Commutative Rings: New Research

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COMMUTATIVE RINGS: NEW RESEARCH

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COMMUTATIVE RINGS: NEW RESEARCH

JOHN LEE EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Commutative rings : new research / [edited by] John Lee. p. cm. Includes index. ISBN 978-1-61728-192-1 (E-Book) 1. Commutative rings. I. Lee, John, 1958 June 21QA251.3.C675 2009 512'.44--dc22 2009008039

Published by Nova Science Publishers, Inc. Ô New York

CONTENTS Preface

vii

Chapter 1

Jonsson Modules over Commutative Rings Greg Oman

1

Chapter 2

Counting Zero - Divisors Shane P. Redmond

7

Chapter 3

Recent Progress on Minimal Ring Extensions and Related Concepts David E. Dobbs

13

Commutative Algebra Applied to Stabilization Problems for Systems over Rings A. Saez-Schwedt

39

Asymptotic Behavior of Associated or Attached Prime Ideals of Certain Ext-modules and Tor-modules K. Khasyarmanesh and F. Khosh-Ahang

53

Linear Algebra over Commutative Rings Applied to Control Theory Miguel V. Carriegos

65

Chapter 4

Chapter 5

Chapter 6

Chapter 7

A Characterization of Commutative Clean Rings Warren Wm. McGovern

Chapter 8

Jordan Automorphisms of Certain Jordan Matrix Algebra over Commutative Rings Ruiping Yao, Dengyin Wang and Yanxia Zhao

Index

105

119 129

PREFACE Commutative rings are a branch of abstract algebra that deals with the multiplication operation. This book examines the question, given any positive integer n, is there a commutative ring with identity that has n zero-divisions? This question is examined in stages through looking at local rings, reduced rings and finally commutative rings in general. In addition, several themes pertaining to the classification of minimal ring extensions are described. Some recent and new results on linear systems theory over commutative rings are also looked at. Finally, this book gives a brief history and summary of the active area of asymptotic stability of associated or attached prime ideals. Some of the old and new results about the asymptotic properties of associated and attached prime ideals related to injective, projective or flat modules, are discussed. Chapter 1 - Let M be an infinite unitary module over a commutative ringR with identity. Then M is called a Jónsson module provided every proper submodule of M has smaller cardinality than M. These modules have been studied by several algebraists, including Robert Gilmer, Bill Heinzer, and the author. In this note, the authors recall the major results on Jónsson modules to bring the reader up to speed on current research. Included are some applications to Artinian and uniserial modules as well as quasi-cyclic groups. There are several wide open problems in this area, some of which may be independent of the usual axioms of set theory. The authors close the article with a discussion of several such problems and outline some possible strategies for solving them. Chapter 2 – Given any positive integer n, is there a commutative ring with identity that hasn zero-divisors? This question, inspired by looking at zero-divisor graphs, is examined in stages through looking at local rings, reduced rings, and finally commutative rings in general. The answer is no in all cases. In the course of the investigation, connections to the Goldbach conjecture are uncovered and additional open questions are posed. The results are based on computer calculations instead of traditional, theory based proofs (although the computation of a counterexample is certainly a proof). Samples of the computer routines used to investigate this question are given. Chapter 3 - A unital extension A ⊂ B of (commutative unital) rings is said to be a minimal ring extension if there is no ring C such that A ⊂ C ⊂ B. The introductory section summarizes the background needed to study this concept. Section 2 describes several themes pertaining to the classification of minimal ring extensions, including the classification in 2006 by Dobbs-Shapiro of the minimal ring extensions of an arbitrary (integral) domain and various recent generalizations of the fact that any minimal domain extension of a non-field

viii

Gaston M. N’Guerekata

must be an overring. Section 3 describes some ways that minimal ring extensions arise naturally. These include chain-theoretic studies, such as the recent result by CoykendallDobbs that, despite the known case for finite chains, a domain R with a saturated chain consisting of integrally closed overrings need not be a Prufer domain; and FIP-theoretic studies generalizing the Primitive Element Theorem of field theory. Among the FIP-theoretic results are the generalization in 2003 by Dobbs-Mullins-Picavet-Picavet-LHermitte of the Primitive Element Theorem by characterizing, for each field K, the commutative unital Kalgebras that have only finitely many unital K-subalgebras; and the recent characterization by Dobbs-Picavet-PicavetLHermitte of the rings having only finitely many unital subrings. Also surveyed in regard to the latter topic are recent results concerning whether composites of minimal ring extensions (resp., ring extensions satisfying FIP)R ⊂ S and R ⊂ T are such that S ⊂ ST is a minimal ring extension (resp., satisfies FIP). The final section summarizes recent results that use variants of the classical Kaplansky transform to characterize minimal ring extensions R ⊂ T in which R is integrally closed in T; and, to illustrate similarities and differences between allied concepts, quotes several results from an unpublished dissertation of M. S. Gilbert on the generalization of minimal ring extensions that concerns ring extensions whose set of intermediate rings is linearly ordered by inclusion. Chapter 4 - In this chapter the authors study some recent and new results on linear systems theory over commutative rings. A linear system over a commutative ring R is a pair of matrices (A, B) of sizes n x n and n x m respectively, with coefficients in R. The system is called reachable if the n x mn block matrix A*B = [B|AB|···|An-1B] is right-invertible. The following three properties of systems are classically related to the problem of stabilization of systems: (i) pole assignment (PA), (ii) coefficient assignment (CA) and (iii) feedback cyclization (FC). It is a well known result that (iii) Ö (ii) Ö (i), and (i) implies that (A,B) is reachable. If R is a field, all four conditions are equivalent. One defines PA rings, CA rings and FC rings as those commutative rings R such that any reachable system over R satisfies (i), (ii) or (iii) respectively. For non necessarily reachable systems the authors extend the above properties, by using the concept of residual rank of a system: The residual rank of a system (A,B), denoted as res.rk(A,B), is defined as the highest nonnegative index i such that the ideal generated by the i x i minors of the matrix A*B is R (reachable systems correspond to the case of residual rank equal to n). This allows us to introduce new classes of rings, satisfying the following (strict) implications: strong FCÖ strong CA Ö strong PA. The authors then prove that the strong FC property is achieved by all rings R satisfying the following two conditions: • UCU property: if B is a matrix with unit content, then the R- module im( B) contains a free rank one direct summand of Rn • stable range one: if( a, b)=R, there exists k such that a + bk is a unit of R. Examples of strong FC rings are, among others: zero-dimensional rings, local-global rings, rings with many units and Bezout domains with stable range one, including certain rings of holomorphic functions. Finally, for all s > 0, the authors prove that UCU rings with stable ranges satisfy the strong form of a new property called FCs. Chapter 5 - Asymptotic behavior of the associated and attached prime ideals represents the interface of two major ideas in the study of modules over a commutative ring. The first, the concepts of associated and attached prime ideals, are important valued tools in

Preface

ix

researcher’s arsenal. The second is the fact that in a Noetherian ring, large powers of an ideal are well behaved, as shown by the Artin-Rees Lemma or the Hilbert polynomial. Although its roots go back further, the interests in asymptotic stability of associated prime ideals are initiated by a question of Ratliff: “What happens to AssR(R/In) as n gets large?” Brodmann answered the question, proving that AssR(M /In M) also stabilizes for large n, where M is a finitely generated module over a commutative Noetherian ring R. Since then, the topic of stability of certain sets of associated (or attached) prime ideals has been growing rapidly. This paper gives a brief history and summary of the active area of asymptotic stability of associated or attached prime ideals. The paper has a section that contains some generalizations (or dual) of Brodmann’s Theorem by using the ‘Hom-functor’ and ‘tensor product’. After that section three lists many of the newer theorems concerning asymptotic stability of certain sets of prime ideals by using the derived functor Tori of the ‘tensor product’ or the derived functor Exti of the ‘Hom-functor’. Finally, last section discusses some of the old and new results about the asymptotic properties of associated and attached prime ideals related to injective, projective or flat modules. Throughout this paper, R denotes a commutative ring with non-zero identity, M is a Noetherian R-module, A is an Artinian R-module and I is an ideal of R. The authors use N0 (respectively N) to denote the set of non-negative (respectively positive) integers. Chapter 6 - In this chapter the authors review some recent results on linear systems theory over commutative rings. The authors use linear algebra over commutative rings as main tool. Some general considerations about the influence of linear algebra on systems theory are given in the Introduction. Then the authors deal with both the linear algebra and control theory background they need in the next sections. Reachability is one of the main subjects in control theory. In the linear case, reachability can be studied as the surjectivity of some linear map. Then the problem can be stated in terms of linear equations and solved by standard methods. Algebraic and feedback equivalence are also studied. Invariant R-modules for both equivalence actions are presented. Consequently the authors can give canonical forms not only in the usual case of constant coefficients (complex numbers) but in the case of commutative rings (real numbers, finite fields, integers, polynomials, real functions,...). The stabilization of linear systems are related with the pole-shifting property, which is the ability of change some poles in the characteristic polynomial. Some results are given in this subject. Finally the authors touch some classes of rings related with systems theory; here the main tool is commutative algebra. Last section is a conclusion of the chapter. Some unsolved research questions are included. Chapter 7 - A commutative ring A is said to be clean if every element of A can be written as a sum of a unit and an idempotent. This definition dates back to 1977 where it was introduced by W. K. Nicholson. In 2002, V. P. Camillo and D. D. Anderson investigated commutative clean rings and obtained several important results. Han and Nicholson show that if A is a semiperfect ring, then A [Z2] is a clean ring. In this paper the authors generalize this argument (for commutative rings) and show that A [Z2] is clean if and only A is clean. The authors also show that if the group ring A [G] is a commutative clean ring, then G must be a torsion group. Our investigations lead us to introduce the class of 2-clean rings.

x

Gaston M. N’Guerekata

Chapter 8 - Let R be a commutative ring with identity 1 and unit 2, and let S be the 2m by 2m Jordan matrix algebra over R. In this article, the authors prove that any Jordan automorphism of S can be uniquely decomposed as a product of inner and extremal automorphisms, respectively.

In: Commutative Rings: New Research Editor: John Lee

ISBN 978-1-60692-614-7 c 2009 Nova Science Publishers, Inc.

Chapter 1

´ J ONSSON M ODULES OVER C OMMUTATIVE R INGS Greg Oman∗ Department of Mathematics, The Ohio State University 231 W. 18th Ave., Columbus, Ohio 43210

Abstract Let M be an infinite unitary module over a commutative ring R with identity. Then M is called a Jónsson module provided every proper submodule of M has smaller cardinality than M. These modules have been studied by several algebraists, including Robert Gilmer, Bill Heinzer, and the author. In this note, we recall the major results on Jónsson modules to bring the reader up to speed on current research. Included are some applications to Artinian and uniserial modules as well as quasi-cyclic groups. There are several wide open problems in this area, some of which may be independent of the usual axioms of set theory. We close the article with a discussion of several such problems and outline some possible strategies for solving them. 2000 AMS Subject Classification: 13C99, 03E10.

All rings in this paper are assumed to be commutative with identity, and all modules are assumed to be unitary.

1 Introduction An old problem posed by Kurosh was to determine if there exists a group of cardinalityℵ1 in which all proper subgroups are countable. In the mid 1970’s, Shelah constructed such a group ([12]). This spurred more interest from the logic community in so-called J´ onsson algebras, which are algebras with countably many finitary operations in which every proper subalgebra has smaller cardinality (see [1] for an excellent survey). These notions piqued the interest of commutative algebraists Robert Gilmer and Bill Heinzer, who translated these ideas to the context of unitary modules over a commutative ring with identity ([6]). They define an infinite module M over a ring R to be a Jónsson ∗ E-mail

address: [email protected]

2

Greg Oman

module iff every proper submodule of M has smaller cardinality than M. Using various ideal-theoretic techniques, they give a complete description of all countable J´ onsson modules over a Prüfer domain, and prove several propositions about general J´ onsson modules. They applied and extended these results in several subsequent papers ([3], [4], [5], [7]). We begin by providing two canonical examples of J´ onsson modules. Example 1. Let F be an infinite field, and consider F as a module over itself. The submodules of F are precisely the ideals of F. Since F has only trivial ideals, it is easy to see that F is a Jónsson module over itself. More generally, if R is any ring with an infinite residue field R/M, then R/M becomes a Jónsson module over R. Example 2. Let p be a prime number. The direct limit of the cyclic groups Z/(pn ) is the socalled quasi-cyclic group of type p∞ , denoted by C(p∞ ). It is well-known that every proper subgroup of C(p∞ ) is finite, whence C(p∞ ) is a Jónsson module over Z. It was proved by W.R. Scott that the quasi-cyclic groups are in fact the only Abelian Jónsson groups (Z-modules). We close the introduction by giving a new proof of this fact. Before doing so, we recall two results from Abelian group theory. The first is exercise 2, p. 67 of [2]. Proposition 1. Every Abelian group is either divisible or contains a maximal subgroup. Proposition 2 (Structure Theorem for Divisible Abelian Groups). Every divisible Abelian group is a direct sum of copies of Q and copies of Z(p∞ ) for various primes p. Proof. See p. 64 of [2]. We now provide a new and simple proof of Scott’s result. Theorem 1 ([11], Remark 1). The only Abelian Jónsson groups are the quasi-cyclic groups C(p∞ ). Proof. Let G be an Abelian Jónsson group. It is easy to see that G is indecomposable: for if L G = M N, then since G is infinite, either |M| = |G| or |N| = |G|. Since G is Jónsson, this forces either M = G or N = G. If G is divisible, it follows from the structure theorem for divisible Abelian groups and the fact that G is indecomposable that G ∼ = Q or G ∼ = Z(p∞ ) for some prime p. As Q is clearly not a Jónsson group, we have G ∼ = Z(p∞ ) and we’re done. Suppose now that G has a maximal subgroup M. Since G is Jónsson, |M| < |G|. It is wellknown that G/M ∼ = Z/(p) for some prime p and thus M must be infinite. Let g ∈ G − M. Then (M, g) = G by maximality of M. However, |G| = |(M, g)| = |M| < |G| and this is a contradiction. This completes the proof.

2 General Results on Jónsson Modules We now review some of the most important general results from the literature. We begin with the following proposition of Gilmer and Heinzer.

Jónsson Modules over Commutative Rings

3

Proposition 3 ([6], Proposition 2.5). Suppose that M is a Jónsson module over the ring R. Let r ∈ R be arbitrary. Then: (1) Either rM = M or rM = 0; (2) Ann(M)= {s ∈ R : (∀m ∈ M)(sm = 0)} is a prime ideal of R. Thus by modding out the annihilator, there is no loss of generality in assuming that a Jónsson module is faithful over an integral domain. The next result states that the nontrivial Jónsson modules are all torsion. Theorem 2 ([10], Theorem 2.1). Suppose that M is a Jónsson module over the ring R. Then either R is a field and M ∼ = R or M is a torsion module. Recall the curious fact (Theorem 1) that every J´ onsson module over the ring Z of integers is countable. A much more general result is true if one assumes the generalized continuum hypothesis. It is an open question if the following proposition can be proved in ZFC (though the result can be proved in ZFC if R is Noetherian). Theorem 3 ([10], Corollary 4). Assume the generalized continuum hypothesis. If M is a Jónsson module over the ring R, then |M| ≤ |R|. We now recall the following fundamental result on countable J´ onsson modules due to Robert Gilmer and Bill Heinzer. Note that this result gives quite a bit of information both about the Jónsson module M and the operator ring R. Theorem 4 ([6], Theorem 3.1). Suppose that M is a countably infinite Jónsson module over the ring R and that M is not finitely generated. Then M is a torsion R-module, and there exists a maximal ideal Q of R such that the following hold: (1) Ann(x) is a Q-primary ideal of finite index for every x ∈ M − {0}; (2) R/Q is finite; (3) The powers of Q properly descend; T i (4) ∞ i=1 Q =Ann(M); (5) If Hi = {x ∈ M : Qi x = 0}, then {Hi }∞ i=1 is a strictly ascending sequence of submodS H . ules of M such that M = ∞ i=1 i

3

Projective and Injective Jo´ nsson Modules

We now turn our attention toward describing the projective and injective J´ onsson modules. If M is a faithful projective Jónsson module over the domain R, then it follows from Proposition 3 and Theorem 2 that R is a field and M ∼ = R. The characterization of the injective Jónsson modules is much more complicated. We have solved the problem over Noetherian rings, but the general problem remains open. Theorem 5 ([9], Corollary 4). Let D be a Noetherian domain, and suppose that M is an infinite injective module over D. Then M is a Jónsson module over D iff one of the following holds: (1) D is a field and M ∼ = D; ∼ (2) M = E(D/J) for some maximal ideal J of D such that D/J is finite and DJ is an almost DVR (E(D/J) is the injective hull of D/J).

4

4

Greg Oman

Which Rings Admit Faithful Jo´ nsson Modules?

As noted in the introduction, if M is a maximal ideal of the ring R and R/M is infinite, then R/M is a Jónsson module over R. Note that if R is not a field, then R/M isn’t faithful. Thus we wonder what can be said of the existence of faithful J´ onsson modules over an arbitrary ring R. Must such modules exist? For many familiar rings, the answer is no. We now review most of what is currently known about this question. Theorem 6 ([8], Corollary 3.9.4). Let F be a finite field, and suppose that {xi : i ∈ I} is a set of indeterminates with |I| ≤ 2ℵ0 . Then both F[xi : i ∈ I] and Z[xi : i ∈ I] admit faithful countable Jónsson modules. In particular, this shows that a domain admitting a faithful J´ onsson module need not be Noetherian, and even in the Noetherian case, there is no finite bound on the Krull dimension of such a domain. The situation where F is infinite is much different. Theorem 7 ([8], Proposition 3.9.6). Let F be an infinite field. The domain F [x1 , . . . , xn ] does not admit a faithful Jónsson module. Suppose now that {xi : i ∈ I} is a set of indeterminates and that |I| ≤ |F|. Then it cannot be proved in ZFC that F[xi : i ∈ I] admits a faithful Jónsson module. We close this investigation with results for power series rings and finite-dimensional valuation rings. Theorem 8 ([8], Proposition 3.9.7). Let F be a field. Then F [[x1 , . . . , xn ]] admits a faithful Jónsson module iff F is finite and n = 1. In this case the Jónsson module is unique and is isomorphic to K/F[[x]] where K is the quotient field of F[[x]]. Theorem 9 ([8], Corollary 3.9.18). Let V be a valuation domain of positive dimension. V admits a faithful Jónsson module iff V is a DVR with a finite residue field. In this case the Jónsson module is also unique and is isomorphic to K/V (again, K is the quotient field of V ).

5

Applications

We begin by recalling that an infinite group G is said to be a Jónsson group provided every proper subgroup of G has smaller cardinality than G. An infinite semigroup S is a Jónsson semigroup if every proper subsemigroup of S has smaller cardinality than S. Our first two results impose bounds on the cardinality of uniserial and Artinian modules. Theorem 10 ([8], Proposition 5.1.2). Let M be an infinite uniserial module over the ring R. Then |M| ≤ |R|. Theorem 11 ([8], Proposition 5.1.4). Let M be an infinite Artinian module over the ring R. Then |M| ≤ |R|. The next result shows that a nonabelian Jónsson group is, in some sense, highly nonabelian.

Jónsson Modules over Commutative Rings

5

Theorem 12 ([8], Corollary 5.2.5). Suppose G is a Jónsson group with derived subgroup G0 . Then G0 = {e} or G0 = G. We now consider an infinite semigroup S. Let S∗ be the subsemigroup of S generated by all commutators of S (if S has no commutators, then we put S∗ := ∅). We obtain the following new characterization of the quasi-cyclic groups: Theorem 13 ([8], Theorem 5.2.4). Assume the generalized continuum hypothesis, and let S be an infinite semigroup. Then S ∼ = Z(p∞ ) for some prime number p iff S satisfies: (i) S is a Jónsson semigroup; (ii) S∗ 6= S. Lastly we state an interesting theorem due to Robert Gilmer and Bill Heinzer whose proof utilizes Jónsson modules. Theorem 14 ([3], Corollary 2). If R admits a proper subring, and if every proper subring of a ring R is Artinian, then R is Artinian.

6

Conclusion

We close this paper with a list of three open problems we feel are interesting. As noted in the introduction, if R has an infinite residue field R/M, then R/M is a Jónsson module over R. It is also easy to show that every finitely generated J´ onsson module over a ring R has the form R/M for some maximal ideal M of R. All known examples of infinitely generated Jónsson modules are countable. Thus we ask: Question 1. Does there exist an infinitely generated uncountable Jónsson module? We have obtained a characterization of countable J´ onsson modules below. Before stating the result, we recall that a module M is said to be almost Noetherian provided M is not finitely generated, but every proper submodule of M is finitely generated. Theorem 15 ([10], Theorem 4.1). Suppose that M is an infinitely generated faithful Jónsson module over the domain D. The following are equivalent: (a) M is countable; (b) M is Artinian; (c) M is almost Noetherian. Considering that the Jónsson modules over Z are all of the form Q/Z(p) for some prime p, it is natural to consider modules of the form D/V where V is a valuation domain and D is a domain containing V as a subring when trying to find examples of uncountable J´ onsson modules. As the following proposition shows, the only possible such J´ onsson modules are of the form K/V where K is the quotient field of V . Proposition 4 ([8], Proposition 3.9.20). Suppose that V is a valuation domain which is not a field, and D is a domain containing V as a subring. If D/V is a faithful Jónsson module over V , then D = K, the quotient field of V .

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Greg Oman

Earlier in this paper, we saw that many familiar rings did not admit faithful J´ onsson modules. We would like to know if it is possible to isolate a ring-theoretic property which is both necessary and sufficient to guarantee the existence of faithful J´ onsson modules. Question 2. Which rings R admit faithful Jónsson modules? We end this paper with what could be the most interesting unsolved question on J´ onsson modules. Recall from Theorem 3 that if one assume the generalize continuum hypothesis, then all Jónsson modules over a ring R cannot be larger than R. Call a module M over a ring R large if |M| > |R|. Question 3. Can the nonexistence of large Jónsson modules be proved in ZFC?

References [1]

Eoin Coleman, Jonsson groups, rings, and algebras, Irish Math. Soc. Bull. No. 36 (1996), 34–45.

[2]

L. Fuchs, Abelian Groups, Pergamon Press, Oxford, 1960.

[3]

R. Gilmer and W. Heinzer, An application of Jónsson modules to some questions concerning proper subrings, Math. Scand. 70 (1992), no. 1, 34–42.

[4]

R. Gilmer and W. Heinzer, Cardinality of generating sets for modules over a commutative ring, Math. Scand. 52 (1983), 41–57.

[5]

R. Gilmer and W. Heinzer, On Jónsson algebras over a commutative ring, J. Pure Appl. Algebra 49 (1987), no. 1-2, 133–159.

