain with quotient field K. A prime ideal P of R is called strongly prime if x, y ∈ K and xy ∈ P imply that x ∈ P or y ∈ P . As an analog of strongly prime, an ideal Q of R is called strongly primary if whenever x, y ∈ K, xy ∈ Q, and x ∈ / Q, then y n ∈ Q for some positive integer n. In this chapter, we study some properties of integral domains containing a strongly primary ideal.

1

Introduction

Let R be an integral domain with quotient field K. A prime ideal P of R is called strongly prime if x, y ∈ K and xy ∈ P imply that x ∈ P or y ∈ P [HH1]. An ideal I of R is called strongly radical if whenever x ∈ K satisfies xn ∈ I for some n ≥ 1, then x ∈ I [AA]. Following [SS], an integral domain R is called rooty if each radical ideal of R is strongly radical (equivalently, each prime ideal of R is strongly radical [AP, Theorem 1.8]). Thus valuation domains are rooty domains [AP, Remark 1.9], and we use this fact in the section 2. R is called a pseudo-valuation domain (PVD) if each prime ideal of R is strongly prime [HH1]. It is known that R is a PVD if and only if (K − R) ∪ U (R) is multiplicatively closed where U (R) is the group of units of R [AA, Theorem 1.2]. On the other hand, R is a valuation domain if and only if K − R is multiplicatively closed. Note that a strongly prime ideal is strongly radical, and so PVDs are rooty. In fact, PVDs are characterized as quasilocal domains (R, M ) with the property that (M : M ) is a valuation domain with maximal ideal M [HH1]. On the other hand, (R, M ) is a quasi-local rooty domain if and only if (M : M ) is root-closed with radical ideal M . Also, if R is a quasi-local domain that is not rooty, then R is root-closed [AP, Theorem 2.1]. For an ideal Q of R and x, y ∈ K, consider the following four conditions. √ (C1 ) If xy ∈ Q and x ∈ Q, then y ∈ Q. 378

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(C2 ) If xy ∈ Q and x ∈ / Q, then y n ∈ Q for some n ≥ 1. √ √ √ √ (C3 ) Q is strongly prime (i.e., if xy ∈ Q, then x ∈ Q or y ∈ Q). √ / Q, then y n ∈ Q for some n ≥ 1. (C4 ) If xy ∈ Q and x ∈ It is clear that the implications (C1 ) ⇒ (C2 ) ⇒ (C4 ) and (C3 ) ⇒ (C4 ) hold. Following [BH], an ideal Q of R is called strongly primary if Q satisfies (C2 ), i.e., if whenever xy ∈ Q with x, y ∈ K, we have x ∈ Q or y n ∈ Q for some n ≥ 1. In particular, an integral domain R is strongly primary if whenever xy ∈ R with x, y ∈ K, we have x ∈ R or y n ∈ R for some n ≥ 1. An integral domain R is called an almost pseudo-valuation domain (APVD) if every prime ideal of R is strongly primary (equivalently, R is a quasi-local domain whose maximal ideal is strongly primary [BH, Theorem 3.4]). Thus any PVD is an APVD. Counting the introduction, this chapter is divided into three sections. In section 2, we study some properties of a strongly primary domain motivated from [BH]. We show that if R is strongly primary, then R is a PVD. In [Do4, Theorem 2.11], Dobbs showed that if P is a nonmaximal prime ideal of R, then the CPI-extension C(R, P ) is a PVD if and only if R/P is a PVD and RP is a valuation domain. As the APVD-analog, in section 3, we show that if P is a nonmaximal prime ideal of R, then C(R, P ) is an APVD if and only if R/P is an APVD and RP is a valuation domain.

2

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We begin this section with an example, which shows that the implications (C2 ) ⇒ (C1 ) and (C4 ) ⇒ (C3 ) do not hold. Example 2.1. Let F be a field and let R = F [[X 2 , X 3 ]] be the formal power series ring over the field F . Then the quotient field of R is F [[X]][ X1 ], and the integral closure of R is F [[X]]. Note that R is a 1-dimensional local Noetherian domain with maximal ideal M = (X 2 , X 3 ). Now, M is strongly primary. To show this, let / M. f, g ∈ F [[X]][ X1 ] such that f g ∈ M and f ∈ Case 1. f ∈ R. Then f is a unit of R, and hence g = f gf −1 ∈ M . Case 2. f ∈ F [[X]] − R. If f ∈ / XF [[X]], then g ∈ X 2 F [[X]]; so g ∈ M . Suppose f ∈ XF [[X]]. Then f = a1 X + a2 X 2 + · · · for some a1 = 0, a2 , ... ∈ F . This yields g(0) = 0. Hence g 2 ∈ M . Case 3. f ∈ / F [[X]]. Then f = X1n h for some positive integer n and h ∈ F [[X]] with h(0) = 0. Thus g ∈ M . 2 / M ; so M is not strongly prime. Therefore, since Note √ that X ∈ M , but X ∈ M = M , M satisfies neither (C1 ) nor (C3 ). Proposition 2.2. Let Q be an ideal of R. √ (1 ) If Q satisfies (C1 ), then Q is strongly radical. (2 ) (cf. [BH, Theorem 2.8]) If Q satisfies (C4 ), then radical ideal of R.

√

Q is comparable to each

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√ Proof. (1): For x ∈ K, if xn ∈ Q, then there is m ≥ 1 such that xmn ∈ Q. √ √ If x ∈ / Q, then xnm−1 ∈ Q by (C1 ); so by induction, we have x ∈ Q ⊆ Q, a √ √ contradiction. Thus x ∈ Q, and hence Q is strongly radical. √ √ (2): Let I be a radical ideal of R with I ⊆ Q. Choose a ∈ I − Q. Then for √ / Q, there exists n ≥ √ 1 such that ( aq )n ∈ Q. So any q ∈ Q, aq a = q ∈ Q. Since a ∈ √ q n ∈ an Q ⊆ I, and hence q ∈ I. Hence Q ⊆ I, and since I = I, Q ⊆ I. Next, we show that if R is rooty, then (C1 ) ⇔ (C2 ) and (C3 ) ⇔ (C4 ). Proposition 2.3. Let R be a rooty domain. (1 ) The conditions (C1 ) and (C2 ) are equivalent. (2 ) The conditions (C3 ) and (C4 ) are equivalent. Proof. It suffices to show that (C2 ) ⇒ (C1 ) and (C4 ) ⇒ (C3 ). (1): Let Q be an ideal of R satisfying (C2 ). For x, y ∈ K, suppose that xy ∈ Q √ / Q, then xn ∈ Q for some n ≥ 1 by (C2 ). Since R is rooty, and x ∈ / Q. If y ∈ √ x ∈ Q, a contradiction. Hence y ∈ Q. √ (2): Suppose that an ideal Q satisfies (C4 ). For x, y ∈ K, assume that xy ∈ Q √ √ and x ∈ / Q. Then xn y n ∈ Q for some n ≥ 1. Since R is rooty, xn ∈ Q implies √ √ / Q, and hence y nk = (y n )k ∈ Q for some k ≥ 1 by (C4 ). that x ∈ Q. So xn ∈ √ √ Since R is a rooty domain, y ∈ Q. Hence Q is strongly prime. Remark. Let V be a 2-dimensional valuation domain with prime ideals (0)  P0  P  V . Since dim(V ) = 2, there is a nonzero nonprimary ideal Q [Gi, √ Exercise 2, p.293]. Since P is maximal, Q = P0 . We show that Q satisfies (C4 ). √ To do this, assume that xy ∈ Q and x ∈ / Q for x, y ∈ K. If x−1 ∈ V , then y = x−1 xy ∈ Q. Otherwise, x ∈ P ; so x ∈ P − P0 . Since P0 is strongly prime, we √ have that xy ∈ Q  P0 yields y ∈ P0 = Q. Thus y n ∈ Q for some n ≥ 1. Hence Q satisfies (C4 ), but it does not satisfy (C2 ) even though V is a valuation (hence rooty) domain. In [BH], Badawi and Houston showed that if R is strongly primary, then R is an APVD. We next show, more strongly, that if R is strongly primary, then R is a PVD. First, we need lemmas. Lemma 2.4. Let R be an integral domain which is not a rooty domain. Then there / R for all n ≥ 1. exists an x ∈ K − R such that ( x1 )n ∈ Proof. Suppose that R is not rooty. Then there is a prime ideal P of R such that P is not strongly radical [AP, Theorem 1.8]. Let x ∈ K − R such that xk ∈ P for some positive integer k, but x ∈ / P . Suppose that ( x1 )n ∈ R for some n ≥ 1. Then 1 kn kn ( x ) ∈ R and x ∈ P , and hence 1 = ( x1 )kn xkn ∈ P , a contradiction. Hence / R for all n ≥ 1. ( x1 )n ∈ Lemma 2.5. If R is strongly primary, then R is quasi-local rooty.