[6]

R. Gilmer and W. Heinzer, On Jónsson modules over a commutative ring, Acta Sci. Math. 46 (1983), 3–15.

[7]

R. Gilmer and W. Heinzer, On the cardinality of subrings of a commutative ring, Canad. Math. Bull. 29 (1986), 102–108.

[8]

G. Oman, A generalization of Jónsson modules over commutative rings with identity, Ph.D. Thesis, The Ohio State University, 2006.

[9]

G. Oman, On modules M for which N ∼ = M for every submodule N of size |M|, The Journal of Commutative Algebra (to appear).

[10] G. Oman, Some results on Jónsson modules over a commutative ring, The Houston Journal of Mathematics 35 (2009), no. 1, 1-12. [11] W.R. Scott, Groups and cardinal numbers, Amer. J. Math. 74 (1952), 187–197. [12] S. Shelah, On a problem of Kurosh, Jonsson groups, and applications, Word Problems II (Proc. Conf. Oxford, 1976), North-Holland (Amsterdam, 1980), 373–394. Reviewed by Professor Alan Loper, The Ohio State University, Newark Branch



Chapter 2

C OUNTING Z ERO -D IVISORS Shane P. Redmond

Abstract Given any positive integer n, is there a commutative ring with identity that has n zero-divisors? This question, inspired by looking at zero-divisor graphs, is examined in stages through looking at local rings, reduced rings, and finally commutative rings in general. The answer is no in all cases. In the course of the investigation, connections to the Goldbach conjecture are uncovered and additional open questions are posed. The results are based on computer calculations instead of traditional, theory based proofs (although the computation of a counterexample is certainly a proof). Samples of the computer routines used to investigate this question are given. Key Words: zero-divisor, ring, reduced ring, Goldbach Conjecture.

This article was inspired by a simple question: Given any positive integer n, is there a commutative ring with identity that has n zero-divisors? In [3], it was shown that a ring R has a finite number n ≥ 2 of zero-divisors only if R is finite, in fact having at most (n + 1)2 elements. So, to answer this question, one only needs to consider finite rings. Of course, the question is somewhat trivial if one removes the requirement that the ring must have an identity. Letting Ak denote the additive group Zk with the trivial multiplication (xy = 0 for all x, y ∈ Ak ), then An has n zero-divisors. Thus, for this article, all rings considered will be finite with identity 16= 0. As in [3], 0 will be counted as a zero-divisor in this article. Also, given a ring R, let Z(R) denote the set of zero-divisors of R, let Fk denote the finite field with k elements, let |X | be the cardinality of a set X , and let X −Y denote the set difference between sets X and Y . Restricting the question to local rings (rings which have a unique maximal ideal, including fields) can give examples for only certain values of n. For a finite local ring R with 1, |R| = ptr and |Z(R)| = p(t−1)r for some prime p and some positive integers t and r, which was proved in [6] using the fact that the maximal ideal of R, the Jacobson radical of R, and Z(R) all coincide. Hence, one must look beyond just local rings to answer this question. Suppose R is a finite commutative ring with 1. Then, trivially, R is Artinian. Thus, either R is a field (and hence of little consequence to the question), or R is local but not a field, or R ∼ = S1 × S2 × · · · × Sk for some integer k ≥ 2, where each Si is a finite local ring

8

Shane P. Redmond

with identity. If the ring R is reduced (that is, it has no nonzero nilpotent elements), then R must be a finite product of finite fields. In [4], a method was given to count the zero-divisors of a direct product. If R ∼ = S1 × S2 × · · · × Sk , then the set of zero-divisors of R is the set of all k-tuples where at least one entry is a zero-divisor. Alternatively, a k-tuple is not a zero-divisor if and only if each entry is not a zero-divisor. Hence, |Z(R)| = |S1 ||S2 | · · · |Sk | − |S1 − Z(S1 )||S2 − Z(S2 )| · · · |Sk − Z(Sk )|.

(1)

In light of the above three paragraphs and Equation (1), one can restate the original question as follows: given any positive integer n, do there exist a positive integer k, prime numbers p1 , p2 , . . . , pk , and positive integers n1 , n2 , . . . , nk and r1 , r2 , . . . , rk such that (n −1)r1

n = pn11 r1 p2n2 r2 · · · pknk rk − (pn11 r1 − p1 1

(n −1)r2

)(pn22 r2 − p2 2

(nk −1)rk

) · · · (pnk k rk − pk

).

(2)

Now, let us restrict the question to reduced rings. First, note that there are no reduced rings with only 2 zero-divisors, but the local rings Z4 and Z2 [X]/(X 2 ) each have precisely m 2 zero-divisors. Next, consider the case where R ∼ = F1 × F2 , where |F1 | = p1 1 and |F2 | = 2 pm 2 for primes p1 and p2 and positive integers m1 and m2 . Then simplifying Equation 2 m2 1 gives |Z(R)| = pm 1 + p2 − 1. In other words, the question has become: for which positive integers n can we find primes p1 and p2 and positive integers m1 and m2 such that m2 1 n + 1 = pm 1 + p2 .

(3)

(It is interesting to note that Equations 2 and 3 are really questions from number theory, with all traces of commutative rings with identity and zero-divisors taken away. Equations 2 and 3 are also well suited to investigation using computers.) If n is odd, then the question could be answered by the Goldbach Conjecture, which asks if every even integer greater than 2 is the sum of two primes. While there is no widely accepted proof of the Goldbach Conjecture, it has been verified for even integers up to 4 · 1014 (see [9]). So, let us focus on even values of n and leave the odd values to the Goldbach Conjecture. Of course, even if the Goldbach Conjecture were not true, this would not necessarily give a negative answer to the original question. However, not every even integer can be written as in Equation 3. In fact, in [5] an infinite subsequence of the Fibonacci numbers is given that does not satisfy this condition. In the course of the investigation of this question, this author used Mathematica routines to find the even integers n ≤ 7500 that do not satisfy Equation 3. Those values are displayed in the following table. The motivation for the table is as follows: If there is a reduced ring with n zero-divisors, find the fewest number of fields necessary to construct such a ring as a direct product of fields. If there is no reduced ring with n zero-divisors, find the fewest number of local rings necessary to construct such a ring as a direct product of local rings and fields.

Counting Zero-Divisors type of ring with n zero-divisors three fields

four fields

five fields six fields one ring, two fields one ring, three fields one ring, four fields one local ring no such ring

9

values of n not satisfying Equation 3 148, 372, 508, 700, 756, 808, 876, 958, 996, 1018, 1086, 1198, 1258, 1270, 1476, 1528, 1540, 1548, 1588, 1596, 1618, 1656, 1718, 1758, 1776, 1828, 1858, 1968, 1972, 2170, 2230, 2278, 2428, 2502, 2578, 2668, 2788, 2842, 2878, 2908, 2992, 2998, 3028, 3118, 3148, 3180, 3214, 3238, 3298, 3340, 3352, 3430, 3432, 3504, 3538, 3664, 3696, 3738, 3778, 3816, 3844, 3876, 3966, 3984, 4000, 4150, 4152, 4228, 4270, 4310, 4572, 4588, 4632, 4648, 4662, 4716, 4780, 4812, 4840, 4854, 4888, 5076, 5098, 5124, 5302, 5404, 5556, 5608, 5616, 5728, 5736, 5760, 5770, 5916, 5950, 6000, 6020, 6118, 6160, 6172, 6192, 6268, 6432, 6448, 6508, 6520, 6538, 6636, 6658, 6730, 6790, 6820, 6852, 6868, 6940, 7108, 7150, 7168, 7176, 7198, 7288, 7296, 7318, 7330, 7342, 7378, 7388, 7392, 7404 330, 906, 1242, 1782, 1866, 1926, 2262, 2292, 2376, 2682, 3162, 3642, 4012, 4194, 4502, 4542, 4690, 4842, 5142, 5730, 5754, 5922, 6072, 6246, 6402, 6672, 6756, 6882, 7416 2982, 3186, 4062, 4326, 4566, 4810, 7266, 7386 5322 6462, 6534 1206, 1806, 6546 7430 2 1210, 3342, 5466

One need not give up on a value that does not satisfy Equation 3 as the original question gives more freedom than just looking at a product of two fields. For example, there is no product of two fields that has 148 zero-divisors, but F2 × F2 × F49 has 148 zero-divisors. There are no products of two or three fields that have 330 zero-divisors. However, F2 × F2 × F2 × F47 has 330 zero-divisors. Similarly, F2 × F3 × F3 × F7 × F29 has 2982 zerodivisors and F2 × F2 × F2 × F2 × F11 × F32 has 5322 zero-divisors, but no products of a fewer number of fields have 2982 or 5322 zero-divisors. This still leaves several values that do not satisfy Equation 3: 1206, 1210, 1806, 3342, 5466, 6462, 6534, 6546, and 7430. Using the same Mathematica routines to check the products of three, four, five, and six fields as used for the previous integers, no solutions to Equation 1 where each Si is a finite field (that is, solutions to Equation 2 where each ni = 1) were found for these values. Further Mathematica routines were written to check the products of seven through thirteen fields, none yielding a solution to Equation 1 for these values. Since the product of thirteen fields with the smallest number of zero-divisors is (F2 )13 with 8191 zero-divisors, there is no need to check products of thirteen or more

10

Shane P. Redmond

fields. Therefore, there are no rings that are products of fields with 1206, 1210, 1806, 3342, 5466, 6462, 6534, 6546, and 7430 zero-divisors. Yet, this still does not give a negative answer to the original question; it only says the answer is false in the reduced case. The ring Z9 × F2 × F7 × F13 has 1206 zero-divisors, but it does have nonzero nilpotent elements (3, 0, 0, 0) and (6, 0, 0, 0). Also, the ring Z9 × F2 × F5 × F27 is not reduced but has 1806 zero-divisors. Similarly, the ring Z27 × F2 × F179 has 6462 zero-divisors, Z27 × F2 × F181 has 6534 zero-divisors, Z9 × F2 × F11 × F47 has 6546 zero-divisors, and Z4 × F2 × F2 × F4 × F128 has 7430 zero-divisors. A variety of computer routines (see the appendix) were written to see if any product of rings has 1210, 3342, or 5466 zero-divisors. However, none of these routines could find any ring that has 1210 or 3342 or 5466 zero-divisors. Thus, the answer to the original question posed in this article is false. 1210, 3342 and 5466 are the smallest positive integers n such that there is no commutative ring with identity that has n zero-divisors. Some interesting open questions still remain. The first several values for some of these questions are given in the table. Question 1 : Given a positive integer m, for which positive integers n are there no reduced rings with n zero-divisors that are the product of m or fewer fields? (Alternatively, for which values of n are there no solutions to Equation 2 when k ≤ m and each ni = 1?) Question 2 : For which positive integers n are there no reduced rings with n zerodivisors? (Alternatively, for which values of n are there no solutions to Equation 2 when each ni = 1?) Question 3 : If n is an integer such that there exists a reduced commutative ring with n zero-divisors, can one always find a product of 6 or fewer fields with n zero-divisors? (Alternatively, if n is a value that has a solution to Equation 2 with each ni = 1, is there a solution to Equation 2 for n with each ni = 1 and k ≤ 6?) If the answer is false, is it the case that given any positive integer m, there is an integer nm such that there is a product of m fields with nm zero-divisors but no product of fewer than m fields has nm zero-divisors? Question 4 : For which positive integers n do there exist only local rings with n zerodivisors? Question 5 : For which positive integers n are there no rings with n zero-divisors?

Appendix Some of the Mathematica routines used in this article are given below. A complete set of all routines used can be found at [7]. The first routine looks for solutions to Equation 3. The user would input a range of odd values from p to q and a value r which gives the largest prime to be used in the search. Typically, r was chosen such that the rth prime was larger than q. The routine returns all the possible ways Equation 3 could be satisfied for values between p and q. If there is no output for m = n + 1, then n is a value that does not satisfy Equation 3. PSearch[m Integer, t Integer] := Module[ {i, j, k, s }, i = 2; k = 1; j = 1; s := Ceiling[Log[2, m]]; Do[If[m == 2 ∧k + (Prime[i]) ∧j, Print[ {m, 2 ∧k, Prime[i], j }]], {i, 2, t }, {k, 1, s }, {j, 1, s }]]

Counting Zero-Divisors

11

BigPSearch[p Integer, q Integer, r Integer] := {a, p, q, 2 }]] Module[ {a}, a := p; Do[PSearch[a, r], The next command defines a function in Mathematica that searches for rings that involve a product of three fields and have n zero divisors. This program can be used in several ways. First, by choosing n to be the desired number of zero-divisors and letting r = s = 1, one can find if there are any rings that are a product of three fields that have n zero-divisors. Once the first instance of a product of three fields with n zero-divisors is found, the message “FOUND” occurs and the program is terminated. Similarly, given any finite ring R1 , assign r to be the number of elements of R1 and s to be the number of non-zero-divisors of R1 . Then, the program returns “FOUND” if there is a ring that has n zero-divisors that is isomorphic to the product of R1 with three fields. The list A consists of the sizes of fields in ascending order (hence, A consists of primes and powers of primes). As one investigates larger and larger values of n, the list A can be increased as needed. This list as given here was stopped at 173 just for space considerations in this article. This routine can be modified to search for products involving any number of fields. threeflds[n Integer, r Integer, s Integer] :=Module[ {i, j, k, m, p, q }, A = {2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173 }; i = 1; k = 1; p = 1; j = True; m = True; q = True; ∧3 - s*(Extract[A, p] - 1) ∧3; Do[While[q, d = r*(Extract[A, p]) If[d > n, q = False]; ∧2) While[m, c = r*Extract[A, p]*((Extract[A, k]) ∧2); If[c > n, m = False]; s*(Extract[A, p] - 1)*((Extract[A, k] - 1) While[j, b = r*Extract[A, p]*Extract[A, k]*Extract[A, i] s*(Extract[A, p] - 1)*(Extract[A, k] - 1)*(Extract[A, i] - 1); Print[ {b}]; If[b == n, Print[ {FOUND!!! }, {Extract[A, i], Extract[A, k], Extract[A, p] }]; j = False; m = False; q = False; v = False]; If[b > n, j = False]; i = i + 1]; k = k + 1; j = True; i = k]; p = p + 1; j = True; m = True; i = p; k = p]] One can speed up the search using the following series of commands. R = {4, 8, 9, 128, 169, 243 S = {2, 4, 6, 156, 162 }; t = 1; While[t

16, 16, 25, 27, 32, 49, 64, 64, 64, 81, 81, 121, 125, }; 12, 8, 20, 18, 16, 42, 56, 48, 32, 72, 64, 110, 100, 64, < 20, threeflds[n, Extract[R, t], Extract[S, t]]; t++]

The lists R and S (respectively) contain the first 19 smallest cases of the number of elements in a local ring and the number of non-zero-divisors of that ring. Hence, this series of commands checks to see if there is any way that n can be the number of zero-divisors in a ring that is a product of a local ring and three fields. As above, the lists R and S have been shortened to save space in this article. The lists R and S can be correspondingly increased as the value of n increases,

12

Shane P. Redmond

Acknowledgements The author would like to thank David F. Anderson of the University of Tennessee-Knoxville for inspiring this paper by posing the original question. The topic came up in conversations about the zero-divisor graph of a commutative ring. Given a commutative ring R with identity, define a graph, Γ(R), whose vertices are the non-zero zero-divisors of R with vertices x and y adjacent if and only if xy = 0. The original question asked was if there was always a commutative ring R with identity such that Γ(R) has n vertices for any positive integer n? This article demonstrates there is no zero-divisor graph of a commutative ring with identity that has 1209 or 3341 or 5465 vertices. For an introduction to the zero-divisor graph of a commutative ring, see [1]. Zero-divisor graphs have also been extended to other algebraic structures in several papers, such as [2] and [8]. The author would also like to thank the members of the Algebra Seminar at Southeastern Louisiana University for their many constructive comments and their careful review of this article.

References [1] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), 434–447. [2] F. DeMeyer, T. McKenzie, and K. Schneider, The zero-divisor graph of a commutative semigroup, Semigroup Forum 65 (2002), no. 2, 206–214. [3] N. Ganesan, Properties of rings with a finite number of zero divisors, Math. Ann. 157 (1964), 215–218. [4] R. Gilmer, Zero divisors in commutative rings, Amer. Math. Monthly 93 (1986), no. 5, 382–387. [5] F. Luca, S. Pantelimon, Fibonacci numbers that are not sums of two prime powers, Proc. Amer. Math. Soc. 133 (2005), no. 7, 1887–1890. [6] R. Raghavendran, Finite associative rings, Composito Mathematica 21 (1969), no. 2, 195–229. [7] S. P. Redmond, Addendum to “Counting zero-divisors” [webpage], http://people.eku.edu/redmonds/addendumtocountingzerodivisors.htm (2008). [8] S. P. Redmond, Structure in the zero-divisor graph of a non-commutative ring, Houston J. Math. 30 (2004), no. 2, 345–355. [9] J. Richstein, Verifying the Goldbach conjecture up to 4 · 1014 , Math. Comp. 70 (2001), no. 236, 1745–1749. Reviewed by G. A. Cannon, L. Kabza, J. A. Lewallen, K. M. Neuerburg of Southeastern Louisiana University



Chapter 3

R ECENT P ROGRESS ON M INIMAL R ING E XTENSIONS AND R ELATED C ONCEPTS David E. Dobbs∗ Department of Mathematics, University of Tennessee, Knoxville

Abstract A unital extension A ⊂ B of (commutative unital) rings is said to be a minimal ring extension if there is no ring C such that A ⊂ C ⊂ B. The introductory section summarizes the background needed to study this concept. Section 2 describes several themes pertaining to the classification of minimal ring extensions, including the classification in 2006 by Dobbs-Shapiro of the minimal ring extensions of an arbitrary (integral) domain and various recent generalizations of the fact that any minimal domain extension of a non-field must be an overring. Section 3 describes some ways that minimal ring extensions arise naturally. These include chain-theoretic studies, such as the recent result by Coykendall-Dobbs that, despite the known case for finite chains, a domain R with a saturated chain consisting of integrally closed overrings need not be a Prufer domain; and FIP-theoretic studies generalizing the Primitive Element Theorem of field theory. Among the FIP-theoretic results are the generalization in 2003 by Dobbs-Mullins-Picavet-Picavet-LHermitte of the Primitive Element Theorem by characterizing, for each field K, the commutative unital K-algebras that have only finitely many unital K-subalgebras; and the recent characterization by Dobbs-Picavet-PicavetLHermitte of the rings having only finitely many unital subrings. Also surveyed in regard to the latter topic are recent results concerning whether composites of minimal ring extensions (resp., ring extensions satisfying FIP) R ⊂ S and R ⊂ T are such that S ⊂ ST is a minimal ring extension (resp., satisfies FIP). The final section summarizes recent results that use variants of the classical Kaplansky transform to characterize minimal ring extensions R ⊂ T in which R is integrally closed in T ; and, to illustrate similarities and differences between allied concepts, quotes several results from an unpublished dissertation of M. S. Gilbert on the generalization of minimal ring extensions that concerns ring extensions whose set of intermediate rings is linearly ordered by inclusion. Key Words: Minimal ring extension, crucial maximal ideal, prime ideal, overring, idealization, direct product, total quotient ring, von Neumann regular ring, re∗ E-mail


14

David E. Dobbs duced ring, chain, Prüfer domain, integrality, FIP property, Primitive Element Theorem, reduced ring, nilpotent element, subalgebra, characteristic, annihilator, composite, Noetherian domain, divided prime ideal, Kaplansky transform, valuation domain, λ-extension. AMS Subject Classification: Primary 13B99, 13A15; Secondary 13G05, 13B21, 13F05.

1 Introduction All rings and algebras considered below are commutative with identity; and all ring/algebra homomorphisms, subrings and modules are unital. IfA is a ring, then tq(A) denotes the total quotient ring of A; Spec(A) the set of all prime ideals of A; Max(A) the set of all maximal ideals of A; and dim(A) the (Krull) dimension of A. By an overring of A, we mean any ring B such that A ⊆ B ⊆ tq(A). If R is a proper subring of a ring T , then R ⊂ T is called a minimal ring extension if the inclusion map R ,→ T is a minimal ring homomorphism in the sense of [20], that is, if there is no ring S such that R ⊂ S ⊂ T . (As usual, ⊂ denotes proper inclusion.) By a minimal overring of R, we mean any overring of R which is a minimal ring extension of R. A key result of Ferrand-Olivier [20, Théorème 2.2 (i)] shows that ifR ⊂ T is a minimal ring extension, then there exists a (necessarily unique) M ∈ Max(R) such that, for each P ∈ Spec(R), the canonical injective ring homomorphism RP → TP := TR\P is an isomorphism if P 6= M and a minimal ring extension if P = M. It has becomes customary to call this M the crucial maximal ideal of the minimal ring extension R ⊂ T . It is easy to see via globalization that a ring extension R ⊂ T has a crucial maximal ideal (that is, an ideal M ∈ Max(R) with the above properties) if and only if R ⊂ T is a minimal ring extension. As the title indicates, this article is devoted to surveying recent work on minimal ring extensions and allied concepts. After the topic of minimal ring extensions was introduced by Ferrand-Olivier [20] in 1970, it was the subject of a number of unpublished dissertations, such as [26] and [8]. For several years afterwards, the study of minimal ring extensions emphasized base rings that were (commutative integral) domains and extension rings that were overrings; of particular interest along these lines is a result [31] on minimal overrings that is stated below as Theorem 2.1. However, the last decade has seen considerable progress in new directions in the study of minimal ring extensions and related notions, and those new results are the focus of this article. The organization and a quick overview of the results can be found in the Abstract. Besides the notation and terminology introduced above, we let X ,Y denote commuting algebraically independent indeterminates over the ambient coefficient ring(s); and we use standard notation for various conductors and annihilators. Any unexplained material is standard, as in [25], [24].