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Proof. First, suppose that R is not rooty. Then there is an x ∈ K − R such that / R for all n ≥ 1 by Lemma 2.4, which is contrary to the fact that x x1 = 1 ∈ R ( x1 )n ∈ and R is strongly primary. Thus R is rooty. Also, R must be quasi-local. Otherwise, R is root-closed by [AP, Theorem 2.1]. Now, for any x ∈ K − R, 1 = xx−1 ∈ R; so (x−1 )n ∈ R for some n ≥ 1. Since R is root-closed, we have x−1 ∈ R. Hence R is a valuation domain; so quasi-local, a contradiction. Theorem 2.6. If R is strongly primary, then R is a PVD. Proof. By Lemma 2.5, R is rooty and has a unique maximal ideal M . Thus M is strongly radical [AP, Theorem 1.8]. Claim: M is strongly prime. /M Proof of claim. Let x, y ∈ K. Suppose that xy ∈ M and x ∈ / M . Then xn ∈ for all n ≥ 1 since R is rooty. Case 1. Let x ∈ R − M . Then since R is strongly primary, x−1 ∈ R; so y = x−1 (xy) ∈ M . Case 2. Let x ∈ / R. Since xy ∈ M ⊆ R and R is strongly primary, y n ∈ R for some n ≥ 1. Since M is strongly radical, we see that y n ∈ M implies y ∈ M . / M . Then y n ∈ R − M is a unit of R. Note (xy)n = xn y n ∈ M ; Suppose that y n ∈ so xn ∈ M . Therefore x ∈ M , a contradiction. It follows that R is a PVD [HH1, Theorem 1.4]. Proposition 2.7. Let R be an integral domain. Then (1 ) R is rooty and a maximal ideal of R is strongly primary if and only if R is a PVD. (2 ) R is root-closed and strongly primary if and only if R is a valuation domain. In particular, if R is integrally closed, then R is strongly primary if and only if R is a valuation domain. Proof. (1): Suppose that R is rooty and a maximal ideal M of R is strongly primary. Then M is strongly prime by Proposition 2.3(2). Also R is quasi-local by Proposition 2.2(2). Hence R is a PVD. The converse is clear. (2): Assume that R is root-closed and strongly primary. Let x ∈ K − R. Since xx−1 = 1 ∈ R and R is strongly primary, (x−1 )n ∈ R for some n ≥ 1. Since R is root-closed, x−1 ∈ R. Hence R is a valuation domain. The converse is clear. We record some facts about strongly primary ideals for future reference. Lemma 2.8. Let R be an integral domain. (1 ) Every primary ideal of a valuation domain is strongly primary. (2 ) If a maximal ideal M of R is strongly primary, then R is an APVD. If, in addition, P  M is a prime ideal, then P is strongly prime. Proof. (1): [BH, Proposition 2.1]. (2): It follows from [BH, Theorem 2.8 and Theorem 3.4].

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A prime ideal P of an integral domain R is called divided if it is comparable to every ideal of R [Do1]. Proposition 2.9. Let R be an integral domain. (1 ) Let S be a multiplicatively closed subset of R. If Q is a strongly primary ideal of R such that Q ∩ S = ∅, then QRS is strongly primary. √ (2 ) Let R be a PVD and let Q be a primary ideal of R such that Q is not maximal. Then Q is strongly primary. / QRS . Then there is Proof. (1): Assume that for x, y ∈ K, xy ∈ QRS and x ∈ an s ∈ S such that xsy ∈ Q. Since Q is strongly primary and x ∈ / Q, we have n n (sy) ∈ Q for some n ≥ 1, and so y ∈ QRS . Hence QRS is strongly primary. √ (2): Let P = Q. Then RP is a valuation domain by [HH1, Proposition 2.6]. Note that Q ⊆ QRP = QRP ∩ P RP = QRP ∩ P = Q. Suppose that xy ∈ Q and x ∈ / Q. Then y ∈ P , for otherwise, x = a contradiction. Hence Q is strongly primary.

xy y

∈ QRP = Q,

The converse of Proposition 2.9(1) does not hold. For example, let R be a Pr¨ ufer domain which is not a valuation domain. Then for any maximal ideal M of R, M RM is strongly primary, but M = M RM ∩ R is not strongly primary. Recall that a prime ideal P of R is branched if there exists a P -primary ideal distinct from P ; if P is the only P -primary ideal of R, then P is said to be unbranched [Gi, p.189]. Let R be a PVD and let P be a prime ideal of R. Suppose that P is branched. n Then there exists a P -primary ideal J(= P ). Thus P0 := ∩∞ n=1 J is prime in R and there are no prime ideals strictly between P0 and P (see the proof of [HH2, Proposition 2.2]). Hence P is not the union of prime ideals properly contained in P . Suppose that P is unbranched. Then P is also unbranched in a valuation overring V in which every prime ideal of R is also a prime ideal of V [HH2, Proposition 1.2(5)]. By [Gi, Theorem 17.3(e)], P is the union of prime ideals of V (hence R) properly contained in P . Therefore we have the result: A prime ideal P of a PVD is unbranched if and only if P is the union of prime ideals properly contained in P . Theorem 2.10. The following statements are equivalent for an integral domain R. (1 ) Every primary ideal of R is strongly primary. √ (2 ) Every primary ideal Q of R with Q maximal is strongly primary. (3 ) R is a valuation domain or R is a PVD whose maximal ideal is unbranched. Proof. (1) ⇒ (2): Clear. (1) ⇔ (3): It follows from [BH, Proposition 3.3]. √ (2) ⇒ (3): Suppose that every primary ideal Q of R with Q maximal is strongly primary. Then each maximal ideal of R is strongly primary, so (R, M ) is

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quasi-local by Proposition 2.2(2). Suppose that M is unbranched. If xy ∈ M and x∈ / M for some x, y ∈ K, then y n ∈ M for some n ≥ 1. Since M is unbranched, n y ∈ P  M for some prime P ; so y ∈ P since P is strongly prime by Lemma 2.8(2). Thus y ∈ M ; so M is strongly prime. Hence R is a PVD whose maximal ideal is unbranched. Next, suppose that M is branched. Then there exists √ an M -primary ideal J distinct from M . Pick x ∈ M − J. We first claim that J ⊆ xR. If a ∈ J, 2 2 / J, xa ∈ / M , and we have ( ax )n ∈ M then a = xa ax ∈ M . Since x ∈ √ √ for some n ≥ 1 2n n ∈ x M ; so a ∈ xR. Therefore J ⊆ xR, as claimed. by assumption. Thus a √ Now xR = M implies that xR is M -primary, whence xR is strongly primary by hypothesis. Hence R = (xR : xR) is a valuation domain by [BH, Corollary 2.12]. Corollary 2.11. Every ideal of R is strongly primary if and only if R is a field or a 1-dimensional valuation domain. Proof. Suppose that every ideal of R is strongly primary. Then aR is strongly primary for 0 = a ∈ R; so R is a valuation domain [BH, Corollary 2.12], and dim(R) = 1 [Gi, Exercise 2, p. 293]. The converse is clear from Theorem 2.10 since if the radical of an ideal I is maximal, then I is primary.

3

Almost Pseudo-Valuation Domains

Recall from [BH] that an integral domain R is called an almost pseudo-valuation domain (APVD) if every prime ideal of R is strongly primary. Badawi and Houston showed that R is an APVD if and only if some maximal ideal of R is strongly primary, if and only if (R, M ) is a quasi-local domain and there is a valuation overring of R in which M is a primary ideal [BH, Theorem 3.4]. A nonzero ideal I of R is called powerful if whenever xy ∈ I for elements x, y ∈ K, we have x ∈ R or y ∈ R [BH]. Note that R is powerful if and only if R is a valuation domain, and that P is a powerful prime ideal if and only if P is a strongly prime ideal [BH, Proposition 1.3]. Let P be a prime ideal of R. Let φ be the canonical homomorphism from R onto R/P , and let π be the epimorphism from RP onto the quotient field (R/P )P/P ∼ = RP /P RP of R/P defined by π(a/b) = φ(a)/φ(b). Then π −1 (R/P ) = R + P RP is called the CPI-extension of R with respect to P , and is denoted by C(R, P ) [BS]. It is easy to see that P RP is the kernel of π; so we have C(R, P )/P RP ∼ = R/P . Also, C(R, P ) = R if and only if every ideal of R is comparable to P [BS, Theorem 2.4], and C(R, P ) = RP if and only if P is a maximal ideal of R [Do4, Proposition 2.8(a)]. Recall from [Do1] that R is said to be divided in case P = P RP for each prime ideal P of R. For each prime P of R, C(R, P ) is R-flat if and only if R is locally divided (i.e., RM is divided for each maximal ideal M of R) [Do3, Theorem 2.4]. Moreover, C(R, P ) inherits certain properties from R/P and RP (cf. [BS, Proposition 3.11 and 3.12]). In [Do4, Theorem 2.11], it was proved that if P is a nonmaximal prime ideal of R, then C(R, P ) is a PVD if and only if R/P is a PVD and RP