2 Classification Results Before classifying a ring-theoretic property such as “minimal ring extension”, one often needs to examine that property more closely in special contexts, for instance, that of do-

Recent Progress on Minimal Ring Extensions

15

mains. For this context, Sato-Yanagihara-Yoshida [31] found the following significant restriction. Theorem 2.1 ([31, page 1738, lines 8–13]). Let R ⊂ T be a minimal ring extension of domains such that R is not a field. Then T is (R-algebra isomorphic to) an overring of R. Moving beyond the context of domains, one can ask if a conclusion like that of Theorem 2.1 is still available. G. Picavet and M. Picavet-L’Hermitte [28] provided a positive answer with the following generalization of Theorem 2.1. Theorem 2.2 ([28, Proposition 3.9]). Let R ⊂ T be a minimal ring extension such that dim(tq(R)) = 0 and either (1) T is integral over R, each non-zero-divisor of R remains a non-zero-divisor in T , and the conductor (R : T ) is not a minimal prime ideal of R or (2) R ,→ T is a flat epimorphism. Then T is (R-algebra isomorphic to) an overring of R. The author noticed that one way to prove Theorem 2.1 (though this was not the approach in [31]) is to use the notion of the crucial maximal ideal of a minimal ring extension. By pursuing such an approach, we obtained the following generalization of Theorem 2.1. Theorem 2.3 ([11, Theorem 2.2]). Let R ⊂ T be a minimal ring extension such that (a) the set of zero-divisors of R is contained in some nonmaximal prime ideal of R and (b) each non-zero-divisor of R remains a non-zero-divisor in T . Then T is (R-algebra isomorphic to) an overring of R. In fact, Theorem 2.3 was obtained shortly after, but independently of, Theorem 2.2. That naturally raised the question of whether and how these two results might be related. In [11, Remark 2.4 (iv)], it was shown that Theorem 2.2 can be used to prove the case of Theorem 2.3 in which dim(tq(R)) = 0. Part of that explanation used a dichotomy that had been noted by Ferrand-Olivier [20, Théorème 2.2]: any minimal ring extension is either an integral ring extension or a flat epimorphism. This dichotomy may explain why two cases were considered in the formulation of Theorem 2.2; we will return to this dichotomy in discussing certain recent classification results at the start of Section 4. However, there is a sense in which Theorem 2.3 is stronger than Theorem 2.2. Indeed, it was shown in [11, Example 2.3] that if we have any preassigned integer n ≥ 2, then there exists a minimal ring extension R ⊂ T such that conditions (a) and (b) of Theorem 2.3 are satisfied, neither R nor T is a domain, dim(R) = n and dim(tq(R)) = n − 1. (In particular, dim(tq(R)) 6= 0.) It was further shown that it can be arranged in such an example that R is a chained ring. Note that any chained ring is a pseudo-valuation ring, or PVR, in the sense introduced by D. F. Anderson, Badawi and the author in [4]. PVRs generalize the pseudovaluation domains, or PVDs, introduced by Hedstrom-Houston [23]. It is interesting to note that the construction in [11, Example 2.3] depended on pullback-theoretic ideas from a second paper on PVRs [2, Example 3.16 (c)]. Despite Theorem 2.1 (and the generalizations of it that were given in Theorems 2.2 and 2.3), not all minimal ring extensions of a domain R are overrings of R (up to isomorphism). In fact, Ferrand-Olivier [20] gave the following classification of the minimal ring extensions of a field K. Note that extensions of the kind that are described in parts (b) or (c) of Theorem 2.4 are not (isomorphic to) overrings of K.

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Theorem 2.4 ([20, Lemme 1.2]). Let K be a field and T a nonzero commutative unital K-algebra, viewed as a ring extension of K. Then K ⊆ T is a minimal ring extension if and only if (exactly) one of the following three conditions holds: (a) T is a field which is a minimal field extension of K; (b) T is K-algebra isomorphic to K × K, which is viewed as a K-algebra via the diagonal structure map K → K × K, a 7→ (a, a); (c) T is K-algebra isomorphic to K[X ]/(X 2 ), the ring of dual numbers over K. Moving beyond the context of Theorem 2.4, one can ask whether it is possible to classify the minimal ring extensions of an arbitrary domain. The answer will be given in Theorem 2.6. First, we would like to note that not all domains admit minimal ring extensions that are domains. Indeed, [31, Theorem 8] gives a sufficient condition for a Noetherian domain which contains an infinite field to fail to have a minimal overring. Thus, it was imperative to know what kinds of minimal ring extensions (if any) an arbitrary domaincan have. Along these lines, we will record the fact that every domain R has a minimal ring extension which is not (isomorphic to) an overring of R. In fact, Theorem 2.5 will not even require the base ring R to be a domain. The key construction is that of idealization, for which background may be found in [27] and [24]. Recall that if R is a ring and E is an R-module, then the idealization T = R(+)E is the R-algebra which has the R-module structure of R ⊕ E and the multiplication given by (r1 , e1 )(r2 , e2 ) := (r1 r2 , r1 e2 + r2 e1 ), for all r1 , r2 ∈ R and e1 , e2 ∈ E. Note that R is viewed as a subring of T via the injective (unital) ring homomorphism R → R(+)E, r 7→ (r, 0). Theorem 2.5 ([10, Remark 2.8 and Corollary 2.5]). Every nonzero ring R has a minimal ring extension. In fact, if E is a nonzero R-module, view the idealization R(+)E as a ring extension of R as above. Then R(+)E is not R-algebra isomorphic to an overring of R. Moreover, R(+)E is a minimal ring extension of R if and only if E is a simple R-module (i.e., if and only if E ∼ = R/M for some maximal ideal M of R). [10] contained two proofs of the important “Moreover” assertion in Theorem 2.5: see [10, Theorem 2.4 and Remark 2.9]. The second of these proofs also established the following useful fact: if E is an R-module, then the R-subalgebras C of R(+)E are in one-to-one correspondence with the R-submodules F of E via F 7→ C = R(+)F. It is no exaggeration to say that the topic of overrings preoccupied the literature on the minimal ring extensions of domains for some time, probably due in part to the recognized importance of Theorem 2.1. However, it was known that not all domains admit minimal ring extensions that are domains; indeed, Sato-Yanagihara-Yoshida themselves [31, Theorem 8] gave a sufficient condition for a Noetherian domain which contains an infinite field to fail to have a minimal overring. (Moreover, if the base domain is an algebraically closed field K, one can see via Theorem 2.4 that no minimal ring extension of K is a field.) The main effect of Theorem 2.5 was to divert attention away from the topic of overrings. The question then became: besides overrings and idealizations, what else must be considered if one is to produce a generalization of Theorem 2.4 that would be valid for base rings that are arbitrary domains? Shapiro and the author [17] found a complete answer by proving the classification result in Theorem 2.6. In retrospect, the formulation of this result is motivated by the parenthetical phrase at the end of Theorem 2.5. Indeed, if K is a field, we note that


17

M := 0 is the only maximal ideal of K, and the K-algebras appearing in the final two parts of Theorem 2.4 can be viewed up to isomo! rphism as K × K ∼ = = K × K/M and K[X ]/(X 2 ) ∼ ∼ K(+)K = K(+)K/M. This observation will find a natural home toward the end of the statement of Theorem 2.6. Recall that a ring is said to be reduced if it has no nonzero nilpotent elements. Theorem 2.6 ([17, Theorem 2.7 and Remark 2.8 (a)]). Let R be a domain. Then the minimal ring extensions S of R are, up to R-algebra isomorphism, of the following three types: (i) Domains S that contain R and are minimal ring extensions of R; (ii) For each maximal ideal M of R, the idealization R(+)R/M; (iii) For each maximal ideal M of R, the ring R × R/M. Moreover, if M and N are distinct maximal ideals of R, then R(+)R/M is not R-algebra isomorphic to R(+)R/N and R × R/M is not R-algebra isomorphic to R × R/N. Up to R-algebra isomorphism, the rings S in (ii) are precisely the non-reduced minimal ring extensions of R and the rings in (iii) are precisely the reduced non-domain minimal ring extensions of R. If R is not a field, then up to R-algebra isomorphism, the rings in (i) are the minimal overrings of R (that is, the minimal ring extensions of R that are overrings of R). Also, if R is not a field, then distinct minimal overrings of R within the same quotient field of R cannot be isomorphic as R-algebras. If R is a field K, then up to K-algebra isomorphism, the rings in (i) are the minimal field extensions of K, the unique ring in (ii) is K[X ]/(X 2 ), and the ring R × R/M in (iii) is K × K. It was clear from [17] that rings S of the kind described by the idealizations and direct products in conditions (ii) and (iii) of Theorem 2.6 are minimal ring extensions of the base ring R even if R is not a domain. It was also clear that the assertions concerning nonisomorphisms in the second part of Theorem 2.6 carry over to arbitrary base rings. Then the question became: since a classification of the minimal ring extensions of an arbitrary base ring does not seem within reach at this time, can one generalize Theorem 2.6 to some natural class of base rings that properly encompasses the class of domains? After some time, Shapiro and the author [18] answered this question by proving the following result. It is clear that Theorem 2.7 generalizes the main contribution in Theorem 2.6, namely, the case where the base domain R is not a field. Theorem 2.7 ([18, Corollary 2.5]). Let R be a ring such that Q, the total quotient ring of R, is a von Neumann regular ring and no maximal ideal of R is a minimal prime ideal of R. Then the minimal ring extensions of R are, up to R-algebra isomorphism, of the following three types: (i) Minimal overrings of R, that is, R-subalgebras of Q that are minimal ring extensions of R; (ii) For each maximal ideal M of R, the idealization R(+)R/M; (iii) For each maximal ideal M of R, the ring R × R/M. Moreover, the above listing is a classification, in the following sense. The above types (i), (ii), (iii) of R-algebras do not overlap; if M and N are distinct maximal ideals of R, then R(+)R/M is not R-algebra isomorphic to R(+)R/N and R × R/M is not R-algebra isomorphic to R × R/N; and if A and B are isomorphic R-subalgebras of Q, then A = B. There was a natural reason to seek a generalization of Theorem 2.6 in terms of the kind of total quotient ring that was considered in Theorem 2.7. To wit: a ring A is a von

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Neumann regular ring if and only if AP is a field for each prime (resp., maximal) ideal P of A. Another formulation of this fact is perhaps better known: a ring A is is a von Neumann regular ring if and only if A is reduced and dim(A) = 0 (cf. [25, Exercise 22, page 64]). Thus, one way that Theorem 2.7 generalized Theorem 2.6 was to replace the requirement that the total quotient ring of the base ring be a field with the requirement that this total quotient ring be a von Neumann regular ring. Note that one of the assumptions in Theorem 2.7 cannot be deleted. Indeed, if we remove the hypothesis that no maximal ideal of the base ring R is a minimal prime ideal of R, then [18, Examples 2.6 and 2.7] show how to construct examples of a base ring R having a von Neumann regular total quotient ring such thatR has a minimal ring extension S that is not R-algebra isomorphic to a ring of type (i), (ii), or (iii) from the statement of Theorem 2.7. Perhaps the easiest such example proceeds by letting K1 , K2 be fields and K1 ⊂ L a minimal field extension, and then putting R := K1 × K2 and S := L × K2 . More generally, if K1 , . . . , Kn is a finite list of fields, it is straightforward to use a similar construction to describe, up to isomorphism, the minimal ring extensions ofK1 × · · · × Kn . In detail, consider direct products of the form K1 × · · · Ki−1 × Ti × Ki+1 × · · · × Kn , where Ti is am minimal ring extension of Ki (as catalogued in Theorem 2.4). (For a proof, use the well known fact that if A and B are rings, then the category of A × B-modules is equivalent to the product of the category of A-modules with the category of B-modules. Another use of this fact will be mentioned following Theorem 3.4.) This is somewhat encouraging, since any ring of the form K1 × · · · × Kn (where each Ki is a field) is a von Neumann regular ring. Nevertheless, we do not know of any classification of the minimal ring extensions of an arbitrary von Neumann regular ring. Note that Theorem 2.7 does not settle this question, for the following reason: if R is a von Neumann regular ring, then it is true that R = tq(R), but each maximal ideal of R is a minimal prime ideal of R. Although a classification of the minimal ring extensions of a von Neumann regular ring does not seem within reach at present, the following result of Shapiro and the author [18] explains how to reduce the study of minimal ring extensions to the case of extensions of reduced rings. First, reduced rings. If √ √ we need the following background about associated A is a ring, we let A denote the nilradical of A and Ared := A/ A the associated reduced ring of A. If A ⊆ B are rings, then the canonical ring homomorphism Ared → Bred is an injection by means of which we view Ared as a subring of Bred . Theorem 2.8 ([18, Theorem 2.1]). Let R ⊂ T be rings. Then T is a minimal ring extension of R if and √ if one of the following two conditions holds: √ only ring extension; (i) R = T and Rred ⊂ Tred is√a minimal √ (ii) There exists an element a ∈ T \ R such that T = R[a], a2 ∈ (R : T ), and (R : T ) ∈ Max(R). √ √ Moreover, if (ii) holds, then R 6= T , Rred = Tred , and (R :R a) = (R : T ). The situation described in condition (ii) of Theorem 2.8 is rather well understood. Indeed, let R ⊂ T be rings. The requirement that T = R[a] for some element a ∈ T such that a2 ∈ (R : T ) (which is part of condition (ii) in Theorem 2.8) implies thatT = R +Rt for some t ∈ T (namely, t = a). Ring extensions R ⊂ T satisfying the latter condition were studied in the unpublished doctoral dissertation of M. S. Gilbert, who showed in [21, Proposition 4.12] that any such data lead to an isomorphism R/(R : T ) → T /R of R-modules, given by


19

r + (R : T ) 7→ rt + R for all r ∈ R; that each R-submodule contained between R and T is a ring; and that one thereby infers an order-isomorphism between the set of rings intermediate between R and T and the set of ideals of R which contain (R : T ). Additional material from [21] will be given below as the major part of Section 4.

3

On FIP and Composites

It seems reasonable to ask for ways that minimal ring extensions arise naturally. Proposition 3.1 provides one such way and leads naturally to a discussion of some classes of domains that are characterized by the behavior of certain types of maximal chains of overrings. Then we turn to the major portion of this section, which is devoted to the FIP property (whose definition is recalled below) and, in particular, to the way in which minimal ring extensions play the role of the fundamental building blocks in maximal chains ofR-subalgebras of ring extensions R ⊂ T that satisfy the FIP property where the base ring R is a (commutative) semisimple ring. The proof of the next result uses a technique that can be applied to study chains in many algebraic settings. In fact, the proof of Proposition 3.1 is plainly an adaptation of a proof of Kaplansky for a result about proper inclusions of prime ideals [25, Theorem 11]. Theorem 3.1. Let R ⊂ T be an extension of distinct rings. Then there exists a minimal ring extension A ⊂ B such that R ⊆ A ⊂ B ⊆ T . Proof. The set of chains of R-subalgebras of T forms a partially ordered set under inclusion. By applying Zorn’s Lemma to this set, one can produce a maximal such chain, say C . Pick any element t ∈ T \ R. Consider the R-subalgebras of T defined by A := ∪{E ∈ C | t 6∈ E} and B := ∩{E ∈ C | t ∈ E}. Note that the union (resp., intersection) that is involved in the definition of A (resp., B) is not empty since t 6∈ R and R ∈ C (resp., t ∈ T and T ∈ C ) by the maximality of C . As t ∈ B \ A, it is clear that R ⊆ A ⊂ B ⊆ T . It remains only to prove that A ⊂ B is a minimal ring extension. Suppose not. Then there exists a ring D such that A ⊂ D ⊂ B. It is easy to see that D is comparable under inclusion with each E ∈ C (as either E ⊆ A or E ⊇ B). Therefore, C ∪ {D} is a chain, contradicting the maximality of C , to complete the proof. Given any ring extension R ⊆ T , the above proof leads one to consider the maximal chains of R-subalgebras of T . If R is a domain with quotient field K = T , this means considering the maximal chains of overrings of R. Many classical results may be reformulated from this point of view. Consider, for instance, the result of Davis [7, Theorem 1] that a domain R (with quotient field K) is a Prüfer domain if (and only if) each overring of R is integrally closed (in K). By using Zorn’s Lemma to produce maximal chains in the spirit of the proof of Proposition 3.1, this result of Davis may be restated as follows. If R is a domain, then: R is a Prüfer domain ⇔ if C is any maximal chain of overrings of R, then each D ∈ C is integrally closed. It seems natural to ask if there is a variant of this characterization of Prüfer domains in which the universal quantification (on C ) is replaced by an existential quantification. As Theorem 3.2 records, the author used minimal ring extensions to find a positive answer to this question for domainsR with only finitely many prime

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ideals. However, some years later, a collaboration with Coykendall [6] produced a negative answer in general: see Theorem 3.3 and the comments that follow it. Theorem 3.2 ([9, Corollary 2.5]). Let R be a domain with quotient field K. Then the following conditions are equivalent: (1) There exists a finite maximal chain R = R0 ⊂ · · · ⊂ Rn = K of overrings of R in which each Ri is integrally closed (in K); (2) R is a Prüfer domain with only finitely many prime ideals; (3) If C is any maximal chain of overrings of R, then C is finite and each D ∈ C is integrally closed (in K). Moreover, if the above equivalent conditions hold and n, {R j } are as in condition (1), then Ri is a Prüfer domain with exactly n + 1 − i prime ideals, for each i = 0, 1, . . . , n. One should note that condition (3) in Theorem 3.2 did not appear in the statement of [9, Corollary 2.5]. However, its inclusion is valid here, as can be seen by combining the rest of Theorem 3.2 (which did appear in [9, Corollary 2.5]) with the above reformulation of Theorem 3.1. For domains with only finitely many prime ideals, the above form of Theorem 3.2 provides the sort of equivalence of universal and existential quantifications that we had sought. To treat the general case, the following definition will be helpful. LetR be a domain with quotient field K, and let C be a nonempty chain of overrings of R; let A (resp., B) be the intersection (resp., union) of the members of C . (Hence, A ⊆ D ⊆ B for each D ∈ C .) We say that C is a saturated chains of overrings of R if C is maximal as a chain of rings D such that A ⊆ D ⊆ B. Note that any maximal chain C of overrings of R is a saturated chain of overrings of R and, moreover, that R, K ∈ C . However, the converse is false, as R and K need not be members of a given saturated chain of overrings ofR; indeed, if D is an overring of R, then the singleton set {D} is trivially a saturated chain of overrings of R. The next result gives some equivalent conditions on a domain R which imply that R has a maximal chain C of overrings such that each D ∈ C is integrally closed. (To see this, consider condition (2) in Theorem 3.3 and recall that each Pr¨ ufer domain is integrally closed.) Theorem 3.3 ([6, Theorem 2.3]). Let R be a domain with quotient field K. Then the following conditions are equivalent: (1) R is the intersection of some chain of Prüfer overrings of R; (2) R is integrally closed and there is a maximal chain C of overrings of R such that each D ∈ C \ {R} is a Prüfer domain; (3) R is the intersection of some saturated chain of Prüfer overrings of R (where we do not necessarily assume that R is a member of the chain). The question remains: if a domain R satisfies the equivalent conditions in Theorem 3.3, must R be a Prüfer domain? Despite expectations that may be raised by Theorem 3.2, the answer to this question is negative. Indeed, it was shown in [6, Corollary 2.4] that if R is the intersection of a countable chain of valuation domains, then R satisfies the equivalent conditions in Theorem 3.3. Thus, each Krull domain with only countably many height 1 prime ideals satisfies the the equivalent conditions in Theorem 3.3 [6, Corollary 2.5]. Perhaps the easiest example of such a domain R which is not a Prüfer domain is R := Q[X ,Y ] [6, Example 2.6]. More spectacularly, it was shown in [6, Example 2.7] that


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for each d, 1 ≤ d ≤ ∞, there exists a d-dimensional integrally closed non-Pr¨ ufer non-Krull domain R which satisfies the equivalent conditions in Theorem 3.3. Thus, we have recently found many new examples of domains R such that some maximal chain of overrings of R consists only of integrally closed domains. However, our understanding has not yet reached the point of characterizing the class of Krull domains with this property (or, for that matter, the possibly smaller class of Krull domains that satisfy the equivalent conditions in Theorem 3.3). For a list of other related open questions, see [6, Remark 2.8 (c)]. We turn next to the main focus of this section. A ring extension R ⊆ T is said to satisfy FIP (for the “finitely many intermediate rings property”) if there are only finitely many rings D such that R ⊆ D ⊆ T . This concept is relevant because it is obvious that any minimal ring extension satisfies FIP. Moreover, in the spirit of Theorem 3.2, we note that if R ⊆ T satisfies FIP, then any maximal chain C of R-subalgebras of T must take the form C = {Ri | i = 0, . . . , n} for some non-negative integer n, where R = R0 ⊂ . . . ⊂ Rn = T and Ri ⊂ Ri+1 is a minimal ring extension for each i = 0, . . . , n − 1. The pedigree of the FIP concept is stalwart and algebraic, for the classical Primitive Element Theorem can be reformulated to state that if K ⊆ L are fields, then K ⊆ L satisfies FIP if and only if there exists an element u ∈ L such that L = K(u) and u is algebraic over K. Indeed, one reason that the FIP concept was introduced by D. D. Anderson, Mullins and the author in [1] was to seek an algebra-theoretic generalization of the Primitive Element Theorem. Theorem 3.4 presents a complete result along these lines which was found subsequently by G. Picavet, M. Picavet-L’Hermitte, Mullins and the author [13]. The special case of Theorem 3.4 in which K is a perfect field was given earlier in [1, Theorem 3.8]. Theorem 3.4 ([13, Theorem III.5]). Let K ⊆ T be a ring extension, where K is a field. Then K ⊆ T satisfies FIP if and only if at least one of the following four conditions holds: (i) K is finite and T is a finite-dimensional K-vector space; (ii) K is infinite, T is a reduced ring, and T = K[α] for some α ∈ T which is algebraic over K; (iii) K is infinite and T = K[α] for some α ∈ T which satisfies α3 = 0; (iv) K is infinite and T = K[α] × K2 × · · · × Kn , where α ∈ T satisfies α3 = 0, and for each i = 2, . . . , n, K ⊆ Ki is a field extension which has FIP. One should point out that Theorem 3.4 is only a special case of [13, Theorem III.5]. Indeed, [13, Theorem III.5] characterizes the ring extensions R ⊆ T that satisfy FIP in case the base ring R is semisimple (and commutative), i.e., in case R ∼ = K1 × · · · × Kn for some finite list of fields K1 , . . . , Kn . Then T is isomorphic as an R-algebra to T1 × · · · × Tn for suitable Ki -algebras Ti , the point being that the category of (commutative algebras) over a ring product A × B is equivalent to the product of the categories of A-algebras and of B-algebras. Then, as shown in [13, Proposition III.4], the proof of [13, Theorem III.5] is reduced to proving Theorem 3.4 for each Ki ⊆ Ti , i.e., for the case of a base ring which is a field. At the beginning of this section, we mentioned that minimal ring extensions could be viewed as playing “the role of the fundamental building blocks in maximal chains” of intermediate rings of ring extensions that satisfy FIP. In conjunction with the proof of the above-mentioned [13, Theorem III.5], this fact has been proven when the base ring is an infinite field. It is stated next as Theorem 3.5. Note that this result makes contact with the

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classification by Ferrand-Olivier of the minimal ring extensions of a field that was given in Theorem 2.4. Theorem 3.5 ([13, Proposition III.10]). Let K be an infinite field and K ⊆ T a ring extension which satisfies FIP. Let T 0 and T 00 be K-subalgebras of T such that T 0 ⊂ T 00 is a minimal ring extension. The case T 0 = K is handled by Theorem 2.4, and so we can assume K ⊂ T 0 . Using product categories, identify T 0 = K1 × · · · × Kn , where n ≥ 1, Ki is a field if n, i ≥ 2 and either (1) K1 is a field or (2) K1 is of the form K1 = K[α] where α3 = 0 6= α. Then T 00 is of the form T 00 = K10 ×· · · ×Kn0 where there exists i ∈ {1, . . . , n} such that K j = K 0j for each j 6= i and Ki0 is of the following form: (a) If (1) holds, then one of the following three conditions holds: (α) Ki0 is a field and Ki ,→ Ki0 is a minimal field extension; (β) Ki0 ∼ = Ki × Ki; 0 (γ) Ki = Ki [β] where β2 = 0. (b) If i 6= 1 and (2) holds, then one of the following two conditions holds: (α) Ki0 is a field and Ki ,→ Ki0 is a minimal field extension; (β) Ki0 ∼ = Ki × Ki; (c) If i = 1, (2) holds and α2 = 0, then either K10 ∼ = K1 ×K or K10 is of the form K10 = K[β], 3 2 where β = 0 and β = aα for some a ∈ K \ {0}. (d) If i = 1, (2) holds and α3 = 0 with α2 6= 0, then K10 ∼ = K1 × K. Besides proving the case of Theorem 3.4 for a perfect base field, [1] also studied the FIP property for ring extensions having arbitrary base rings. Some results along those lines are collected next. Theorem 3.6. Let R ⊆ T be a ring extension that satisfies FIP. Then: (a) ([1, Proposition 2.2 (a), (b)]) Each element of T is the root of a suitable polynomial in R[X ] with a unit coefficient; in particular, T is algebraic over R. Moreover, T is a finitetype R-algebra. (b) ([1, Theorem 2.4 (a)]) Suppose, in addition, that T is a domain. Then either R and T are each fields or T is (R-algebra isomorphic to) an overring of R. Theorem 3.6 (b) illustrates that, as was the case for the study of minimal ring extensions that was reported in Section 2, the study of FIP has paid special attention to the context of overrings. As another result in this vein, we mention the following fact that has received two proofs [1, Corollary 2.3 and Remark, page 606]: if R ⊆ T is an extension of domains that satisfies FIP such that R is integrally closed in T , then T is (R-algebra isomorphic to) an overring of R. We next announce a recent generalization of Theorem 3.6 (b). Theorem 3.7 ([12, Theorem 2.1]). Let R ⊂ T be domains, with corresponding quotient fields K ⊂ L. If R 6= K, then there exists a denumerable (strictly) descending chain of R-subalgebras of T . Examples in [12, Remark 2.2] show that the assertion in Theorem 3.7 fails if one deletes either of the assumptions that R 6= K or K 6= L or strengthens the “denumerable” part of the conclusion. A partial generalization of Theorem 3.6 is given in [12, Proposition 2.3] for rings with nontrivial zero-divisors. That partial generalization, like most of the proof of


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Theorem 3.7, is couched in terms of an integral ring extension R ⊂ T . Indeed, the proof of Theorem 3.7 reduced quickly to the case where T is integral over R and, from there, to the case where T is a finitely generated R-module. In that regard, we next announce a related result. In recent work in collaboration with G. Picavet and M. Picavet-L’Hermitte, the author has found the following companion for Theorem 3.7. Theorem 3.8 ([16]). Let R ⊂ T be a module-finite ring extension such that the conductor (R : T ) = 0. If there exists a nonzero element of T which is neither a unit of T nor a zerodivisor of T , then there exists a denumerable (strictly) descending chain of R-subalgebras of T . We turn next to a property related to FIP. Let R be a ring, and let F denote the prime (sub)ring of R. It will be convenient to say that a ring R satisfies FSP (for the “finitely many subrings property”) if R has only finitely many (unital) subrings, i.e., if F ⊆ R satisfies FIP. In Theorem 3.11, we will determine all the rings that satisfy FSP. This task should not be confused with the far easier problem of determining all the rings possibly without identity having only finitely many subrings possibly without identity. The latter question was settled more than forty years ago by Rosenfeld [30], who found the answer to the latter question to be the class of finite rings possibly without identity. According to Theorem 3.9, the characterization of the rings of positive characteristic that satisfy FSP has the same flavor as the result of Rosenfeld. Theorem 3.9 ([13, Proposition V.1]). Let R be a ring of positive characteristic. Then R satisfies FSP (if and) only if R is finite. It was much more difficult to obtain the characteristic 0 analogue of Theorem 3.9. Motivated in part by the Primitive Element Theorem, Mullins, M. Picavet-L’Hermitte and the author first settled the characterization of the rings that satisfy FSP for the class of singly generated (commutative unital) rings. This characterization was the main result of [14], and it is stated in Theorem 3.10. As a matter of definition, a (necessarily commutative) ring is called singly generated if it is generated by a set of the form {0, 1, s}, for some element s. Theorem 3.10 ([14, Theorem 3.12]). Let R be a singly generated unital ring. Then R satisfies FSP if and only if (exactly) one of the following four conditions holds: (1) R is a finite ring; (2) R ∼ = Z[ 1n ] for some positive integer n; (3) R is a module-finite ring extension of Z which is not a domain and the conductor (Z : R) := {r ∈ R | rR ⊆ Z} is nonzero; (4) R = Z[t] ⊃ Z is not integral over Z, R is not a domain, there exist integers a, b ≥ 2 such that at = b, and the greatest common divisor d := (a, b) of the minimal such a and a t is integral over Z, and the corresponding b satisfies the following conditions: d > 1, (a,b) there does not exist a prime number p such that ker(ϕ) ⊆ pZ[X], where ϕ is the (unital) ring homomorphism Z[X ] → R sending X to t. Most of the effort in proving Theorem 3.10 had to do with developing what became its condition (4). To appreciate the subtlety of that condition, we next mention two relevant examples. Consider the rings R1 := Z[X ]/(4X − 2, 2X 2 − X ) and R2 := Z[X ]/(4X − 2). If

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we take t to be the coset represented by X , then each of these rings fits into the general context that is pertinent here, with a = 4, b = 2, d = 2, α := da = 2, and αt = 2t integral over Z (in fact, idempotent). However, R1 satisfies FSP while R2 does not satisfy FSP. Indeed, the conclusion for R2 follows from (Theorem 3.10 and) the fact that its ker(ϕ) ⊆ 2Z[X]; for the conclusion about R1 , observe similarly that there is no prime number p such that (4X − 2, 2X 2 − X ) ⊆ pZ[X ]. At the time that Theorem 3.10 was published, it was known that not every ring that satisfies FSP has to be singly generated: cf. [14, Example 3.13 (a), Remark 3.14 (c)]. Therefore, the question remained: what (apart from the singly generated case) are the rings that satisfy FSP? One line of attack was suggested by the fact (an easy consequence of Theorem 3.6 (a)) that any ring that satisfies FSP must be of finite type over its prime ring. After much technical work (such as Theorem 3.19 below) involving the crucial maximal ideals of some relevant minimal ring extensions, the characterization of the rings satisfying FSP was reduced to the singly generated case (and hence settled because of Theorem 3.10) by G. Picavet, M. Picavet-L’Hermitte and the author. Their result is stated next. Theorem 3.11 ([15, Theorem 3.20]). Let R be a ring with prime ring F. Then the following conditions are equivalent: (1) R satisfies FSP; (2) R is a finite-type F-algebra and whenever {t1 , . . . ,tn } is a finite set such that R = F[t1 , . . . ,tn ], then F[ti ] satisfies FSP for each i; (3) There exists a finite set {t1 , . . . ,tn } such that R = F[t1 , . . . ,tn ] and F[ti ] satisfies FSP for each i; (4) Either (i) R is a finite-type ring extension of Z and whenever {t1 , . . . ,tn } is a finite set such that R = Z[t1 , . . . ,tn ], then Z[ti ] satisfies FSP for each i or (ii) R is finite; (5) Either (λ) R is a ring extension of Z such that there exists a finite set {t1 , . . . ,tn } for which R = Z[t1 , . . . ,tn ] and Z[ti ] satisfies FSP for each i or (µ) R is finite. Recall from Theorems 3.4 and 3.5 that FIP-related studies often encounter ring direct products. In this regard, the work in [15] also managed to solve a related problem, namely, the question of determining all the ring direct products that satisfy FSP. The complete answer to this question is obtained by combining the next result with the characteristic 0 case of Theorem 3.11 (when the latter is suitably buttressed by the characteristic 0 case of Theorem 3.10). To further motivate the implications (1) ⇒ (2) and (1) ⇒ (3) in Theorem 3.12, it is easy to see that the class of rings that satisfy FSP is stable under ring homomorphic images [14, Remark 3.14 (b)]. Theorem 3.12 ([15, Theorem 2.3]). Let I be a nonempty index set and Ri a nonzero ring for each i ∈ I. Consider the ring direct product A := ∏i∈I Ri . Then the following conditions are equivalent: (1) A satisfies FSP; (2) I is finite, R j is finite (and hence is of positive characteristic and satisfies FSP) for all but at most one index j ∈ I, and if i ∈ I is such that Ri has characteristic 0, then Ri satisfies FSP; (3) I is finite, R j satisfies FSP for each j ∈ I, and there is at most one i ∈ I such that Ri has characteristic 0.


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Some of the “technical work” alluded to prior to the statement of Theorem 3.11 depended on studying the behavior of composites of minimal ring extensions. In the next six results, we collect some key facts about such composites that are drawn from a recent paper of Shapiro and the author [19]. We begin with a cautionary note in Example 3.13, showing that minimal ring extensions of a common base ring need not have a composite. Example 3.13 ([19, Example 2.1]). There exist minimal ring extensions R ⊂ S and R ⊂ T for which there does not exist a (commutative unital) R-algebra containing both S and T as R-subalgebras. To produce such data, it is enough to take a domain R with a minimal overring S such that the crucial maximal ideal for the minimal ring extension R ⊂ S is N and there exists a nonzero element n ∈ N such that n−1 ∈ S (we note below one way to construct such a Bézout domain R having Krull dimension 1 and such an overring S); next, take T to be another minimal ring extension of R having crucial maximal ideal N such that T is not R-algebra isomorphic to S (for instance, by Theorem 2.6, take T to be either the idealization R(+)R/N or the direct product R ×R/N). One way to produce R and S as above is the following. By taking R to be the intersection of two incomparable valuation domains of Krull dimension 1 having the same quotient field, we obtain R as a one-dimensional Bézout domain with exactly two maximal ideals, say M and N; then S := RM is a minimal ring extension of R, the crucial maximal ideal for R ⊂ S is N, and any element n ∈ N \ M satisfies n−1 ∈ S. Despite Example 3.13, many distinct minimal ring extensions R ⊂ S and R ⊂ T do have a composite. Theorem 3.14 gives some field-theoretic sufficient conditions for the composite ST to be such that both S ⊂ ST and T ⊂ ST are again minimal ring extensions. The motivation for such a study comes from the standard isomorphisms theorems for modules. Indeed, if A is a (unital but not necessarily commutative) ring with (say, left) A-modules M and N such that both M ∩ N ⊂ M and M ∩ N ⊂ N are what may be termed “minimal module extensions” (in the sense that M/(M ∩ N) and N/(M ∩ N) are simple A-modules) and if the “module-theoretic composite” M + N exists (as an A-module), then M ⊆ M + N is a minimal module extension (since (M + N)/M ∼ = N/(M ∩ N) is a simple A-module) and, similarly, so is N ⊆ M + N. Theorem 3.14 ([19, Proposition 2.2]). Let K ⊂ F and K ⊂ L be distinct field extensions that are contained in some algebraic closure of K. Then: (a) If [F : K] and [L : K] are distinct prime numbers, then K ⊂ F, K ⊂ L, F ⊂ FL and L ⊂ FL are each minimal field extensions. (b) If K ⊂ F and K ⊂ L are minimal field extensions such that F and L are each splitting fields over K, then F ⊂ FL and L ⊂ FL are each minimal field extensions. Moving beyond the context of fields, one has a number of positive ring-theoretic results giving sufficient conditions for composites of minimal ring extensions to produce minimal ring extensions. These positive results are collected in Theorem 3.15. For part (c) of Theorem 3.15 and the subsequent material, we will need the following information and definitions. If B is an integral minimal ring extension of R with crucial maximal ideal M, then (by a standard isomorphism theorem and Theorem 2.4) B/M is isomorphic as an R/M-algebra to exactly one of the following three possibilities: (1) a minimal field extension of R/M, (2) R/M × R/M, (3) (R/M)[X ]/(X 2 ). It will be convenient to say that an integral minimal

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overring B of R (inside K = tq(R)) is of type 1, type 2, or type 3 according as to whether B satisfies condition (1), (2), or (3) above. Given two distinct integral minimal overrings S, T ⊂ K of R, we will say that S, T are a type (a, b) example if S is of type a and T is of type b, while R ⊂ S and R ⊂ T have the same crucial maximal ideal M. Occasionally, we will relax this terminology by not requiring the integral minimal ring extensionsS, T to be overrings of R. Theorem 3.15. (a) ([19, Theorem 2.5]) Let R ⊂ S and R ⊂ T be minimal ring extensions, with crucial maximal ideals M and N, respectively. If M 6= N and if the composite ST exists in some ring extension of R, then both S ⊂ ST and T ⊂ ST are minimal ring extensions. (b) ([19, Corollary 2.6]) Let R be a ring such that tq(R) is a von Neumann regular ring (for instance, take R to be a domain). Let R ⊂ S and R ⊂ T be distinct minimal ring extensions such that R is integrally closed in both S and T . If the composite ST exists in some ring extension of R, then both S ⊂ ST and T ⊂ ST are minimal ring extensions. (c) ([19, Proposition 3.4 (a), (b)]) Let R be any ring. Suppose that S, T are a type (1, b) example of integral minimal ring extensions of R, each with crucial maximal ideal M. Suppose also that the composite ST exists. If b ∈ {2, 3}, then ST is a minimal ring extension of S of type b. On the other hand, if b = 1 and, in addition, if S and T are overrings of R that are each contained in tq(R), then ST is a minimal ring extension of S if and only if S/M ⊂ (S/M)(T /M) is a minimal extension of fields. One question that may have been raised by the hypotheses in Theorem 3.15 (b) can be quickly answered, as follows. Theorem 3.16 ([19, Theorem 2.8]). Let R be a ring such that a total quotient ring of R is a von Neumann regular ring (for instance, take R to be a domain). Then there do not exist minimal overrings S and T of R (possibly inside different total quotient rings of R) such that the composite ST exists in some ring extension of R with R being integrally closed in S and T being integral over R. Despite the tone of Theorem 3.15, it is “usually” not the case that composable minimal ring extensions R ⊂ S and R ⊂ T lead to S ⊂ ST being a minimal ring extension. Example 3.17 states matters more precisely. Example 3.17 ([19, Proposition 3.4 (c), Example 3.8, Example 3.9]). Suppose that either a = b = 1 or (a, b) ∈ {2, 3} × {2, 3}. Then there exist a Noetherian domain R and distinct integral minimal overrings S, T of R that are inside the same quotient field of R, have the same crucial maximal ideal, and give an example of type (a, b) such that S ⊂ ST is not a minimal ring extension. Proof. (Sketch) We treat first the case a = b = 1. Let K be a field with algebraic closure K, and let F, L be distinct minimal field extensions of K inside K. Let V = K + M be a DVR with nonzero maximal ideal M (for instance, V = K[[X ]]). Then the domain R := K + M is Noetherian, as are its integral overrings S := F + M and T := L + M. Note that S, T (as overring extensions of R) give an example of type (1, 1). It follows from the second assertion in Theorem 3.15 (c) that S ⊂ ST = V is a minimal ring extension if and only if F ⊂ FL is a minimal field extension. Accordingly, it suffices to find K, F, L as above such


27

that F ⊂ FL is not a minimal field extension. The following example of such data was established in the proof of [19, Proposition 3.4 (c)]. Let p ≥ 5 be a prime number, ω a primitive pth root of unity (in the complex numbers) and α the real pth root of 2. Then the fields K := Q, F := Q(α) and L := Q(ωα) have the asserted properties. The other cases are handled by ad hoc constructions. Their favor is typified by the following construction, which works for the case a = 2, b = 3. Let F be an arbitrary field, with X and Y (as usual) algebraically independent indeterminates overF, and define R := F[{X n (X 2 − X ), X n(X 2 − X )Y, X sY 2 , X tY 3 | n, s,t ≥ 0}]. The fact that R is Noetherian can be seen most easily via Eakin’s Theorem (cf. [25, Exercise 15, page 54]), which applies since F[X ,Y ] is a module-finite ring extension of R. Note that X 6∈ R (for, otherwise, the substitution Y 7→ 0 would lead to X being the sum of a constant in F and a polynomial of degree at least 2, a contradiction). Also, Y 6∈ R (for, otherwise, the substitution X 7→ 1 would lead to Y ∈ F[Y 2 ,Y 3 ], a contradiction). Next, consider the subrings of F[X ,Y ] defined by S := R[X ] (= F[X ,Y 2 ,Y 3 , (X 2 − X )Y ]) and T := R[Y ]. It is proved in [19, Example 3.9] that the above definitions of R, S, T give an example of type (2, 3) with the asserted properties. We omit the details of that proof here, noting only that the verification that S, T are integral minimal overrings of R of the asserted types depends on Theorem 4.4 in the next section. In view of the negative tone of Example 3.17, one may well ask what can be concluded in general about the ring extension S ⊂ ST when one is given distinct composable minimal ring extensions R ⊂ S and R ⊂ T . Theorem 3.18 summarizes what is known in this regard. As usual, if E is a vector space over a field k, then dimk (E) denotes the k-vector space dimension of E. Theorem 3.18 ([19, Proposition 3.3]). Let R be a ring. Let R ⊂ S and R ⊂ T be distinct integral minimal ring extensions with the same crucial maximal ideal M. Suppose also that S and T are each R-algebra isomorphic to overrings of R (possibly inside different total quotient rings of R) and that S and T are not isomorphic as R-algebras. Suppose that the composite ST exists in some ring extension of R. Let k denote the field R/M. Put d := dimk (S/M) and n := dimk (T /M). Let C be any chain of rings contained between S and ST , and let D be any chain of rings contained between T and ST . Then: (a) |C | and |D | are each finite. In fact, |C | ≤ (n − 1)d + 1 and |D | ≤ (d − 1)n + 1. (b) Suppose also that S/M (resp., T /M) is a field. Then |C | ≤ n (resp., |D | ≤ d). (c) Suppose also that S/M and T /M are each fields and that S, T are each contained in K = tq(R). Then |C | ≤ n and |D | ≤ d. Moreover, if M ∈ Spec(M :K M), then |C | ≤ log2 (n) and |D | ≤ log2 (d). (d) If neither S/M nor T /M is a field, then |C | ≤ 3 and |D | ≤ 3. By sharpening the focus in Theorem 3.18 to the case of the base ring Z, one can get more precise information. Indeed, building on [19], the authors of [15] were able to obtain Theorem 3.19, which played a significant role in their proof of Theorem 3.11. Theorem 3.19 ([15, Proposition 3.5 (a)]). Let R = Z[t, s] be a two-generated ring extension of Z such that Z ⊂ Z[t] and Z ⊂ Z[s] are distinct minimal ring extensions with crucial

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maximal ideals pZ and qZ, respectively. Then: (a) R has FSP. (b) Suppose that the prime numbers p, q are unequal. Then each maximal chain of rings from Z to R consists of exactly three elements. Also, there exists x ∈ R such that Z[x] = R. (c) Suppose that the prime numbers p, q are equal. Then R is integral over Z. We can illustrate the conclusion in Theorem 3.19 (a) for the case p = q = 2 with an example that appeared in [14, Remark 3.14 (a)]. As usual, let X and Y be commuting algebraically independent indeterminates overZ. Consider R := Z[X ,Y ]/(X 2 , 2X ,Y 2 , 2Y ) = Z[x, y], where x and y are the cosets represented by X and Y , respectively. It was shown in [14] that Z ⊂ Z[x] =: T and Z ⊂ Z[y] =: S are each minimal ring extensions; and that (as predicted by Theorem 3.19 (a)) R = T S has FSP. Indeed, [14, Remark 3.14 (a)] gives the complete list of the 12 subrings of R. Some of the proofs leading to Theorem 3.19 could have been couched more generally by, for instance, replacing Z with an arbitrary principal ideal domain. However, apart from the need to be relevant for the characteristic 0 case of Theorem 3.11, there was another good reason for our focus on the base ring Z. Indeed, another example in [14, Remark 3.14 (a)] shows, despite expectations that may have been raised by Theorem 3.19, that there exist composable minimal ring extensions A ⊂ B and A ⊂ C such that A ⊂ BC does not have FIP. The details are given next. Example 3.20. There exist a field R and elements s,t such that the ring extension R ⊂ R[s,t] does not satisfy FIP even though each of the extensions R ⊆ R[s] and R ⊆ R[t] satisfies FIP. It can be further arranged that both R ⊆ R[s] and R ⊆ R[t] are minimal field extensions (and, hence, minimal ring extensions). Proof. Let R := F(X p ,Y p ), where F is the field F with p elements for some prime number p and X ,Y are algebraically independent indeterminates overF. Let s := X and t := Y . By a standard homework problem in a first graduate course on field theory,R ⊆ T := F(X ,Y ) = R[X ,Y ] = R(X ,Y ) is a p2 -dimensional field extension which does not satisfy FIP (by the Primitive Element Theorem, since each element in T has its p-th power in R). Note, however, that R ⊆ R[s] = R[X ] = R(X ) = F(X ,Y p ) and R ⊆ R[t] = R[Y ] = R(Y ) = F(X p ,Y ) each satisfy FIP, by the Primitive Element Theorem. Indeed, R ⊆ R[s] and R ⊆ R[t] are each minimal field extensions since [R[s] : R] = [R[t] : R] = p, a prime number. To close this summary of results concerning the FIP property, we note that the behavior of nilpotent elements has played an important role in studying FIP. This is perhaps already clear from conditions (iii) and (iv) in Theorem 3.4. In addition, an entire section of [13] was devoted to studying whether ring extensions of the form R ⊆ R[u] satisfy FIP in case the element u is nilpotent. We pause to collect a couple of statements from that section of [13]. The dichotomy that is described below in Theorem 3.21 (b) can be illustrated by the examples in [13, Example IV.4]. Theorem 3.21. (a) ([13, Corollary IV.3]) Let R be a residually finite domain which is not a field, and let u be a nilpotent element belonging to some ring extension of R. Then R ⊆ R[u] satisfies FIP if and only if (0 : u) 6= 0.


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(b) ([13, Proposition IV.6]) Let R be a domain which is not a field, and let u be a nilpotent element of nilpotency index 2 that belongs to some ring extension of R. Then R ⊆ R[u] satisfies FIP if and only if R/(0 : u) has only finitely many ideals. Nilpotent elements also played a crucial role in [14]. Indeed, enroute to Theorem 3.10, it was shown in [14, Proposition 3.11] that if R = Z[t] is a non-integral ring extension such that the element t satisfies at = b for some integers a 6= 0 and b, then R satisfies FSP if and only if Rred satisfies FSP. Before the earliest work on FIP, the role of nilpotent elements and their nilpotency index was recognized by M. S. Gilbert in studying a different generalization of the concept of “minimal ring extension”. One of Gilbert’s results in this vein will be stated in Theorem 4.13 below.