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is a valuation domain. In this section, we show that R/P is an APVD and RP is a valuation domain if and only if C(R, P ) is an APVD. Lemma 3.1. (cf. [Do2, Lemma 4.5(iii)]) Let R be an APVD and let P be a nonmaximal prime ideal of R. Then RP is a valuation domain. Proof. Let M be the maximal ideal of R. Then M 3 is powerful [BH, Corollary 2.6]. Moreover, since M 3 RP = RP , RP is a valuation domain by [BH, Proposition 1.13]. Proposition 3.2. (cf. [BH, Corollary 3.6]) Let R be an APVD with unique maximal ideal M . If T is an overring of R such that M T = T , then T is a valuation domain. Proof. The hypotheses imply that T is strongly primary by [BH, Theorem 2.10]; so T is a PVD by Theorem 2.6. Let Q be the maximal ideal of T and let P = Q ∩ R. Then P  M because M T = T . So RP → T , which implies that T is a valuation domain by Lemma 3.1. Proposition 3.3. Let R be an APVD with maximal ideal M . If M is unbranched, then R is a PVD. Proof. For x, y ∈ K, assume that xy ∈ M and x ∈ / M . Since M is unbranched, there is a prime ideal P of R such that xy ∈ P and P  M . Then P is strongly prime by Lemma 2.8(2); so y ∈ P ⊆ M . Hence R is a PVD. Hence if R is an APVD which is not a PVD, then there is a prime ideal P such that dim(R/P ) = 1. Theorem 3.4. Let P be a prime ideal of R such that P = P RP and RP is a valuation domain. Then R/P is an APVD if and only if R is an APVD. Proof. Note that (R/P )P/P ∼ = RP /P RP = RP /P [Gi, Proposition 5.8]. So RP /P can be considered as the quotient field of R/P . Also, since P is a divided prime ideal of R, C(R, P ) = R, and R is quasi-local if and only if R/P is quasi-local. Thus we assume that R is quasi-local with maximal ideal M . (⇒): For x, y ∈ K, assume that xy ∈ M and x ∈ / M . We will show that y n ∈ M 1 for some n ≥ 1. If x ∈ / RP , then x ∈ P RP = P . Thus y = xy x1 ∈ P ⊆ M . So / M , x1 ∈ RP ; we assume that x ∈ RP . Since RP is a valuation domain and x ∈ so y = xy x1 ∈ RP . For z ∈ RP , z¯ denotes z + P ∈ RP /P . Thus x¯y¯ ∈ M/P and x ¯ ∈ / M/P . Since M/P is strongly primary, there is an integer n ≥ 1 such that (¯ y )n ∈ M/P . So y n ∈ M . Thus M is strongly primary and hence R(= C(R, P )) is an APVD by Lemma 2.8(2). (⇐): For x ¯ = x + P and y¯ = y + P in RP /P (with x, y ∈ RP ), assume that x ¯y¯ ∈ M/P and x ¯∈ / M/P . Then xy ∈ M and x ∈ / M . Since R is an APVD, M is y )n ∈ M/P . Thus M/P is strongly primary and y n ∈ M for some n ≥ 1. So y¯n = (¯ strongly primary, and hence R/P is an APVD by [BH, Theorem 3.4].