4

A Miscellany

This section concerns two themes. The first of these stems from the following dichotomy which was observed by Ferrand-Olivier [20]: if R ⊂ T is a minimal ring extension, then either R is integrally closed in T or T is integral over R. As evidence that these two cases are studied by quite different methods, we note that the role of the crucial maximal ideal was observed only for integral minimal ring extensions in the dissertation of Modica [26], while the subsequent dissertation of Déchene [8] recognized that crucial maximal ideals exist for both cases of the above dichotomy. In the spirit of Section 2, we next summarize some characterization results for these two cases, beginning with the case whereR is integrally closed in T . The first work on characterizing minimal ring extensions R ⊂ T in the relatively integrally closed case was done by Ayache [3], who showed that if a domain R is integrally closed but not a field, then each minimal overring of R is the Kaplansky transform of a maximal ideal that satisfies certain properties. Rather than state such characterization results from [3] more precisely, we note that they are all subsumed by Theorem 4.3 below. To prepare for this result, we begin in Theorem 4.1 with the case of a quasilocal base ring. Many of the ensuing statements need the following definitions. A prime idealP of a ring R is called (a) divided (prime ideal of R) if, for each ideal I of R, either I ⊂ P or P ⊆ I. By a regular element, we mean simply a non-zero-divisor. Theorem 4.1 ([18, Theorem 3.1]). Let (R, M) be a quasilocal ring. Then the following two conditions are equivalent: (1) There exists a minimal ring extension T of R such that R is integrally closed in T ; (2) There exists a divided prime ideal P of R such that R/P is a one-dimensional valuation domain and there exists an element u ∈ M \ P which is a regular element of R. Moreover, if the above conditions hold, then each element of M \P is a regular element of R, Rad(u) = M, and T ∼ = Ru is R-algebra isomorphic to an overring of R. In particular, = RP ∼ any two minimal ring extensions of R in which R is integrally closed must be isomorphic as R-algebras. Theorem 4.2 will collect a number of results that were proved in [18] as applications of Theorem 4.1. Note that Corollary 4.2 (c) generalizes a result of G. Picavet and M. PicavetL’Hermitte [28] on the minimal overrings of a local Noetherian ring.

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Corollary 4.2. (a) ([18, Corollary 3.2]) Let R be a quasilocal ring and Q a given total ring of quotients of R. If T is a minimal ring extension of R such that R is integrally closed in T , then there exists a unique ring A such that R ⊂ A ⊆ Q and T ∼ = A as R-algebras. Thus, there do not exist distinct R-subalgebras A1 , A2 of Q such that R ⊂ Ai is a minimal ring extension and R is integrally closed in Ai for i = 1, 2. (b) ([18, Remark 3.9 (c)]) Let R be a chained ring (in the sense that R is a ring whose set of ideals is linearly ordered by inclusion); let M denote the maximal ideal of R. Assume also that R is one-dimensional and that its minimal prime ideal is divided. Then either (i) R is integrally closed and tq(R) is a minimal ring extension of R or (ii) each element of M is a zero-divisor of R (in which case, R = tq(R) is not a reduced ring). (c) ([18, Proposition 3.11]) Let (R, M) be a local Noetherian ring. Then there exists a minimal ring extension T of R such that R is integrally closed in T if and only if R is a DVR. Moreover, if these equivalent conditions hold, then T is R-algebra isomorphic to the quotient field of R. If S is a minimal overring of R, then either R is a DVR with quotient field S or S is integral over R. If (R, M) is (an) integrally closed (local Noetherian ring), then there does not exist a minimal ring extension B of R such that R ⊂ B ⊂ tq(R). To go beyond the case of a quasilocal base ring (and obtain a complete generalization of the characterization results of Ayache), Shapiro and the author had to assume in [18] that the base ring R is such that its total quotient ring is von Neumann regular. (Of course, any domain R satisfies this property.) It was also necessary to go beyond the classical domainbased concept of a Kaplansky transform. Fortunately, with the above hypothesis on tq(R), we could appeal to Shapiro’s recent introduction of the generalized Kaplansky transform [32]. We next recall what will be needed of that generalization for our later discussion. Let R be a ring such that Q := tq(R) is von Neumann regular, and let M be a maximal ideal of R. Then the (generalized) Kaplansky transform of (R with respect to) M is defined to be Ω(M) := {q ∈ Q | Rad(R : q) = M} ∪ R; in the spirit of [3], it is useful to note the equivalent formula Ω(M) =

∞ \ [

(R :Q yn ).

y∈M n=1

Our generalization of the studies in [3] is contained in the next result. In the statement of Theorem 4.3 (b), it will be convenient to let [T ] denote the R-algebra isomorphism class of an R-algebra T . Theorem 4.3 ([18, Theorem 3.7]). Let R be a ring such that the total quotient ring, Q, of R is von Neumann regular. Then: (a) Let T be an extension ring of R and let M be a maximal ideal of R. Then the following three conditions are equivalent: (1) T is a minimal ring extension of R such that R is integrally closed in T and M is the crucial maximal ideal of R ⊂ T ; (2) T is a minimal overring of R such that R is integrally closed in T and M is the crucial maximal ideal of R ⊂ T ; (3) The following three conditions hold:


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(i) M is the radical of some finitely generated ideal of R, (ii) There exists a prime ideal P ⊂ M such that PRM is a divided prime ideal of RM , there exists a regular element u ∈ MRM \ PRM , and (R/P)M/P is a one-dimensional valuation domain (hence, M is not a minimal prime ideal of R), and (iii) T ∼ = Ω(M) as R-algebras. (b) The class consisting of the R-algebra isomorphism classes of minimal ring extensions of R in which R is integrally closed is in one-to-one correspondence with the set of maximal ideals M of R that satisfy conditions (i) and (ii) of condition (3) above. The asserted one-to-one correspondence can be obtained by sending [T ] to the crucial maximal ideal of R ⊂ T ; the inverse of this bijection sends any M (satisfying conditions (i) and (ii) above) to [Ω(M)]. The question remains: if tq(R) is not a von Neumann regular ring, what is the structure of a minimal ring extension R ⊂ T such that R is integrally closed in T ? In work in preparation, Cahen, Lucas and the author [5, Theorem 3.5] have found a satisfactory answer to this question. This new work is valid for any base ring R. In particular, it generalizes Theorem 4.3. In doing so, it introduces a further generalization of the notion of a Kaplansky transform, replaces “crucial maximal ideals” with a new concept that we have termed “critical ideals”, and replaces valuation domains with valuation pairs (in the sense of Manis). To say more about this here would take us too far afield. It remains to discuss characterization results for the other case of the dichotomy that was noted in [20], namely, the integral minimal ring extensions. As noted prior to Theorem 3.15, if R ⊂ T is an integral minimal ring extension with crucial maximal ideal M, then there are three possibilities for T /M, namely, (1) a minimal field extension of R/M, (2) R/M × R/M, (3) (R/M)[X ]/(X 2 ). An ideal-theoretic rendering of these three cases was given in [13, Corollary II.2]. We next state a generator-and-relations variant. Theorem 4.4 ([18, Proposition 2.12]). Let R be a ring with total quotient ring Q. Let T be an integral ring extension of R. (Hence, if R 6= Q, then T 6⊇ Q). Then T is a minimal ring extension of R if and only if there exists a maximal ideal M of R such that one of the following three conditions holds: (1) M is a maximal ideal of T and T /M is a minimal field extension of R/M; (2) There exists q ∈ T \ R such that T = R[q], q2 − q ∈ M, and Mq ⊆ R; (3) There exists q ∈ T \ R such that T = R[q], q2 ∈ R, q3 ∈ R, and Mq ⊆ R. If any of the above three conditions holds, then M is uniquely determined as (R : T ), the crucial maximal ideal of the extension R ⊂ T . Furthermore, conditions (1), (2), (3) are mutually exclusive. Indeed, if T is an integral minimal ring extension of R, then (2) (resp., (3)) is equivalent to T /M being isomorphic as an R/M-algebra to R/M × R/M (resp., (R/M)[X ]/(X 2 )). The above-mentioned work in progress [5] also includes an extensive study of the integral minimal ring extensions. We will forgo any further meaningful discussion of this matter here. The second theme in this section has to do with a generalization of “minimal ring extension” that was introduced and developed in the unpublished doctoral dissertation of M. S. Gilbert. For reasons of space, we cannot do justice here to the depth and breadth of [21], and

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so we will touch mostly on aspects of [21] that relate to some of the earlier material in this survey. The key definition is the following. A ring extension R ⊆ T is called a λ-extension (for “linearly ordered intermediate rings”) if the set of rings A such that R ⊆ A ⊆ T is linearly ordered by inclusion. It is clear that any minimal ring extension is aλ-extension; the converse fails, in view of any trivial ring extension R ⊆ R. One should note that although they both generalize the concept of “minimal ring extension”, the λ-extensions are not the same as the ring extensions that satisfy FIP. Perhaps the easiest example of a √ ring√ extension that satisfies FIP but is not a λ-extension is a field extension such as Q ⊂ Q( 2, 3). The next result gives a striking example of a λ-extension which does not satisfy FIP. Example 4.5 ([21, Example 4.6]). There exists an integral λ-extension R ⊂ T which does not satisfy FIP. It can be arranged that R is a one-dimensional valuation domain (in fact, a DVR), T is a domain but not (R-algebra isomorphic to) an overring of R, and the set of rings A such that R ⊆ A ⊆ T is denumerable. √ Proof. (Sketch) Let p be √ a prime number, d a square-free integer, and L := Q( d). As √ usual, let ω be either (1 + d)/2 or d according as to whether d ≡ 1 (mod 4) or d ≡ 2, 3 (mod 4). It is well known that Z + Zω is the ring of algebraic integers of the algebraic number field L. Put S := Z \ pZ. Then R := ZS and T := (Z + Zω)S have the asserted properties. Indeed, R is a DVR, R ⊂ T inherits integrality from Z ⊂ Z + Zω, and T is not (isomorphic to) an overring of R since the quotient field of T is L. Finally, it is shown in [21, Example 4.6] that the rings between R and T are T = B0 ⊃ B1 ⊃ B2 ⊃ . . . ⊃ Bn ⊃ Bn+1 ⊃ · · · ⊃ R, where Bn := (Z + Zpn ω)S for each non-negative integer n. We termed the behavior in Theorem 4.5 “striking” because it does not involve an overring. Theorem 4.6 shows that if we avoid some of the features of the construction in Theorem 4.5 (such as integrality or a base ring that is a valuation domain), then λ-extensions of domains exhibit the same behavior that we saw in Theorem 3.6 (b) for extensions that satisfy FIP. Theorem 4.6. (a) ([21, Proposition 4.3]) Let R ⊆ T be a λ-extension of domains such that R is integrally closed in T . Then T is (R-algebra isomorphic to) an overring of R. (b) ([21, Theorem 4.7]) Let R be an integrally closed domain which is not a valuation domain. If R ⊆ T is a λ-extension such that T is a domain, then T is (R-algebra isomorphic to) an overring of R. Part (a) of the next result continues the “overring” theme from Theorem 4.6, while part (b) of this result makes contact with the “minimal overring” theme from Section 2. To facilitate the statement of Theorem 4.7 (b), we make the following definition. If R is a domain with quotient field K, we say that R is a λ-domain if R ⊆ K is a λ-extension. Theorem 4.7. (a) ([21, Proposition 1.3]) Let R ⊆ K be a λ-extension such that R is not a field and K is a field. Then K is the quotient field of R and the integral closure of R is a valuation domain. (b) ([21, Corollary 1.12]) Let R be a domain with integral closure R0 . If R0 is a valuation domain and a minimal overring of R, then R is a λ-domain.


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As suggested by Theorem 4.7, the connection between λ-domains and valuation domains runs deep. For instance, a domain R is a valuation domain if and only if it is an integrally closed λ-domain [21, Corollary 1.5]. More generally, a domain R is a λ-domain if (and only if) the following three conditions hold: the integral closureR0 of R is a valuation domain, each overring of R is comparable to R0 with respect to inclusion, and the set of rings A such that R ⊆ A ⊆ R0 is linearly ordered by inclusion [21, Theorem 1.9]. As we have seen that a λ-extension need not be integral (consider the inclusion of a valuation domain which is not a field in its quotient field), it makes sense to ask if a λ-extension must at least be algebraic. The next result states that the answer is in the affirmative, exactly as it was in Theorem 3.6 (a) for extensions that satisfy FIP. Theorem 4.8 ([21, Lemma 1.1]). Let R ⊆ T be a λ-extension. Then each element of T is the root of a suitable polynomial in R[X ] with a unit coefficient; in particular, T is algebraic over R. Recall that Theorem 3.12 characterized the ring direct products that satisfy FSP. The next two results address analogues for λ-extensions. Theorem 4.9 ([21, Proposition 2.8]). Let I be a index set such that |I| ≥ 2 and Ti a ring for each i ∈ I. Consider the ring direct product T := ∏i∈I Ti and a λ-extension R ⊆ T . For each i, let πi : T → Ti denote the canonical surjection, and set Ii := ker(πi ) ∩ R. Suppose that Ii + I j 6= R for each pair i, j of indices. Then |I| = 2 and R 6= T . We next state one of the consequences resulting from the line of reasoning in [21] that began with Theorem 4.9. Note that the second assertion in Theorem 4.10 is the “λextension” analogue of the assertion concerning condition (b) in Ferrand-Olivier’s characterization in Theorem 2.4 of the minimal ring extensions of a field. Theorem 4.10 ([21, Corollary 2.14 (a), (b)]). Let I be a index set such that |I| ≥ 2 and Ti a nonzero ring for each i ∈ I. Consider the ring direct product T := ∏i∈I Ti and a ring extension K ⊆ T , where K is a field. If K ⊆ T is a λ-extension, then |I| = 2. Indeed, K ⊆ T is a λ-extension if and only if T is isomorphic as a K-algebra to K × L for some field L such that K ⊆ L is a λ-extension. In view of Theorem 4.10, one may ask what can be said about arbitrary λ-extensions of a field. Some basic facts along these lines are stated in Theorem 4.11. Recall that dim(A) denotes the Krull dimension of a ring A and dimk (E) denotes the vector space dimension of a vector space E over a field k. Theorem 4.11 ([21, Theorem 3.1]). Let K ⊆ T be a λ-extension, where K is a field. Then: (a) T is algebraic over K and dim(T ) = 0. (b) If B is a ring such that K ⊆ B ⊂ T , then dimK (B) < ∞. (c) If dimK (T ) < ∞, then the rings B such that K ⊆ B ⊆ T form a finite chain. (d) If dimK (T ) = ∞, then the rings B such that K ⊆ B ⊆ T form an increasing denumerable chain K = T0 ⊂ T1 ⊂ T2 ⊂ · · · ⊂ Tn ⊂ · · · ⊂ T such that dimK (Tn ) < ∞ for each n ≥ 0, Ti ⊂ Ti+1 is a minimal ring extension for each i ≥ 0, and ∪∞ n=0 Tn = T .

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It is important to note, in contrast to the behavior of the FIP property, that the situation described in Theorem 4.11 (c) can arise. For instance, one need only consider J´ onssonextensions of fields, such as those in [22, Examples (E1) and (E2) (pages 81–82) and Examples 2.7 and 2.15]. In particular, a λ-extension of fields K ⊆ L can be such that [L : K] = ∞. Additional examples of this phenomenon for field extensions are available via Theorems 4.16 and 4.17 below. Recall that an algebra is said to be decomposable if it is isomorphic to a direct product of nonzero algebras relative to an index set with at least 2 elements. Theorem 4.12 reduces the study of the decomposable λ-extensions of a field K to the study of the λ-extensions K ⊆ L where L is a field, thus giving additional importance to the subject of field extensions that are λ-extensions. Note also that Theorem 4.12 reconnects with the ring extensions that were characterized in Theorem 4.10. Theorem 4.12 ([21, Theorem 3.4]). Let K ⊆ T be a ring extension where K is a field. Then T is a decomposable λ-extension of K if and only if T is isomorphic as a K-algebra to K ×L for some field L such that K ⊆ L is a λ-extension. At the close of Section 3, we noted that nilpotent elements have played an important role in the study of the FIP property. To some extent, that work was motivated by the earlier results of Gilbert on λ-extensions, such as Theorem 4.13. Theorem 4.13 ([21, Corollary 3.6]). Let K ⊆ T be a λ-extension where K is a field. If u is a nilpotent element of T and n is the index of nilpotency of u, then n ≤ 3. In [29], Quigley classified the subfields of an algebraically closed fieldL which are maximal among the subfields of L not containing a given element of L. The relevant definition for that work is the following. One says that an extension K ⊆ L of fields is a µ-extension, or that L is a µ-extension of K, if there is an element x ∈ L such that K is maximal among the subfields of L not containing x. An example [22, Page 96, lines 6–11] shows that a µ-extension need not be a λ-extension. However, as Theorem 4.14 records, µ-extensions exhibit the behavior described in Theorem 4.11 (a) and there are other positive connections between the “µ-extension” and “λ-extension” concepts. Theorem 4.14 ([21, Proposition 3.15]). Let K ⊆ L be an extension of fields. Then: (a) If K ⊆ L is a µ-extension, then L is algebraic over K. (b) If K ⊆ L is a λ-extension and K 6= L, then K ⊆ L is a µ-extension. If K ⊆ L is a field extension, it will be convenient to say that K is purely inseparably closed in L if each element of L that is purely inseparable over K must belong to K. Theorem 4.15 records that this concept is one of the alternatives in a dichotomy that applies to any field extension that is a λ-extension. Theorem 4.15 ([21, Proposition 3.17 (2)]). Let K ⊂ L be an extension of distinct fields. Then K ⊆ L is a λ-extension if (and only if) the sets of fields F such that K ⊆ F ⊆ L is linearly ordered by inclusion. If these equivalent conditions hold, then L is algebraic over K and either L is purely inseparable over K or K is purely inseparably closed in L.


35

If K is a field of characteristic p > 0, it is known that K has a perfect purely inseparable algebraic field extension Π, which is unique up to K-algebra isomorphism. Following Bourbaki, we call Π a perfect closure of K. If K is an algebraic closure of K, then we can take Π = {u ∈ K | u is purely inseparable over K}. Theorem 4.16 ([21, Theorem 3.29]). Let K be a non-perfect field of characteristic p > 0, and let Π be a perfect closure of K. Then the following conditions are equivalent: (1) K ⊆ Π is a λ-extension; (2) K ⊆ Π is a µ-extension; (3) [K : K p ] = p; (4) There exists u ∈ K such that K = K p (u). It can be shown, as a consequence of Theorem 4.16, that if K be any field of characteristic p > 0 such that [K : K p ] = p and if Π denotes a perfect closure of K, then K ⊆ Π is a purely inseparable λ- (and µ-) extension such that dimK (Π) = ∞. Thus, decidedly non-FIP behavior of λ-extensions, of the kind described in Theorem 4.11 (d), is a consequence of a condition (namely, condition (3) in Theorem 4.16) which falls under the rubric of the Primitive Element Theorem. Recall that the structure of splitting field extensions can be described by using the purely inseparable and the Galois subcases. Therefore, as a companion for Theorem 4.16, we close this summary of results on λ-extensions by describing the algebraic (not necessarily finite dimensional) Galois field extensions that are λ-extensions. Theorem 4.17. (a) ([21, Theorem 3.36]) Let K ⊂ L be a finite-dimensional Galois extension of distinct fields. Then the following three conditions are equivalent: (1) K ⊆ L is a λ-extension; (2) K ⊆ L is a µ-extension; (3) Gal(L/K) is cyclic of prime-power order. (b) ([21, Theorems 3.37 and 3.38]) Let K ⊂ L be an infinite-dimensional algebraic Galois extension of fields. Then the following three conditions are equivalent: (1) K ⊆ L is a λ-extension; (2) K ⊆ L is a µ-extension; (3) Gal(L/K) is isomorphic as a topological group to the additive group of p-adic integers for some prime number p.

5

Conclusion

We close by pinpointing some open questions that arise from the material in the preceding sections. Relative to the material in Section 2, the most important open question is to classify the minimal ring extensions of a ring R that does not necessarily satisfy the hypotheses of Theorem 2.7. To do so, one will need to go beyond the known types of minimal ring extensions in that result (minimal overrings, certain idealizations, certain direct products). As mentioned in Section 2, the most natural type of ring R to consider next is probably an arbitrary von Neumann regular ring (or, failing that, one should consider prominent types

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of such rings). New concepts, possibly of a topological nature, will likely be needed for further progress in this area. As for the first topic in Section 3, FIP and related properties, one important open problem is to find a FIP-theoretic analogue of the characterization of the FSP property in Theorem 3.11. In doing so, one should be mindful of examples, such as Example 3.20, which show that behavior of the singly-generated subalgebras is not always predictive of behavior of a given algebra. In the same spirit, another open question is to find a FIP-theoretic analogue or generalization of Theorem 3.10, where one would seek to replace the base ring Z with appropriate base rings R and then classify the faithful unital R-algebras T = R[u] such that R ⊆ T satisfies FIP. Another interesting technical problem concerning FIP is to generalize Theorem 3.5 by describing the “steps” in any maximal chain of R-subalgebras for a ring extension R ⊆ T that satisfies FIP where the base ring R is not semisimple. We next mention one open question concerning the second theme in Section 2, composites of minimal ring extensions. This concerns finding results of a “dual” nature to those in the results 3.13–3.19. There are several ways to seek such “dualizations”, but we would focus here on the following question. If both S ⊆ R and T ⊆ R are minimal ring extensions, what are some sufficient conditions for S ∩ T ⊆ S and/or S ∩ T ⊆ T to be minimal ring extensions? As for the first topic in Section 4, characterizations of minimal ring extensions in the relatively integrally closed case and in the integral case, the obvious open questions would be to generalize Theorem 4.3 in a way that removes the hypothesis that the base ring has a total quotient ring that is von Neumann regular; and to present companions for Theorem 4.4 that find new ways to describe the integral case. As mentioned in Section 4, both of these questions will be answered in a work that is in preparation [5]. Finally, we discuss open questions concerning the second topic in Section 4, λ-extensions. In fact, the treatment of λ-extensions in Section 4 was effectively limited to analogues of behavior studied in earlier sections. As a result, a considerable amount of [21] was not reflected in Section 4, and in fact, that omitted material does lead to a number of open questions. For instance, [21, Lemma 4.1 and Theorem 4.2] show the extent to which λextensions exhibit “crucial maximal ideal” behavior, in the sense of [20, Théor` eme 2.2 and Lemme 3.2]. (That extent is limited, as we noted in the Introduction that minimal ring extensions are characterized by the existence of a crucial maximal ideal.) We would ask whether ring extensions satisfying FIP (or FSP) also exhibit related behavior. Additional domaintheoretic questions arise concerning λ-extensions. We will mention three such questions. For instance, [21, Corollary 4.10 (2)] established that if R is a one-dimensional Prüfer domain that satisfies property (#) and if T is a proper overring of R, then R ⊆ T is a λ-extension (if and) only if R ⊆ T is a minimal ring extension. We would ask for generalizations of this result to either higher Krull dimensions or appropriate non-Pr¨ ufer domains. Next, we would ask whether the extensive study in [21, Chapter I] of properties related toλ-domains in the context of pseudo-valuation domains can be extended to the context of more general pullbacks. Finally, with an eye to possible applications to orders in algebraic number fields, we would ask for generalizations, to monogenic algebras of degree exceeding 2, of the following result [21, Corollary 4.13 (2)]: if R ⊆ T is a (necessarily integral) ring extension such that T = R + Rt for some t ∈ T , then R ⊆ T is a λ-extension if and only if R/(R : T ) is a chained ring.


37

References [1] Anderson, D. D.; Dobbs, D. E.; Mullins, B. Houston J. Math. 25, 603–623 (1999). Corrigendum, Houston J. Math. 28, 217–219 (2002). [2] Anderson, D. F.; Badawi, A.; Dobbs, D. E. Boll. Un. Mat. 8, 535–545 (2000). [3] Ayache, A. Comm. Algebra 31, 5693–5714 (2003). [4] Badawi, A.; Anderson, D. F.; Dobbs, D. E. In Commutative Ring Theory; Cahen, P.-J.; Fontana, M.; Houston, E.; Kabbaj, S.-E.; Ed.; Lecture Notes Pure Appl. Math. 185; Dekker: New York, NY, 1997; 241–250. [5] Cahen, P.-J.; Dobbs, D. E.; Lucas, T. G. Rocky Mountain J. Math., to appear. [6] Coykendall, J.; Dobbs, D. E. JP J. Algebra, Number Theory and Appl., to appear. [7] Davis, E. D. Trans. Amer. Math. Soc. 110, 196–212 (1964). [8] Dechéne, L. I. Adjacent Extensions Of Rings; Ph. D. thesis; University of California at Riverside: Riverside, CA, 1978. [9] Dobbs, D. E. Internat. J. Commut. Rings 1, 173–179 (2002). [10] Dobbs, D. E. Comm. Algebra 34, 3875–3881 (2006). [11] Dobbs, D. E. Comm. Algebra 35, 773–779 (2007). [12] Dobbs, D. E. Comm. Algebra 37, 604–608 (2009). [13] Dobbs, D. E.; Mullins, B.; Picavet, G.; and Picavet-L’Hermitte, M. Comm. Algebra 33, 3091–3119 (2005). [14] Dobbs, D. E.; Mullins, B.; Picavet-L’Hermitte, M. Comm. Algebra 36, 2638–2653 (2008). [15] Dobbs, D. E.; Picavet, G.; and Picavet-L’Hermitte, M. J. Algebra and its Appl. 7, 601–622 (2008). [16] Dobbs, D. E.; Picavet, G.; and Picavet-L’Hermitte, M. Int. Electron. J. Algebra 5, 121–134 (2009). [17] Dobbs, D. E.; Shapiro, J. J. Algebra 305, 185–193 (2006). [18] Dobbs, D. E.; Shapiro, J. J. Algebra 308, 800–821 (2007). [19] Dobbs, D. E.; Shapiro, J. JP J. Algebra, Number Theory and Appl. 9, 241–275 (2007). [20] Ferrand, D.; Olivier, J.-P. J. Algebra 16, 461–471 (1970). [21] Gilbert, M. S. Extensions Of Commutative Rings With Linearly Ordered Intermediate Rings; Ph. D. dissertation; University of Tennessee, Knoxville: Knoxville, TN, 1996.

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[22] Gilmer, R.; Heinzer, W. Pacific J. Math. 128 81–116 (1987). [23] Hedstrom, J.; Houston, E. G. Pacific J. Math. 75, 137–147 (1978). [24] Huckaba, J. A. Commutative Rings with Zero Divisors; Dekker: New York, NY, 1988. [25] Kaplansky, I. Commutative Rings, rev. ed.; Univ. Chicago Press: Chicago, IL, 1974. [26] Modica, M. L. Maximal Subrings; Ph. D. dissertation; University of Chicago: Chicago, IL, 1975. [27] Nagata, M. Local Rings; Wiley-Interscience: New York, NY, 1962. [28] Picavet, G.; Picavet-L’Hermitte, M. In Multiplicative Ideal Theory In Commutative Algebra; Brewer, J.; Glaz, S.; Heinzer, W.; Olberding, B.; Ed.; Springer-Verlag: New York, NY, 2006; 369–386. [29] Quigley, F. Proc. Amer. Math. Soc. 13, 562–566 (1962). [30] Rosenfeld, A. Scripta Math. 28, 51–54 (1967). [31] Sato, J.; Sugatani, T.; Yoshida, K. I. Comm. Algebra 20, 1746–1753 (1992). [32] Shapiro, J. Rocky Mountain J. Math. 38, 267–289 (2008).



Chapter 4

C OMMUTATIVE A LGEBRA A PPLIED TO S TABILIZATION P ROBLEMS FOR S YSTEMS OVER R INGS A. Sáez-Schwedt∗ Departamento de Matemáticas, Universidad de León, Campus de Vegazana, 24071 León, Spain

Abstract In this chapter we study some recent and new results on linear systems theory over commutative rings. A linear system over a commutative ring R is a pair of matrices (A, B) of sizes n × n and n × m respectively, with coefficients in R. The system is called reachable if the n × mn block matrix A∗ B = [B|AB| · · ·|An−1 B] is right-invertible. The following three properties of systems are classically related to the problem of stabilization of systems: (i) pole assignment (PA), (ii) coefficient assignment (CA) and (iii) feedback cyclization (FC). It is a well known result that (iii) ⇒ (ii) ⇒ (i), and (i) implies that (A, B) is reachable. If R is a field, all four conditions are equivalent. One defines PA rings, CA rings and FC rings as those commutative rings R such that any reachable system over R satisfies (i), (ii) or (iii) respectively. For non necessarily reachable systems we extend the above properties, by using the concept of residual rank of a system: The residual rank of a system (A, B), denoted as res.rk(A, B), is defined as the highest nonnegative index i such that the ideal generated by the i × i minors of the matrix A∗ B is R (reachable systems correspond to the case of residual rank equal to n). This allows us to introduce new classes of rings, satisfying the following (strict) implications: strong FC ⇒ strong CA ⇒ strong PA. We then prove that the strong FC property is achieved by all rings R satisfying the following two conditions: • UCU property: if B is a matrix with unit content, then the R- module im(B) contains a free rank one direct summand of Rn . • stable range one: if (a, b) = R, there exists k such that a + bk is a unit of R. ∗ E-mail


40

A. Sáez-Schwedt Examples of strong FC rings are, among others: zero-dimensional rings, local-global rings, rings with many units and Bezout domains with stable range one, including certain rings of holomorphic functions. Finally, for all s > 0, we prove that UCU rings with stable range s satisfy the strong form of a new property called FC s .

1

Introduction

Let R be a commutative ring with 1. An m-input, n-dimensional system (or a system of size (n, m)) over R will be a pair of matrices (A, B), with A ∈ Rn×n and B ∈ Rn×m. The residual rank of the system (A, B), denoted by res.rk(A, B), is defined as the residual rank of the reachability matrix A∗B = [B|AB| · · ·|An−1B], i. e. res.rk(A, B) = max{i : Ui (A∗ B) = R}, where Ui (A∗B) denotes the ideal of R generated by the i × i minors of the matrix A∗ B, with the convention U1 (B) = R. The system (A, B) is reachable or controllable if and only if res.rk(A, B) = n, and res.rk(A, B) ≥ 1 if and only if U1 (B) = R, i.e. B has unit content [9, Lemma 2.3]. Also, if two matrices M, M 0 satisfy im(M) ⊆ im(M 0), then it is clear from the properties of the ideals of minors that res .rk(M) ≤ res.rk(M 0 ), and one has an equality if the image modules are isomorphic. In particular, given a system (A, B), one has that res.rk(A + BK, BU) ≤ res.rk(A, B) for all matrices K,U. In Control Theory, the linear systems studied have coefficients in a field, see for example [20]. Also, [1] is an excellent reference, which motivates the study of linear systems with scalars in a commutative ring. The topics from Commutative Algebra and Linear Algebra over Commutative rings needed to develop this theory are covered in [13] and [14]. This chapter starts describing the work initiated in the 1980s [2, 3, 8], until the more recent articles [9, 15, 17, 18]. The concepts of pole assignability (PA), coefficient assignability (CA) and feedback cyclization (FC) are classically defined for reachable systems, as can be seen in [1, 8], we recall it here: a system (A, B) of size (n, m) over a ring R is called: (i) pole assignable if, given arbitrary scalars x1 , . . ., xn in R, there exists a matrix K such that χ(A + BK), the characteristic polynomial of A + BK, is equal to f (x) = (x − x1 ) · · ·(x − xn ); (ii) coefficient assignable if, given an arbitrary monic polynomial f (x) of degree n, there exists a matrix K such that χ(A + BK) = f (x); and (iii) feedback cyclizable if there exist a matrix K and a vector u such that the single-input system (A + BK, Bu) is reachable. It is known that (iii)⇒(ii)⇒(i), and (i) implies that (A, B) is a reachable system. The ring R is called a PA (resp. CA) (resp. FC) ring if all reachable sytems over R are pole assignable (resp. coefficient assignable) (resp. feedback cyclizable). By means of the residual rank, we are able to extend all these stability properties to non-reachable systems, and new classes of rings are derived. Let (A, B) be a system over R with res.rk(A, B) = r. (A, B) is said pole assignable if, given arbitrary scalars x1 , . . ., xr , there exists a matrix K such that χ(A + BK) is a multiple of f (x) = (x − x1 ) · · ·(x − xr ). In [9], the class of PS rings is introduced as those rings for which every system is pole assignable. We will also denote these rings by strong PA rings. Similarly, in [18] (A, B) is said coefficient assignable if given any monic polynomial f (x) of degree r, there exists K

Commutative Algebra Applied to Stabilization Problems for Systems over Rings 41 such that χ(A + BK) is a multiple of f (x), and R is called a strong CA ring if any system over R is coefficient assignable. To complete the strong stability properties, in [17] (A, B) is called feedback cyclizable if there exist K, u such that res.rk(A + BK, Bu) = r, and a strong FC ring is a ring for which any system is feedback cyclizable. We say that two systems (A, B) and (A0 , B0 ) are feedback equivalent if there exist invertible matrices P ∈ GLn (R), Q ∈ GLm (R) and a matrix K ∈ Rm×n such that (A0 , B0 ) = (PAP−1 + PBK, PBQ). In particular, B and B0 = PBQ are equivalent matrices. There are other equivalent formulations for this equivalence. For example, (A, B) and (A0 , B0 ) are equivalent if and only if there exist P, Q, K such that PA − A0 P = B0 K, PB = B0 Q. Also, (A, B) and (A0 , B0) are equivalent if and only if (A0, B0 ) = (P(A + BK)P−1, PBQ), for some (other) P, Q, K. The feedback classification problem for systems over rings is unsolved: indeed, it contains as special cases the similarity classification A ∼ PAP−1 (case K = 0) and the equivalence classification B ∼ PBQ, both unsolved questions. The residual rank is invariant under feedback [9], a fact that will be used repeatedly throughout this chapter. As an easy consequence, all the studied stability properties are also invariant under feedback [18, Lemma 3.1], which suggest the systematic use of feedback equivalence. Let us illustrate how this technique can be used. Suppose that we have a system (A, B) of size (n, m) in either of the following forms: 1 0 0 0 , B= , (1) A= 0 B2 b1 A1 B A1 0 , B= 1 , (2) A= B2 A2 A3 where in (2) (A1 , B1 ) is a reachable pair of size (r, m). The following argument is used in the proof of [3, Theorem 1] to obtain the PA property. If (A, B) is a reachable system in the form (1), then by Eising’s Lemma [10, Lemma 1] the (n − 1)-dimensional system Σ1 = (A1 , [b1 |B1 ]) is reachable. Let f (x) = (x − r1 ) · · ·(x − rn) be a monic polynomial of degree n over R. If Σ1 is assumed to be pole assignable, there exist matrices K1 , K2 such that A1 + b1 K1 + B2 K2 has characteristic polynomial (x − r2 ) · · ·(x − rn ). It is not difficult to construct a feedback equivalence which replaces A1 by A1 + b1 K1 + B2 K2 , and with a further feedback we can put the pole r1 in the first row of A. Concretely, we have obtained an equivalent system: r1 1 0 0 0 0 , B = A = 0 B2 b1 A1 + b1 K1 + B2 K2 where χ(A0 ) = (x − r1 )χ(A1 + b1 K1 + B2 K2 ) = f (x), as desired. Since pole assignability is invariant under feedback, it follows that the system (A, B) is pole assignable. Now suppose that (A, B) has residual rank r and is of the form (2), such that the reachable pair (A1 , B1 ) of size (r, m) is coefficient assignable, i.e. for any monic polynomial f (x) of degree r over R, there exists a matrix K1 such that χ(A1 + B1 K1 ) = f (x). Defining K = [K1 0], we have that the matrix A + B1 K1 0 A + BK = 1 ∗ A3

42

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has characteristic polynomial χ(A1 + B1 K1 )χ(A3 ), a multiple of f (x), i.e. (A, B) is coefficient assignable. This is the argument used in [18] to solve the CA problem for nonreachable systems, provided the CA problem is solvable for reachable sytems and any system is feedback equivalent to some system in the form (2). Several questions arise, for example: 1. For which rings is any reachable system feedback equivalent to a system like (1)? 2. For which rings is any system of residual rank r feedback equivalent to a decomposition like in (2) with a reachable subsystem of size r? 3. If Σ = (A, B) is in the form (1) and Σ1 = (A1 , [b1|B2 ]) is feedback equivalent to some system (A01 , [b01|B02 ]), does there exist a feedback equivalence which replaces the blocks A1 , b1 , B2 in (A, B) by A01 , b01 , B02 ? This motivates the introduction of some new terminology. We say that a ring R has the UCU (Unit-content Contains Unimodular) property if, whenever U1 (B) = R, there exists a vector u such that Bu is unimodular, or equivalently, the image of B contains a free rank 1 summand of Rn . UCU rings are called BCU rings in [4] and [5]. It is clear that if (A, B) is a reachable system, then the matrix B must necessarily have unit content. If B is such that there exists a matrix A with (A, B) reachable, then B is called a good matrix. We will see later that not all unit-content matrices are good. Also, a UCU ring R has the GCU (Good Contains Unimodular) property: for any good matrix B, there exists a vector u such that Bu is unimodular, or equivalently im(B) contains a free rank 1 summand of Rn ; and the BCS or UCS (Unit-content Contains Summand) property (see [4] and [2, p. 267]). The class of GCU rings characterizes those rings for which question 1 has a positive answer, and in a similar way, UCU rings give a solution to question 2. A partial solution to question 3 turns out to be very useful in our development. Section 2 ends by showing several examples of strong FC rings, in all cases we obtained UCU rings with stable range 1. Finally, Section 3 generalizes the FC problem by allowing multiple inputs, i.e., by transforming a system (A, B) into a system (A + BK, BU), where U has a fixed number of columns s ≥ 1. The class of FCs and strong FCs rings are introduced, and the strong FC s problem is solved for UCU rings with stable range s.

2

Strong Stabilization Properties

We start with a useful technique for manipulating systems inductively. Given an n-dimensional system (A1 , [b1|B1 ]), we define its ‘Eising’ augmentation as the (n + 1)-dimensional system given by: 1 0 0 0 , B= . A= 0 B1 b1 A1 If the column vector b1 is zero, this is the usual dynamic augmentation of size 1 (see [12]). The name Eising comes from the famous Eising’s Lemma from [10], which states that the augmented system (A, B) is reachable if and only if the reduced system (A1 , [b1 |B1 ]) is

Commutative Algebra Applied to Stabilization Problems for Systems over Rings 43 reachable. An analogous result is proved in [17, Lemma 2.6] for non-reachable systems, in terms of the residual rank: res.rk(A, B) = res.rk(A1 , [b1 |B1 ]) + 1. We will see how we can relate the feedback equivalence between reduced and augmented systems, and explain how this can be used as a powerful induction argument in order to construct canonical forms and to prove stabilization properties inductively. The techniques used here are similar to those of [12] while studying the relation between static and dynamic feedback, although the results obtained are of a different nature: we do not find a class of commutative rings for which the equivalence of the augmented systems is equivalent to that of the reduced systems. Instead of this, we prove that none of the equivalences imply the other, but exhibit a class of systems for which both equivalences are the same. We start proving that neither of the equivalences implies the other. Example 2.1. Consider the systems (A, B) and (A0, B), where       0 0 0 1 0 0 0 0 A =  1 0 0  , A0 =  1 0 0  , B =  0 1  . −1 1 0 0 0 0 1 0 It is easy to see that (A0, B) = (PAP−1 , PBQ), where   1 0 0 1   P= 1 1 0 , Q= −1 0 0 1

0 , 1

therefore (A, B) and (A0 , B) are equivalent systems. However, the reduced systems 0 0 1 1 0 0 1 1 , , , 1 0 0 0 1 0 −1 0 cannot be equivalent because their second matrices are not equivalent: one has determinant 0 while the other not. Example 2.2. Consider the systems (A, B) and (A0, B0 ), where 0 0 1 0 0 0 1 0 0 0 , B = . A= , B= , A = 1 0 0 1 1 0 0 0 Now the reduced systems (0, [1|0]) and (0, [1|1]) are feedback equivalent, while the augmented systems are clearly not equivalent. The previous two examples are valid for systems over any commutative ring, and show that neither of the systems equivalences implies the other. After these two ‘negative results’, we shall exhibit a ‘positive result’, that is, a situation where the equivalence of the reduced systems implies the equivalence of the augmentations, and vice versa. The key idea is to put restrictions on the structure of the matrices (P, Q, K) involved in the feedback equivalence.

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Lemma 2.3. Consider the following matrices with blocks of appropriate sizes: 0 0 1 0 1 0 0 0 0 0 , B= , A = 0 , B = , A= b1 A01 0 B02 0 B2 b1 A1 P12 K12 Q11 Q12 K P , Q= , K = 11 , P = 11 0 P22 0 Q22 K21 K22 X1 T1 0 , Q1 = . K1 = X2 T2 T3 (i) If the augmented systems (A, B), (A0 , B0) are equivalent via PA − A0 P = B0 K , PB = B0 Q then the reduced systems (A1 , [b1|B1 ]) and (A01 , [b01 |B01 ]) are equivalent. (ii) If the reduced systems (A1 , [b1 |B1 ]) and (A01 , [b01 |B01 ]) are equivalent via P1 A1 − A01 P1 = [b01 |B01 ]K1 , P1 [b1 |B1 ] = [b01 |B01 ]Q1 then the augmented systems (A, B) and (A0, B0 ) are equivalent. Proof. Equating blocks in the equality PA − A0 P = B0 K, we get P11 b1 = K11 ,

(3)

P12 A1 = K12 ,

(4)

P22 b1 − b01 P11 P22 A1 − b01 P12 − A01 P22

= =

B01 K21 , B01 K22

(5) (6)

and PB = B0 Q yields P11 = Q11 ,

(7)

P12 B1 = Q12 ,

(8)

0 = 0, P22 B1 =

(9)

B01 Q22 .

(10)

Combining (5), (6) and (10), one gets the matricial equations  P12  0 0 0  ,  P22 A1 − A1 P22 = [b1 |B1 ] K22 P11 0   .  P22 [b1 |B1 ] = [b01 |B01 ] K21 Q22 Since P11 and Q22 are invertible, this proves (i). To prove (ii), just define P, Q, K block by block, as follows: T1 X1 B1 Tb T1 X1 , Q= , K = 10 1 P= 0 P1 0 T3 B1 T2

X1 A1 . X2

With this construction, equations (3) · · · (10) hold, hence we have PA − A0 P = B0 K, PB = B0 Q, and (ii) follows.

Commutative Algebra Applied to Stabilization Problems for Systems over Rings 45 A very important point where this technique is used is in the proof of the following result. Theorem 2.4. For a commutative ring R, the following statements are equivalent: (i) R is a UCU ring. (ii) Any system Σ = (A, B) of size (n, m) over R with res.rk(A, B) ≥ r is feedback equivalent to a system in form (2): B A1 0 , B= 1 , A= A2 A3 B2 where (A1 , B1 ) is a reachable pair of size (r, m), and the remaining blocks are of appropriate sizes. Proof. It is proved in [18, Proposition 2.5]. The key fact is that if a system can be decomposed in this form via a feedback equivalence (P, Q, K), then (P, K) gives also a valid decomposition (see [18, Lemma 2.2]), and by Lemma 2.3, any equivalence on the reduced system without Q can be lifted to an equivalence on the augmented system. Next, we prove the non-reachable version of the known implications FC ⇒CA⇒PA, and that the PA, CA and FC properties are equivalent to their strong forms for UCU rings. Theorem 2.5. Strong FC rings are strong CA rings, and strong CA rings are PS (strong PA) rings. Moreover, the inclusions are strict. Proof. See [18, Theorem 3.5]. Proposition 2.6. If R is a UCU ring, then R is a PA (resp. CA) (resp. FC) ring if and only if R is a strong PA (resp. strong CA) (resp. strong FC) ring. Proof. See [18, Proposition 3.6]. Before showing examples of rings satisfying our stability properties, we recall the notion of stable range of a ring. Let s be a positive integer. Following [11, 21], we say that a ring R “has s in its stable range”, or is s-stable, if given (a1 , . . ., as , b) = R, there exists scalars y1 , . . ., ys in R such that (a1 + by1 , . . ., as + bys ) = R. The following is a characterization of s-stable rings. Lemma 2.7. For a commutative ring R, the following statements are equivalent: (i) R has s in its stable range. (ii) If A ∈ Rn×(n+s−1), B ∈ Rn×m are matrices such that im(A|B) = Rn , there exists a matrix K such that im(A + BK) = Rn . Proof. It is a straightforward adaptation of [12, Theorem 5] and [21, Theorem 3 0 ]. The next two results yield several examples of rings satisfying the strong FC property, and thus also the strong forms of CA and PA.