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Corollary 3.5. Let P be a nonmaximal prime ideal of R. Then R/P is an APVD and RP is a valuation domain if and only if the CP I-extension C(R, P ) is an APVD. Proof. Note that P RP is not a maximal ideal of C(R, P ). Thus the result follows from Theorem 3.4, Lemma 3.1, and the facts that P RP is a divided prime ideal of C(R, P ), C(R, P )P RP = RP [Do3, Lemma 2.2], and C(R, P )/P RP ∼ = R/P . Since any APVD is a divided domain [BH, Proposition 3.2], Lemma 3.1, Proposition 3.3 and Theorem 3.4 show that to study an APVD which is not a PVD, it suffices to study a 1-dimensional APVD. Theorem 3.6. Let R be a local Noetherian domain with maximal ideal M = 0. Then R is an APVD if and only if each of the following conditions holds: (1 ) R is a 1-dimensional local domain. (2 ) The integral closure R of R is a rank one discrete valuation domain and M R = M . Proof. (⇒): Note that M is comparable to every radical ideal by Proposition 2.2(2), and that the set of radical ideals of R which are contained in M is linearly ordered under inclusion by [BH, Theorem 2.8]. Moreover, since Noetherian domains satisfy the principal ideal theorem [Ka, Theorem 11 and Theorem 88], M is a unique maximal ideal of R and dim(R) = 1.  ideal of R . Thus √ By [BH, Corollary 2.6], M R  = M ; so M is a strongly primary  M is a prime ideal of R ; so R is local. Moreover, since R is a Dedekind domain [Ka, Ex. 13, p. 73], R is a rank one discrete valuation domain. (⇐): Since dim(R ) = 1, M is a primary ideal of R , and hence by Lemma 2.8(1) M is strongly primary. Thus R is an APVD. Example 3.7. (1) Let F [[X]] be the power series ring over a field F . Let V = 1 ]+M be an n-dimensional valuation domain where M is the maximal ideal F [[X]][ X and either n ≥ 1 or n = ∞ [Gi, Ex. 16, p. 372]. Let R = F [[X 2 , X 3 ]] + M . Then R is an (n + 1)-dimensional APVD which is not a PVD. (2) Let R = F [[X 6 , X 7 , X 8 , X 9 , X 10 , X 11 ]] = F + X 6 F [[X]]. Then R is a 1dimensional Noetherian APVD, the integral closure of R is F [[X]], and there is an overring of R which is not an APVD. Proof. (1): First, we note that F [[X 2 , X 3 ]] is not a PVD because if N := (X 2 , X 3 ), then (N : N ) = F [[X]] does not have maximal ideal N . Thus F [[X 2 , X 3 ]] is neither a field nor a PVD with quotient field F [[X]][ X1 ]. Hence R is not a PVD by [Do2, Proposition 4.9(i)]. It is clear that RM = V, M RM = M , and R/M ∼ = F [[X 2 , X 3 ]]. 2 3 2 3 Now, since (X , X ) is a strongly primary ideal of F [[X , X ]] (see Example 2.1), F [[X 2 , X 3 ]] ∼ = R/M is an APVD by Lemma 2.8(2), and hence R is an APVD by Theorem 3.4. (2): It is easy to show that R is a 1-dimensional Noetherian domain whose integral closure is F [[X]], and that R is an APVD. Let D = F [[X 3 , X 4 ]] = F +

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(X 3 , X 4 ). Then D is an overring of R. Note that (X 3 , X 4 )F [[X]] = X 3 F [[X]], / (X 3 , X 4 ). Hence D is not an APVD by Theorem 3.6. X 5 ∈ X 3 F [[X]], but X 5 ∈ Remark. In Example 3.7(2), each overring of R is quasi-local because R is a quasilocal domain whose integral closure, F [[X]], is a valuation domain [Pa, Corollary 2.15 and Proposition 2.34].

Acknowledgment: We express our sincere thanks to the referee for helpful suggestions on the improvement of this chapter. The third author gratefully acknowledges support under a grant from the Inha University Research Grant (INHA-31615).

Bibliography [AA] D.D. Anderson and D.F. Anderson, Multiplicatively closed subsets of fields, Houston J. Math. 13 (1989), 1–11. [AP] D.F. Anderson and J. Park, Rooty and root-closed domains, Advances in Commutative ring theory, Lecture Notes Pure Applied Math., Marcel Dekker 205 (1999), 87–99. [BH] A. Badawi and E.G. Houston, Powerful ideals, strongly primary ideals, almost pseudo-valuation domains, and conductive domains, Comm. Algebra 30 (2002), 1591–1606. [BS] M.B. Boisen and P.B. Sheldon, CPI-extensions: overrings of integral domains with special prime spectrums, Canad. J. Math. 29 (1977), 722–737. [Do1] D. E. Dobbs, Divided rings and going down, Pacific J. Math. 67 (1978), 353–363. [Do2] D. E. Dobbs, Coherence, ascent of going-down and pseudo-valuation domains, Houston J. Math. 4 (1978), 551–567. [Do3] D. E. Dobbs, On locally divided integral domains and CPI-overrings, Internat. J. Math. & Math. Sci. 4 (1981), 119–135. [Do4] D. E. Dobbs, On flat divided prime ideals, Factorization in integral domains, Lecture Notes Pure Applied Math., Marcel Dekker 189 (1997), 305–315. [Do5] D. E. Dobbs, On Henselian pullbacks, Factorization in integral domains, Lecture Notes Pure Applied Math., Marcel Dekker 189 (1997), 317–326. [Gi ] R. Gilmer, Multiplicative ideal theory, Marcel Dekker, New York, 1972. [HH1] J.R. Hedstron and E.G. Houston, Pseudo-valuation domains, Pacific J. Math. 75 (1978), 137–147. [HH2] J.R. Hedstron and E.G. Houston, Pseudo-valuation domains II, Houston J. Math. 4 (1978), 199–207. [Ka] I. Kaplansky, Commutative Rings, revised edition, Univ. of Chicago Press, 1974. 387