46

A. Sáez-Schwedt

Theorem 2.8. Let R be a commutative ring with 1 in its stable range. Then R has the strong FC property if and only if R has the UCU property. In particular, all 1-stable UCU rings are strong FC rings. Proof. See [17, Theorem 3.1]. Proposition 2.9. The following commutative rings are strong FC rings: (i) Local-global rings. (ii) Rings with many units. (iii) Zero-dimensional rings, in particular von Neumann regular rings and artinian rings. (iv) Bezout domains with stable range 1, in particular the ring of all algebraic integers and the ring H (Ω) of holomorphic functions on a noncompact Riemann surface. Proof. See [17, Proposition 3.4]

3

Rings with Finite Stable Range

We start this section by giving characterizations of s-stable rings among all GCU and UCU rings, but before we need a simple lemma which illustrates the relationship between unimodular vectors and good matrices. Lemma 3.1. For a commutative ring R and a matrix B ∈ Rn×m, consider the following statements: (i) PB has its first row unimodular, for some invertible matrix P. (ii) B has a basic vector in its image. (iii) B has a good vector in its image. (iv) B has a unimodular vector in its image. (v) B is a good matrix. (vi) B has unit content. Then, one has (i)=(ii)=(iii)⇒(v)⇒(vi) and (iii)⇒(iv). Moreover, (iv)⇒(i) for Hermite rings in the sense of Lam (unimodular vectors can be completed to invertible matrices ), (v)⇒(i) over GCU rings, and (vi)⇒(i) over UCU rings. In particular, a ring R is a UCU ring if and only it is a GCU ring with the property that unit-content matrices are good matrices. Proof. (i)⇒(ii): Pick P, λ such that PBλ =

1 . v

By elementary row transformations, there

exists an invertible matrix P0 such that P0 PBλ = column of the invertible matrix (P0 P)−1 .

1 0

, which means that Bλ is the first

Commutative Algebra Applied to Stabilization Problems for Systems over Rings 47 (ii)⇒(iii): if there exists a basis {v, v2 , . . ., vn } of Rn , we can define an endomorphism f : Rn → Rn which “cyclically permutes” the basic vectors, in this form: f

f

f

f

v → v2 → · · · → vn → 0, therefore the vectors {v, f (v), . . ., f n−1 (v)} span Rn . Taking matrices A, b which represent f , v in this basis, it is clear that the system (A, b) is reachable with im(b) ∼ = im(v). (iii)⇒(i): take a reachable system of the form (A, v), for some v = Bλ, and consider the invertible matrix P−1 = [v|Av| · · ·|An−1v]. Right multiplication by P implies that PBλ has its first entry equal to 1, so that the first row of PB must be unimodular. The implications (iii)⇒(v)⇒(vi) are immediate, and the equivalence(iv)=(iii) holds precisely if the ring is Hermite in the sense of Lam. Furthermore, (v)⇒(iv) and (vi)⇒(iv) are the definitions of GCU and UCU rings respectively, but since both classes of rings are Hermite rings, we have also (iv)⇒(i). Finally, the last assertion is immediate because the property “unit-content matrices are good” allows jumping from (vi) to (v), and the GCU property permits the passage from (v) to (iv). Remark 3.2. We have seen in the last lemma that, for a unimodular vector v of Rn , the concepts of being good and being completable to an invertible matrix are equivalent. Note that in this situation we have an exact sequence of R-modules: v0

0 → Ker(v0 ) → Rn → R → 0 which splits because v0 is right-invertible (where 0 denotes transpose), therefore one has Rn ∼ = R ⊕ Ker(v0 ), and Ker(v0 ) is a stably-free R-module. It is easy to see that Ker(v0 ) is free if and only if the row vector v0 ∈ R1×n is completable to an invertible n × n matrix (see [14]). This yields immediate examples of matrices with unit content which are not good, for example, take a ring R such that there exists a unimodular vector v which is not completable to an invertible matrix. Let R = R[x, y, z]/(x2 + y2 + z2 − 1) be the ring associated to the real ¯ y, ¯ z¯] is unimodular in R, but is not completable to sphere in R3 . Then, the vector v = [x, an invertible matrix, because this would imply the existence of a continuous nonvanishing tangent field to the sphere, which is impossible. As we will see, there is a standard argument which shows that for any stably-free module P, one can construct a reachable system (A, B) with im(B) ∼ = P, see for example [3, Lemma 1]. Suppose that P is an R-module such that Rt ∼ = Rr ⊕ P, for some positive integers t, r. By Gabel’s Theorem [13, IV.44], we have that a finite direct sum of copies of P is free, that is, Pm ∼ = Rn , for some m ≤ n. Denote by B : Rn → Rn the projection onto the first copy of P, and A : Rn → Rn the endomorphism which cyclically permutes the copies of P: A maps the first copy onto the second, the second onto the third, . . . , and the last onto the first. Then, it is clear that im(B) ∼ = P and im(B) + Aim(B) + · · ·+ Am−1 im(B) + · · · + An−1 im(B) = Rn , which means that (A, B) is a reachable system, i.e. P is a good R-module. Next, we recall the concept of “feedback cyclization with s inputs” introduced in [15].

48

A. Sáez-Schwedt

Definition 3.3. Let s be a positive integer. A system (A, B) over a ring R is s-cyclizable if there exist matrices K,U, where U has s columns, such that the s-input system (A+BK, BU) is reachable. In particular, (A, B) must be necessarily a reachable system. The ring R is called an FCs ring if any reachable system over R is s-cyclizable. In analogy with [2, Theorem 5] for the case of 1-stable rings, in [15, Theorem 3.3] it is proved that s-stable rings with the GCU property are FC s rings. The main technique used in the cited theorem is a proper use of the feedback equivalence, by applying only specific transformations which preserve the studied properties and which can be applied inductively. Continuing with our systematic use of the residual rank, there is an immediate adaptation of the FC s property to non-reachable systems, and as a consequence we will obtain that s-stable UCU rings satisfy this strong form of FC s . Definition 3.4. A system (A, B) with res.rk(A, B) = r over a ring R is s-cyclizable if there exist matrices K,U, where U has s columns, such that res.rk(A + BK, BU) = r. R is a strong FCs ring if any system over R is s-cyclizable. Proposition 3.5. Let R be a UCU ring with the FC s property. Then, R is a strong FC s ring. In particular, s-stable UCU rings have the strong FC s property. Proof. The proof of this result is given in [16]. The last part of this section will be devoted to obtaining characterizations of rings with bot the GCU and s-stable conditions, and with both the UCU and s-stable conditions. Proposition 3.6. For a ring R, the following statements are equivalent. (i) R is an s-stable GCU ring. (ii) For any reachable system (A, B), where B = [B1 |B2 ], B1 ∈ Rn×s, there exists V such that B 1 + B2 V has a unimodular vector in its image. (iii) With the previous notations, there exists a unimodular vector of the form B 1 λ + B2 µ, where λ ∈ Rs×1 is unimodular. (iv) Given a reachable system P invertible matrices (A, B) of size (n, m) over R, there exist 1 0 · · ·0 ∗ 0 Q11 , where , with Q11 ∈ GLs (R), such that PBQ = and Q = 0 ∗ ∗ Q21 Q22 0 · · ·0 denotes s − 1 zeros. Proof. We will prove that (i)⇒(ii)⇒(iii)⇒(i) and (ii)⇒(iv)⇒(ii). (i)⇒(ii). let R be an s-stable GCU ring and consider a reachable system (A, B) of size (n, m), where m ≥ s and B1 ∈ Rn×s. By Lemma 3.1, there exists an invertible matrix P such that PB has its first row unimodular, therefore by the s-stable property there exists a matrix V with PB1 + PB2V having its first row unimodular, from which it follows easily that P(B1 + B2V ) has a unimodular vector in its image, and the same holds for B1 + B2V . s×1 such that (B + B V )λ is (ii)⇒(iii). There exist a matrix V and a vector 1 2 λ ∈ R λ hence (iii) holds. unimodular, but this vector is of the form [B1 |B2 ] Vλ

Commutative Algebra Applied to Stabilization Problems for Systems over Rings 49 (iii)⇒(i) To see that R is GCU, for any reachable system (A, B) one has to find a unimodular vector in the image of B. Adding zero columns if necessary, we may assume that B has m ≥ s columns, so that we can partition B =[B1 |B2 ] as in (iii), to conclude that there λ in the image of B. exists a unimodular vector (B1 + B2V )λ = [B1 |B2 ] Vλ For the s-stable property, consider (b1 , b2 ) = R, where b1 ∈ Rn×s and b2 ∈ R. Applying (iii) to the one-dimensional reachable system (0, [b1|b2 ]) yields a unimodular row b1 + b2 v, which means that R is s-stable. (ii)⇒(iv). The existence of a unimodular vector in the image of B1 + B2V implies that B1 + B2V is a good matrix by Lemma 3.1. But (ii) implies in particular that R is GCU, therefore by [3, Lemma 2] there exist invertiblematrices P and Q11 such that P(B1 + 1 0 Q11 0 we have that PBQ has the desired Now, defining Q = B2V )Q11 = 0 ∗ V Q11 I form. (iv)⇒(ii) If B01 denotes the first n × s block of PBQ, it is clear that B01 has a unimodular vector in its image. Therefore, P−1 B01 Q−1 11 must also have a unimodular vector in its image, and is of the form −1 P−1 (PB1 Q11 + PB2 Q21 )Q−1 11 = B1 + B2 Q21 Q11 ,

whence (ii) follows. It is immediate to derive a “strong version” of the previous proposition. Proposition 3.7. For a ring R, the following statements are equivalent. (i) R is an s-stable UCU ring. (ii) For any matrix B = [B1 |B2 ] with unit content, where B 1 ∈ Rn×s, there exists V such that B 1 + B2 V has a unimodular vector in its image. (iii) With the previous notations, there exists a a unimodular vector of the form B 1 λ + B2 µ, where λ ∈ Rs×1 is unimodular. P (iv) Given a matrix B ∈ Rn×m over R with unit content, there exist invertible matrices 1 0 · · ·0 ∗ 0 Q11 , where , with Q11 ∈ GLs (R), such that PBQ = and Q = 0 ∗ ∗ Q21 Q22 0 · · ·0 denotes s − 1 zeros. (v) If B1 ∈ Rn×r+s−1 and Ur (B1 |B2 ) = R, there exists a matrix V such that im(B1 + B2V ) contains r basic vectors of R n . Proof. The equivalences (i)=(ii)=(iii)=(iv) are completely analogous to the corresponding equivalences of last proposition. We will prove (i) ⇒(v)⇒(ii). [3, Lemma 2] r times we can find Assume (i). If Ur (B) = R, by iterating the result 1r 0 , where 1r is an r × r identity block. If invertible matrices P, Q such that PBQ = 0 ∗ [A1 |B1 ] denotes the upper r ×n block of P(A|B), it consists of the firts r rows of the invertible

50

A. Sáez-Schwedt

matrix Q−1 , therefore by Lemma 2.7 there exists a matrix K such that im(A1 + B1 K) = Rr , i.e. there exists a matrix V such that (A1 + B1 K)V = 1r . Now look at the matrix B A + B1 K A . P(A + BK) = PA + PBK = 1 + 1 K = 1 ∗ ∗ ∗ 1 Therefore, we have that P(A + BK)V = r are r basic vectors of Rn , which can be com∗ 0 . Applying the isomorphism P−1 , we have pleted to a basis by adjoining the block 1n−r found r basic vectors in the image of A + BK. Finally, (v) implies (ii) by considering the case r = 1.

4

Conclusion

In this chapter we have studied a very interesting application of Commutative Algebra to Control Theory, specifically to the stabilization problem for linear systems with coefficients in a commutative ring. The systematic use of the residual rank allowed us to translate many results from reachable to arbitrary system, and a proper use of the feedback equivalence permitted some useful inductive arguments. Finally, the relation obtained between the stable range of a ring and the number of inputs needed to stabilize a system, appears to open up a promising line of research.

References [1] Brewer, JW; Bunce, JW; Van Vleck, FS. Linear systems over commutative rings; Lecture Notes in Pure and Applied Mathematics 104. New York: Marcel Dekker; 1986. [2] Brewer, J; Katz, D; Ullery, W. Pole assignability in polynomial rings, power series rings and Prüfer domains. J. Algebra 1987, 106, 265–286. [3] Brewer, J; Katz, D; Ullery, W. On the pole assignability property over commutative rings. J. Pure Appl. Algebra 1987, 48, 1–7. [4] Brewer, J; Klingler, L. Rings of integer-valued polynomials and the bcs-property. In Commutative ring theory and applications ; Lecture Notes in Pure and Applied Mathematics 231; Marcel Dekker: New York, 2003; pp. 65–75. [5] Brewer, J; Klingler, L; Minaar, F. Polynomial rings which are BCS-rings. Comm. Algebra 1990, 18, 209–223. [6] Brewer, J; Klingler, L; Schmale, W. C[y] is a CA-ring and coefficient assignment is properly weaker than feedback cyclization over a PID. J. Pure Appl. Algebra 1994, 97, 265–273. [7] Brewer, J; Schmale, W. (A, B)-cyclic submodules. Linear Algebra Appl. 1999, 301, 63–80. [8] Bumby, R; Sontag, ED; Sussmann, HJ; Vasconcelos, WV. Remarks on the poleshifting problem over rings. J. Pure Appl. Algebra 1981, 20, 113–127.

Commutative Algebra Applied to Stabilization Problems for Systems over Rings 51 [9] Carriegos, M; Hermida-Alonso, JA; Sánchez-Giralda, T. Pole-shifting for linear systems over commutative rings. Linear Algebra Appl. 2002, 346, 97–107. [10] Eising, R. Pole-assignment for systems over rings. Systems Control Lett. 1982, 2, 225–229. [11] Estes, D; Ohm, J. Stable range in commutative rings. J. Algebra 1967, 7, 343–362. [12] Hermida-Alonso, JA; López-Cabeceira, MM; Trobajo, MT. When are dynamic and static feedback equivalent? Linear Algebra Appl. 2005, 405, 74–82. [13] McDonald, BR. Linear algebra over commutative rings . Marcel Dekker: New York, 1984. [14] Northcott, D. Finite free resolutions . Cambridge University Press: London, 1976. [15] Sáez-Schwedt, A. Feedback cyclization for rings with finite stable range. Linear Algebra Appl. 2007, 427, 234–241. [16] Sáez-Schwedt, A. Matricial decomposition of systems over rings. Electronic J. Linear Algebra 2008, 17, 493–507. [17] Sáez-Schwedt, A; Sánchez-Giralda, T. Strong feedback cyclization for systems over rings. Systems Control Lett. 2008, 57, 71–77. [18] Sáez-Schwedt, A; Sánchez-Giralda, T. Coefficient assignability and a block decomposition for systems over rings. Linear Algebra Appl. 2008, 429, 1277–1287. [19] Schmale, W; Sharma, PK. Cyclizable matrix pairs over C[y] and a conjecture on To¨ plitz pencils. Linear Algebra Appl. 2004, 389, 33-42. [20] Sontag, ED. Mathematical Control Theory: Deterministic Finite Dimensional Systems; Springer Verlag: New York, 1998. [21] Vasershtein, LN. Stable rank of rings and dimensionality of topological spaces. Funct. Anal. Appl. 1971, 5, 102–110.



Chapter 5

A SYMPTOTIC B EHAVIOR OF A SSOCIATED OR ATTACHED P RIME I DEALS OF C ERTAIN E XT- MODULES AND T OR - MODULES K. Khashyarmanesh and F. Khosh-Ahang∗ Ferdowsi University of Mashhad, Department of Mathematics, Centre of Excellence in Analysis on Algebraic Structures (CEAAS), P.O.Box 1159-91775, Mashhad, Iran

1 Introduction Asymptotic behavior of the associated and attached prime ideals represents the interface of two major ideas in the study of modules over a commutative ring. The first, the concepts of associated and attached prime ideals, are important valued tools in researcher’s arsenal. The second is the fact that in a Noetherian ring, large powers of an ideal are well behaved, as shown by the Artin-Rees Lemma or the Hilbert polynomial. Although its roots go back further, the interests in asymptotic stability of associated prime ideals are initiated by a question of Ratliff: “What happens to AssR (R/I n ) as n gets large?” Brodmann answered the question, proving that AssR (M/I n M) also stabilizes for large n, where M is a finitely generated module over a commutative Noetherian ring R. Since then, the topic of stability of certain sets of associated (or attached) prime ideals has been growing rapidly. This paper gives a brief history and summery of the active area of asymptotic stability of associated or attached prime ideals. The paper has a section that contains some generalizations (or dual) of Brodmann’s Theorem by using the ‘Hom-functor’ and ‘tensor product’. After that section three lists many of the newer theorems concerning asymptotic stability of certain sets of prime ideals by using the derived functor Tori of the ‘tensor product’ or the derived functor Exti of the ‘Hom-functor’. Finally, last section discusses some of the old and new results about the asymptotic properties of associated and attached prime ideals related to injective, projective or flat modules. ∗ E-mail

addresses: [email protected] and [email protected]

54

K. Khashyarmanesh and F. Khosh-Ahang

Throughout this paper, R denotes a commutative ring with non-zero identity, M is a Noetherian R-module, A is an Artinian R-module and I is an ideal of R. We use N0 (respectively N) to denote the set of non-negative (respectively positive) integers.

2 Asymptotic Behavior of Sets Related to Hom-Functor and Tensor Product In 1976, Ratliff essentially initiated the study of asymptotic behavior of certain sets of associated primes. In [28] he stated the following condition: (PD0 ): If R is a Noetherian domain and p is a prime ideal of R containing I such that p ∈ AssR (R/I k ) for some k ∈ N, then p ∈ AssR (R/I n ) for all large n. This is known to hold when I is principal [25, (12.6)], and the original purpose of him in [28] was to prove that (PD0 ) holds for all ideals in all Noetherian domains. Although he showed that many ideals do satisfy (PD0 ), he was unable to show that all ideals do. In this regard, he proposed condition (PD) as follows, which is more general than (PD0 ). (PD): For each prime ideal p in a Noetherian ring R such that IRp is not nilpotent and p ∈ AssR (R/I k ) for some k ∈ N, it holds that p ∈ AssR (R/I n ) for all large n. To this end, he first gave information concerning prime divisors of integral closures ofI n which one of them is generalization of a result of Rees ([31, Theorem 6.7]) in this context. In fact, in [28], Ratliff prepared different circumstances under which(PD) or (PD0 ) holds. Subsequently, Brodmann [3] showed that if R is Noetherian and M is a finitely generated R-module, then the sequence AssR ( I nMM ) is ultimately constant for large n ∈ N. In fact, he In M ) is ultimately constant for large n ∈ N, and then dealt showed that the sequence AssR ( I n+1 M M with AssR ( I n M ) by showing that it is not much larger. It is very easy consequence of Brodmann’s results that the same conclusions hold if we relax the hypotheses and assume only that R is a commutative ring and M is a Noetherian R-module. After that, in 1979, McAdam and Eakin presented an exposition of Brodmann’s argument in slightly greater generality. Now, let As∗R (I, M) and Bs∗R (I, M) denote respectively their ultimate constant values. The main results of [21] show that As∗R (I, R) − Bs∗R (I, R) ⊆ AssR (R). Also, they determined As∗ (I, R) explicitly in certain conditions. Sharp in [35] has generalized the result of [21] to Noetherian modules. He has shown that As∗R (I, M) − Bs∗R (I, M) ⊆ AssR (M) for all Noetherian R-modules M. Dual to the above results, Sharp has also shown in [34] and [35] that both the sequences of sets AttR (0 :A I n ) and AttR ((0 :A I n+1 )/(0 :A I n )) are ultimately constant for large n ∈ N and if, similarly, At∗R (I, A) and Bt∗R (I, A) denote respectively their ultimate constant values, then At∗R (I, A) − Bt∗R (I, A) ⊆ AttR (A).

Asymptotic Behavior

55 n

I M Simple proofs of stability of the sets AssR ( I nMM ), AssR ( I n+1 ), AttR (0 :A I n ) and AttR ((0 :A M I n+1 )/(0 :A I n )) were given in [22]. In [32] Rush has further expanded upon this by showing that if N is an R-submodule of M, then AssR ( IM n N ) is ultimately constant for large n ∈ N, and n if B is an R-submodule of A, then AttR (B :A I ) is ultimately constant for large n ∈ N. On a different course of investigation, Katz, McAdam and Ratliff have shown in [12, Corollary 1.8(c)] that if R is Noetherian, I1 , . . . , Ig are regular ideals of R, and (am (1), . . . , am (g))m∈N is a sequence of g-tuples of non-negative integers which is non-decreasing in the a (1) a (g) sense that ai ( j) 6 ai+1 ( j) for all j = 1, . . . , g and all i ∈ N, then AssR (R/I1 n . . . Ig n ) is independent of n for all large n. Afterwards, in [20], McAdam has put forward the question “Can we drop the assumption that I1 , . . . , Ig are regular?”. Let I1 , . . . , Ig be ideals of R and let (am (1), . . . , am (g))m∈N be a sequence of g-tuples of non-negative integers which is nondecreasing. Kingsbury and Sharp, in [17], have shown that ifN is a submodule of M and B is a submodule of A, then a (1) a (g) AssR (M/I1 n . . . Ig n N)

and

a (1)

AttR (B :A I1 n

a (g)

. . . Ig n

)

are both ultimately constant for large n, which is an affermative answer to the question of McAdam in [20]. Katzman, in [10], has shown that, for two elements a, b of R, the set ∪n∈N AssR (M/(an , bn )M) need not be finite. Let a, b ∈ R be such that b is a non-zerodivisor on M and there exists k ∈ N with am M ∩ bn M = bn−k (am M ∩ bk M) for all m ∈ N and all n > k. Then Brodmann, Rotthaus and Sharp in [10], over a Noetherian ring R, showed that the set ∪m,n∈N AssR (M/(am , bn )M) is finite. Recently, Nhan in [26] introduced the notion of generalized regular sequence as some extension of the known regular sequence. In a Noetherian local ring(R, m), she said / p, that a sequence x1 , . . . , xr of elements of m is a generalized regular sequence on M if xi ∈ M for all p ∈ AssR ( (x1 ,...,xi−1 )M ) satisfying dimR (R/p) > 1, for all i = 1, . . . , r. Then she proved that if x1 , . . . , xr is a generalized regular sequence on M, then [ M AssR (xn11 , . . . , xnr r )M n ,...,n ∈N 1

r

is a finite set.

3 Asymptotic Behavior of Sets Related to Derived Functors Melkersson and Schenzel, in [24], have proved the stability results for the associated respectively attached prime ideals for the derived functor Tori of the tensor product respectively Exti of the Hom-functor as follows. Theorem 3.1. Let R be a Noetherian ring. Then

56

K. Khashyarmanesh and F. Khosh-Ahang (i) for a given i ∈ N0 the sequence of finite sets of associated prime ideals AssR (TorRi (R/I n , M)) resp AssR (TorRi (I n−1 /I n , M)), n ∈ N, becomes, for large n, independent of n.