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[Pa] I. J. Papick, Topologically defined classes of going-down rings, Trans. Amer. Math. Soc. 219 (1976), 1–37. [SS] J. Sato and T. Sugatani, On the radical ideal of seminormal rings, Comm. Algebra 18 (1990), 441–451.

Index A + XB[X] construction, 22, 30 admissible set, 190, 191 affine toric variety, 146, 160, 161, 166, 168 almost Dedekind domain, 363–365, 368, 369, 374, 375 almost GCD domain, 173 almost generating set, 101, 102, 105, 107, 109 almost pseudo-valuation domain (APVD), 379, 383 arithmetic ring, 278 arithmetical ring, 272, 274, 276, 278 asymptotic, 228, 230, 231 B´ezout domain, 364, 370, 371, 373, 375 Binomial Lemma, 162, 163 block monoid, 326–328, 332, 333 Boolean ring, 195, 196 Boolean space, 364, 369, 370, 374, 375 boundary map, 65, 66, 337, 338, 348 boundary valuation domain, 66

Cale variety, 161, 162, 165–169 catenary degree, 118 CK domains, 49 class group, 22, 23, 31–33, 35–37, 39, 47, 48, 51–53, 59, 62, 64, 71, 73–75, 87 class number, 46, 47, 50, 51, 55, 63 class semigroup, 72, 75, 82, 83, 102, 119 Cohen-Macaulay ring, 284, 286–288 comaximal factorization, 2, 3, 11–14, 360 complete integral closure, 71, 72, 85 conductor, 56 congruence monoid, 71, 72, 93–96 CPI-extension, 379, 383 critical maximal ideal, 363–365, 368, 373 D + M construction, 58, 59 Davenport constant, 48, 59, 64–66 Dedekind domain, 46, 50, 52, 53, 59, 213, 214, 219–221, 223 Dickson’s Finiteness Theorem, 100 Diophantine monoid, 144, 159, 160, 166, 167, 169 divided prime, 198, 200, 201, 207, 209 divisible module, 207 divisor class group, 325, 326

C0 -monoid, 71, 72, 81–83, 87, 90–92, 102, 106, 115, 135 C-monoid, 71, 72, 80–83, 85, 87, 88, 90, 91, 99, 100, 102, 115, 118, 135, 139 Cale base, 143, 146, 147, 151, 155, 157, 160–163, 165–168 Cale basis, 172–177, 234, 235, 237– 239, 242–248 Cale domain, 143, 144, 146, 156–158, 160, 165, 166, 168, 169 Cale monoid, 143–145, 150–156, 159– 162, 165, 166, 168

elasticity, 31, 32, 64, 67, 72, 73, 91, 325–328, 333 extraction degree, 147 factorization, 114–125, 128, 129, 132, 135, 139, 227, 230, 294, 295, 389

390 298–300, 337, 340, 341, 347, 353 factorization homomorphism, 117 factorization monoid, 103, 117, 121– 123, 128 finitary monoid, 101 finitely generated monoid, 294, 296– 298 Frobenius number, 261, 267

Index irreducible element, 294 Jaffard-Kaplansky-Ohm Theorem, 364, 370, 371, 373 Kneser, 316, 319 Krull domain, 25–30, 32–37, 39, 51, 54, 55, 325, 326 Krull monoid, 26, 27, 29, 35, 36, 99, 100, 144, 145, 150–156

Galois extension, 65 lcm splitting, 184, 186 Gaussian ring, 273–279 locally finite intersections of localizaGCD-domain, 25–29, 31 tions, 2 generalized unique factorization domain locally half-factorial domain, 53 (GUFD), 15 locally tame, 99, 104, 107, 118, 122, generalized weakly factorial domain (GWFD), 134, 136 175, 176, 182 graded integral domain, 22–35, 37 m-complement, 180, 181 group of divisibility, 2, 4 M int (D), 256, 257 n

h-local domain, 8, 11 half-factorial, 316, 320–323 half-factorial domain, 30–32, 38, 46– 48, 62, 337, 338, 340–344, 347, 351, 353 homogeneous element, 2, 11 idealization, 199, 201, 204, 206, 207, 210 inert extension, 25–27, 31, 32, 34, 35, 37 infinite product, 188–196 inside factorial domain, 172–175, 177 inside factorial integral domain, 234– 251 integer-matrix-valued polynomial, 253 integer-valued polynomial, 253, 255 integral closure, 47, 52, 54, 56, 60–63, 65, 66 integral morphism, 338–340, 343, 344, 350, 351 integrally closed, 25, 26, 28–31, 33– 37, 61 interpolation domain, 253–256 invertible ideal, 357

M¨ obius inversion formula, 228 multiplicative subset, 180–182, 184, 185 non-Noetherian ring, 283, 287 nonnil ideal, 199, 203, 209 null ideal of a matrix, 257 numerical semigroup, 260–269 order, 50, 54, 56, 62–67, 316, 320–322 overring, 338, 344–349, 352, 354 partition identity, 227 PF ring, 277, 278 φ-Noetherian, 198–205, 210 φ-ring, 198, 201, 205–207, 209, 210 Picard group, 23, 28, 32, 33 polynomial, 213–217, 219–222, 224 polynomial ring, 54, 62 power series ring, 52, 55 PP ring, 277, 278 Principal Ideal Theorem, 283, 285 probability, 226–229 product-space, 316 Pr¨ ufer domain, 235, 246–249, 285, 304, 312, 313, 364, 365, 368 Pr¨ ufer integral closure, 311, 312

Index Pr¨ ufer ring, 273, 274, 278, 279 Pr¨ ufer v-multiplication domain (PVMD), 304, 310 pseudo-uniformizer, 190–192 quadratic field, 63, 64 random monic polynomial, 226, 227 rational generalized Krull domain, 173, 177, 234, 236, 240, 245, 246, 248, 249 rigid element, 2, 10 ring of function, 254, 255 root extension, 172–175, 177, 178, 236, 240, 241, 243–246, 249, 250 rooty domain, 378, 380 S-ideal, 356–361 saturated submonoid, 119, 129, 131 scattered space, 374, 375 semigroup ring, 22–26, 28–30, 32, 33, 35, 38 semihereditary ring, 272, 274–277 seminormal, 28, 29, 31, 33–35, 37–39, 66 separation of points, 254, 255, 257 set of lengths, 117, 118, 121, 294, 297 SP -domain, 363–365, 368–370, 373– 375 strong half-factorial domain, 53 strongly primary, 379–385 strongly prime, 378–381, 383 strongly radical, 378–381 Structure Theorem for Sets of Lengths, 117, 118, 136 t-closedness, 66 t-invertible, 310 T -block monoid, 129 t-class group, 2, 3, 172, 173, 176, 177, 234, 240 t-pure, 2, 9–11 tame degree, 118 torsionless grading monoid, 24, 26, 29, 33, 35–37

391 transfer homomorphism, 106, 115, 120– 124, 128, 130–134, 136, 138 treed domain, 237, 241, 244–246 trigonometric polynomials, 145 2-generated invertible ideal, 356, 358, 361 UMT-domain, 304, 310–313 UMV-domain, 304, 305, 308–313 unidirectional, 8, 9, 11 unique comaximal factorization domain (UCFD), 2, 12, 13 unique factorization domain (UFD), 1, 4, 7, 12, 13, 15, 24–33, 39, 46, 48 unique representations domain (URD), 16 unit fraction, 226 unmixedness condition, 282 upper to zero, 304, 307–313 v-domain, 304, 309–312 v-invertible, 306–310, 312 v-invertible ideal, 306 v-noetherian, 71, 87, 91 v-noetherian G-monoid, 101 valuation domain, 188–190, 193, 194, 196 value group, 189, 190, 194, 196 Von Neumann regular ring, 277–279 wB-ht-unmixed ring, 282, 285, 286, 288, 290 wB-unmixed ring, 282, 284 weak factoriality, 66 weak global dimension, 274, 277 weakly factorial domain, 2, 6, 7, 12 weakly Krull domain, 8, 176