(ii) for a given i ∈ N0 the sequence of finite sets of attached prime ideals AttR (ExtiR (R/I n , A)) resp AttR (ExtiR (I n−1 /I n , A)), n ∈ N, becomes, for large n, independent of n. Of course, they conjectured that Theorem 3.1(ii) is true also for a finitely generated ideal I in an arbitrary commutative ring R. Also, they asked whether the sets of prime ideals [ [

AssR (TorRi (R/I n , M))

i∈N0 n∈N

and

[ [

AttR (ExtiR (R/I n , A))

i∈N0 n∈N

are finite? This seems to be an interesting question related to homological properties ofM and A respectively. Moreover they asked about the stability of the sequence AssR (ExtiR (R/I n , M)), n ∈ N, for a given i ∈ N0 . S In fact, n∈N AssR (ExtiR (R/I n , M)) is not a finite set in general, and therefore the set AssR (ExtiR (R/I n , M)) depends on n for large n. Indeed, Katzman [11, Corollary 1.3] gave an example of a Noetherian local ring (R, m) with two elements x, y ∈ m such that 2 (R)) is an infinite set, where H 2 (R) is the second local cohomology module AssR (H(x,y) (x,y) of R with respect to the ideal generated by x and y. Now, since ExtiR (R/I n , M) HIi (M) ∼ = lim −→ n∈N0

S

for all i ∈ N0 , the set n∈N AssR (Ext2R (R/(x, y)n , R)) is infinite (see also [36]). The first present author and Salarian, in [16], have shown that when I is finitely generated, two sequences AttR (TorR1 (R/I n , A)) and AssR (Ext1R (R/I n , M)) are ultimately constant. Furthermore, some researchers are interested in studying the stability of certain subsets of these sequences. For example in [13], it is shown that over a commutative Noetherian local ring R when M is a finitely generated R-module and n ∈ N such that for all j < i the support of local cohomology module HIj (M) is finite, the set (

[

AssR (ExtiR (R/I n , M))) ∩ {p ∈ Spec(R) | dimR R/p > 1}

n∈N

is finite (see also [15, Theorem B(α)]). Recently, in [4], Brodmann and Nhan generalized the concept of regular sequence. Over the commutative Noetherian ring R, for s ∈ N0 , the sequence x1 , . . . , xr of elements of R is

Asymptotic Behavior

57

called to be an M-sequence in dimension > s if x1 , . . . , xr is a poor Mp -sequence for all p ∈ Spec(R) with dimR (R/p) > s. For convenience, for a subset T of Spec(R) and an integer i ∈ N0 , we set (T )i := {p ∈ T | dimR (R/p) = i}, (T )>i := {p ∈ T | dimR (R/p) > i} and (T )>i := {p ∈ T | dimR (R/p) > i}. The main results of [4] are the following two theorems. Theorem 3.2. Let R be a commutative Noetherian ring, s ∈ N0 and r ∈ N. Assume that dimR (SuppR (HIi (M))) 6 s for all i < r. Then for any system of generators a1 , . . . , ak of I and for all integers t 6 r, the sets [ AssR (ExttR (R/I n , M)) >s

n∈N

and

[

AssR (ExttR (R/(an11 , . . . , ank k ), M))

>s

n1 ,...,nk ∈N

are contained in the finite set

St

i i=0 AssR (ExtR (R/I, M)).

Theorem 3.3. Let R be a commutative Noetherian ring, s ∈ N0 and r ∈ N. Assume that dimR (SuppR (HIi (M))) 6 s for all i < r. Let x1 , . . . , xr ∈ I be a sequence which is at the same time an unconditioned M-sequence in dimension > s and an unconditioned I-filter regular sequence with respect to M. Then for any system of generators a1 , . . . , ak of I and for all integers t 6 r, the sets [ AssR (ExttR (R/I n , M)) >s

n∈N

and

[

AssR (ExttR (R/(an11 , . . . , ank k ), M))

n1 ,...,nk ∈N

>s

are contained in the finite set

AssR

M (x1 , . . . , xt )M

>s+1

∪

t [ i=0

AssR

M (x1 , . . . , xt )M

.

s

After that, in [14], we improved or regained the results of [4], [13] and [16]. In fact, at first we proved that whenever R is a commutative Noetherian ring and s, n ∈ N0 such that S dimR (SuppR (HIi (M))) 6 s for all i < n, the sets ( j∈N SuppR (ExtiR (R/I j , M)))s for all i < n S and the set ( j∈N AssR (ExtiR (R/I j , M)))s for all i 6 n, are finite (see [14, Theorem 2.5]). Then, by using a method similar to the previous one, we gained the next proposition which shows that Theorem 2.4 in [13] is not so considerable, because the considered set is empty. Proposition 3.4. Let s, n ∈ N0 such that dimR (SuppR (Hai (M))) 6 s for all i with i < n. Then S the set ( j∈N AssR (ExtiR (R/I j , M)))>s is empty for all i with i 6 n.

58


As an immediate consequence, we can regain the first parts of Theorems 3.2 and 3.3. Now, we present the following theorem which provides a partial answer to the Melkersson and Schenzel’s question. Theorem 3.5. Assume that n ∈ N0 , R is semi-local commutative Noetherian ring and SuppR (HIi (M)) is finite for all i with i < n. Then (i) the set

S

i j j∈N SuppR (ExtR (R/I , M))

(ii) the set

S

i j j∈N AssR (ExtR (R/I , M))

is finite for all i with i < n;

is finite for all i with i 6 n.

S

Proof. Since j∈N SuppR (ExtiR (R/I j , M)) is contained in SuppR (M/IM), if dimR (M/IM)6 1, then we have that S

i j j∈N SuppR (ExtR (R/a , M))

⊆ m-Spec(R) ∪ AssR (M/IM).

So, we may assume that dimR (M/IM) > 1. By [14, Remark 2.3], dimR (SuppR (HIi (M))) 6 1 for all i with i < n. Hence, in view of S Lemma 2.4, the set ( j∈N SuppR (ExtiR (R/I j , M)))>1 is empty for all i with i < n. Now, since R is semi-local, it is easy to see that part (i) immediately follows from [14, Theorem S 2.5(i)]. By (i), it is enough to show that j∈N AssR (ExtnR (R/I j , M)) is finite. To achieve this, since R is semi-local, in the light of [14, Theorem 2.5(ii)], we only need to show that the set S ( j∈N AssR (ExtnR (R/I j , M)))>1 is finite. Now, Proposition 3.4 completes the proof. It is routine to check that, by using the ideas in [16], one can reduce the study of Melkersson and Schenzel’s question to the case that M is a-torsion-free, when i = 1. So, in the case that R is semi-local, Theorem 3.5 implies the dual of the main result of [16]. We end this section by the following corollary which is an immediate consequence of Theorem 3.5. Corollary 3.6. Assume that R is semi-local. Set g := depth(a, M). Then the set [

AssR (ExtgR (R/I j , M))

j∈N

is finite. Note that, for an Artinian R-module A and a fixed integer i, by using Matlis duality functor, one can put all our various results in [14] together to obtain some corollaries about S the finiteness properties of j∈N AttR (TorRi (R/I j , A)).

4

Asymptotic Behavior of Sets Related to Projective, Injective and Flat Modules

Since the class of all R-modules which possess secondary representation can be more extensive than the class of all Artinian R-modules and in [33] it is shown that every injective module over the commutative Noetherian ring R, possesses a secondary representation. Ansari-Toroghi and Sharp, in [2], generalized the results of [34] for injective modules. In fact, they proved the following theorem:

Asymptotic Behavior

59

Theorem 4.1. Let E be an injective module over a commutative Noetherian ring R. Then (i) the sequences AttR (0 :E I n ) and AttR ((0 :E I n+1 )/(0 :E I n )) are ultimately constant. (ii) At∗R (I, E) = {p0 ∈ As∗R (I, R) | p0 ⊆ p for some p ∈ OccR (E)} and Bt∗R (I, E) = {p0 ∈ Bs∗R (I, R) | p0 ⊆ p for some p ∈ OccR (E)}, where OccR (E) is explained as follows: By well-known work of Matlis and Gabriel, L there is a family {pα }α∈Λ of prime ideals of R for which E = α∈Λ E(R/pα ) (we use E(L) to denote the injective envelope of an R-module L), and the set {pα | α ∈ Λ} is ultimately determined by E; we denote it by OccR (E). (iii) Bt∗R (I, E) ⊆ At∗R (I, E) and At∗R (I, E) − Bt∗R (I, E) ⊆ Att(E). Melkersson and Schenzel, in [23], have shown that if E is an injective R-module, then both the sequences of AttR (0 :HomR (M,E) I n ), n ∈ N, and AttR ((0 :HomR (M,E) I n+1 )/(0 :HomR (M,E) I n )), n ∈ N are, for sufficiently large n, independent of n. Dual to the above results, Nishitani in [27] has shown that the sets AssR (HomR (A, E)/I n HomR (A, E)), n ∈ N, and AssR (I n HomR (A, E)/I n+1 HomR (A, E)), n ∈ N, are, for sufficiently large n, independent of n. Also, he has shown that n−1 HomR (A, E) I HomR (A, E) = AssR ∪ p ∈ AssR (HomR (A, E)) | I ⊆ p AssR n n I HomR (A, E) I HomR (A, E) for sufficiently large n. Both these works are based on using the exact functor HomR (., E). In [7], Divaani-Aazar and Tousi provide a generalization of [32] and Theorem 3.1. They have proved that if P is a projective R-module and N is a submodule of M, then the sets AssR (HomR (P, M)/I n HomR (P, N)), n ∈ N, and AssR (I n HomR (P, M)/I n+1 HomR (P, N)), n ∈ N, both become eventually constant. Moreover, they have shown that, for eachi ∈ N0 , the sets AssR (TorRi (R/I n , HomR (P, M))), n ∈ N,

60


and AssR (TorRi (I n /I n+1 , HomR (P, M))), n ∈ N, are, for sufficiently large n, independent of n. Also, in a dual manner they have proved that, if B is a submodule of A and C is a submodule of B, then two sequences of attached prime ideals AttR (HomR (P, B) :HomR (P,A) I n ), n ∈ N, and AttR ((HomR (P, B) :HomR (P,A) I n )/(HomR (P,C) :HomR (P,A) I n )), n ∈ N, become eventually constant. In addition, they have proved that, for each i ∈ N0 , two sequences of sets AttR (ExtiR (R/I n , HomR (P, A))), n ∈ N, and AttR (ExtiR (I n /I n+1 , HomR (P, A))), n ∈ N, are, for sufficiently large n, independent of n. Now, let F be a flat R-module. In [8], Divaani-Aazar and Tousi have proved that if B is a submodule of A and C is a submodule of B, then both of the sequence AttR (F ⊗R B :F⊗R A I n ), n ∈ N, and AttR ((F ⊗R B :F⊗R A I n )/(F ⊗R B :F⊗R A I n+1 )), n ∈ N, are eventually constant. Also, they have showed that ifR is Noetherian, then for each i ∈ N0 , the sets AttR (ExtiR (R/I n , F ⊗R A)) and AttR (ExtiR (I n /I n+1 , F ⊗R A)) are independent of n for sufficiently large n. These mentioned results in the above paragraph are based on using a certain exact functor. In [9], Divaani-Aazar and Tousi studied asymptotic behavior of prime ideals related to exact functors in general. There have been four attempts to dualize the theory of associated prime ideals by Macdonald [18], Chambless [6], Zöshinger [38] and Yassemi [37]. The concept of coassociated prime ideals was introduced by Chambless, Z¨ oshinger and Yassemi in different ways. However, these concepts are equivalent (see [38], [37, 1.6] and [37, 1.7]). In [37] the concept of coassociated prime ideals is introduced in terms of cocyclic modules: A moduleM over a Noetherian ring R is cocyclic if it is a submodule of E = ER (R/m) for some maximal ideal m of R (for an R-module L, we use E(L) to denote the injective hull of L). Then a prime ideal of R is said to be a coassociated prime of M if there exists a cocyclic homomorphic image L of M such that AnnR (L) = p. The set of coassociated primes of M is denoted by CoassR (M). Note that whenever an R-module M has a secondary representation, then AttR (M) = CoassR (M). In 1998, Ansari-Toroghi, in [1], showed that when R is commutative Noetherian ring and N is a finitely generated R-module, the sequences of sets CoassR (0 :HomR (N,M) I n ), n ∈ N,

Asymptotic Behavior (0 :HomR (N,M) I n ) , n ∈ N, CoassR (0 :HomR (N,M) I n−1 ) N ⊗R M AssR n , n ∈ N, I (N ⊗R M) and

61

I n (N ⊗R M) , n ∈ N, AssR n+1 I (N ⊗R M) are ultimately constant in the following cases:

(i) M is a finitely generated R-module; (ii) M is an injective R-module; (iii) M is an Artinian R-module; (iv) M is a flat R-module. Also, he has proved that if N, M, F and E are respectively a finitely generated, an Artinian, a flat and an injective R-module, then for a given i ∈ N0 , the sequences of sets AttR (ExtiR (N/I n N, M)), n ∈ N, AttR (ExtiR (I n N/I n+1 N, M)), n ∈ N, CoassR (ExtiR (R/I n , HomR (N, E))), n ∈ N, CoassR (ExtiR (I n /I n+1 , HomR (N, E))), n ∈ N, AssR (TorRi (R/I n , N ⊗R F)), n ∈ N, and AssR (TorRi (I n /I n+1 , N ⊗R F)), n ∈ N, are eventually constant. Moreover, he generalized the results of Rush, in [32], as follows. Theorem 4.2. (i) Let M and F be respectively an Artinian and a flat R-module. Let M0 and M 00 be submodules of M such that M00 ⊆ M 0 . Then the sequences of sets AttR (HomR (F, M 0 ) :HomR (F,M) I n ), n ∈ N, and AttR

(HomR (F, M 0 ) :HomR (F,M) I n ) , n ∈ N, (HomR (F, M 00 ) :HomR (F,M) I n )

are ultimately constant. (ii) Let N, M and F be respectively a finitely generated, an Artinian and a flat module over a commutative Noetherian ring R. For a given i ∈ N0 , the sequences of sets AttR (ExtiR (N/I n N, HomR (F, M))), n ∈ N, AttR (ExtiR (I n N/I n+1 N, HomR (F, M))), n ∈ N, AttR (ExtiR (F/I n F, M)), n ∈ N, and AttR (ExtiR (I n F/I n+1 F, M)), n ∈ N, are ultimately constant.

62


References [1] Ansari-Toroghy, H. Associated and coassociated primes.Comm. Algebra 26 (1998), no. 2, 453–466. [2] Ansari-Toroghy, H., & Sharp, R. Y. Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings. Proc. Edinburgh Math. Soc. (2) 34 (1991), no. 1, 155–160. [3] Brodmann, M. Asymptotic stability of Ass(M/I n M). Proc. Amer. Math. Soc. 74 (1979), no. 1, 16–18. [4] Brodmann, M., & Nhan, L. T. A finiteness result for associated primes of certain Ext-modules. Comm. Algebra. (to appear). [5] Brodmann, M., Rotthaus, Ch., & Sharp, R. Y. On annihilators and associated primes of local cohomology modules. J. Pure Appl. Algebra 153 (2000), no. 3, 197–227. [6] Chambless, L. Coprimary decompositions, N-dimension and divisibility: application to Artinian modules. Comm. Algebra 9 (1981), no. 11, 1131-1146. [7] Divaani-Aazar, K., & Tousi, M. Asymptotic associated and attached prime ideals related to projective modules. Comm. Algebra 25 (1997), no. 4, 1129–1142. [8] Divaani-Aazar, K., & Tousi, M. Asymptotic attached primes related to flat modules. Algebra Colloq. 6 (1999), no. 2, 193–204. [9] Divaani-Aazar, K., & Tousi, M. Asymptotic prime ideals related to exact functors. Comm. Algebra 27 (1999), no. 8, 3949–3968. [10] Katzman, M. Finite criteria for weak F-regularity. Illinois J. Math. 40 (1996), no. 3, 453–463. [11] Katzman, M. An example of an infinite set of associated primes of a local cohomology module. J. Algebra 252 (2002), no. 1, 161–166. [12] Katz, D., McAdam, S., & Ratliff, L. J., Jr. Prime divisors and divisorial ideals. J. Pure Appl. Algebra 59 (1989), no. 2, 179–186. [13] Khashyarmanesh, K. On the finiteness properties of Algebra 34 (2006), no. 2, 779–784.

S

n i i AssR ExtR (R/a , M).

Comm.

[14] Khashyarmanesh, K., & Khosh-Ahang, F. Asymptotic behaviour of certain sets of associated prime ideals of Ext-modules. Manuscripta Math. 125 (2008), no. 3, 345– 352. [15] Khashyarmanesh, K., & Salarian, Sh. On the associated primes of local cohomology modules. Comm. Algebra 27 (1999), no. 12, 6191–6198. [16] Khashyarmanesh, K., & Salarian, Sh. Asymptotic stability of AttR TorR1 ((R/an ), A). Proc. Edinb. Math. Soc. (2) 44 (2001), no. 3, 479–483.

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[17] Kingsbury, A. K., & Sharp, R. Y. Asymptotic behaviour of certain sets of prime ideals. Proc. Amer. Math. Soc. 124 (1996), no. 6, 1703–1711. [18] Macdonald, I. G. Secondary representation of modules over a commutative ring. Symposia Mathematica, Vol. XI, 23-43, (Convegno di Algebra Commutativa, INDAM, Rome, 1971). Academic Press, London, 1973. [19] McAdam, S. Asymptotic prime divisors. Lecture Notes in Mathematics, 1023. Berlin, Springer-Verlag, 1983. [20] McAdam, S. Primes associated to an ideal. Contemporary Mathematics, 102. American Mathematical Society, Providence, RI, 1989. [21] McAdam, S., & Eakin, P. The asymptotic Ass. J. Algebra 61 (1979), no. 1, 71–81. [22] Melkersson, L. On asymptotic stability for sets of prime ideals connected with the powers of an ideal. Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 2, 267–271. [23] Melkersson, L., & Schenzel, P. Asymptotic attached prime ideals related to injective modules. Comm. Algebra 20 (1992), no. 2, 583–590. [24] Melkersson, L., & Schenzel, P. Asymptotic prime ideals related to derived functors. Proc. Amer. Math. Soc. 117 (1993), no. 4, 935–938. [25] Nagata, M. Local rings. Interscience Tracts in Pure and Applied Mathematics, No. 13. Interscience Publishers a division of John Wiley & Sons, New York-London, 1962. [26] Nhan, L. T. On generalized regular sequences and the finiteness for associated primes of local cohomology modules. Comm. Algebra 33 (2005), no. 3, 793–806. [27] Nishitani, I. Associated prime ideals of the dual of an Artinian module relative to an injective module. Comm. Algebra 22 (1994), no. 7, 2651–2667. [28] Ratliff, L. J., Jr. On prime divisors of I n , n large. Michigan Math. J. 23 (1976), no. 4, 337–352. [29] Ratliff, L. J., Jr. A brief survey and history of asymptotic prime divisors. Rocky Mountain J. Math. 13 (1983), no. 3, 437–459. [30] Ratliff, L. J., Jr. On asymptotic prime divisors. Pacific J. Math. 111 (1984), no. 2, 395–413. [31] Rees, D. Valuations associated with a local ring. II.J. London Math. Soc. 31 (1956), 228–235. [32] Rush, D. E. Asymptotic primes and integral closure in modules. Quart. J. Math., Oxford Ser. (2) 43(172) (1992), 477–499. [33] Sharp, R. Y. Secondary representations for injective modules over commutative Noetherian rings. Proc. Edinburgh Math. Soc. (2) 20 (1976), no. 2, 143–151.

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K. Khashyarmanesh and F. Khosh-Ahang [34] Sharp, R. Y. Asymptotic behaviour of certain sets of attached prime ideals. J. London Math. Soc. (2) 34 (1986), no. 2, 212–218. [35] Sharp, R. Y. A method for the study of Artinian modules, with an application to asymptotic behavior. Commutative algebra (Berkeley, CA, 1987), 443–465, Math. Sci. Res. Inst. Publ., 15, Springer, New York, 1989. [36] Singh, A., & Swanson, I. Associated primes of local cohomology modules and of Frobenius powers, Int. Math. Res. Not. 2004, no. 33, 1703–1733, [37] Yassemi, S. Coassociated primes. Comm. Algebra 23 (1995), no. 4, 1473–1498. [38] Zöshinger, H. Linear-kompakte Moduln u¨ ber noetherschen Ringen. Arch. Math. (Basel) 41 (1983), no. 2, 121-130.



Chapter 6

L INEAR A LGEBRA OVER C OMMUTATIVE R INGS A PPLIED TO C ONTROL T HEORY∗ Miguel V. Carriegos Universidad de León. Spain

Abstract In this chapter we review some recent results on linear systems theory over commutative rings. We use linear algebra over commutative rings as main tool. Some general considerations about the influence of linear algebra on systems theory are given in the Introduction. Then we deal with both the linear algebra and control theory background we need in the next sections. Reachability is one of the main subjects in control theory. In the linear case, reachability can be studied as the surjectivity of some linear map. Then the problem can be stated in terms of linear equations and solved by standard methods. Algebraic and feedback equivalence are also studied. InvariantR-modules for both equivalence actions are presented. Consequently we can give canonical forms not only in the usual case of constant coefficients (complex numbers) but in the case of commutative rings (real numbers, finite fields, integers, polynomials, real functions,...). The stabilization of linear systems are related with the pole-shifting property, which is the ability of change some poles in the characteristic polynomial. Some results are given in this subject. Finally we touch some classes of rings related with systems theory; here the main tool is commutative algebra. Last section is a conclusion of the chapter. Some unsolved research questions are included.

1. Introduction Linear algebra deals with matrices, linear equations and vector spaces or modules. Results in linear algebra have been applied to many research fields for a long time. Control theory is one of the research fields where that assertion is more suitable [29]. ∗

A version of this chapter was also published in Linear Algebra Research Advances edited by Gerald D. Ling, published by Nova Science Publishers. It was submitted for appropriate modifications in an effort to encourage wider dissemination of research.

66

Miguel V. Carriegos

This chapter deals with linear algebra applied to control theory but our purpose is not to list a sequence of advances or even to set up a bridge from one field passing into the another; our main interest is to review some very general results involving both linear algebra over commutative rings and control theory. The generalization is given by using a commutative ring of scalars R instead of the field C of complex numbers or even a general field k. This gives us an obvious generalization from the mathematical point of view and, in the control-theoretic side, allows us to consider linear control systems with constant real coefficients, systems with coefficients in a finite field (coding, convolutional codes); digital systems (coefficients in the ring Z of integers or in a finite ring [21]); delay systems (with coefficients in a polynomial ring [26] or [28]); families of real systems parametrized by some topological space Λ (coefficients in the ring C (Λ, R) of real continuous functions defined on Λ, see [10] or [13]); and non-linear systems linearized at an equilibrium point [3]. For general reading on the subject the reader is referred to [3], [12], [18] and [34] as some main references in mathematical systems theory. There is a correspondence between algebraic properties of matrices and properties of linear systems described by those matrices. For instance the reachability property of linear systems is stated in terms of the surjectivity of some linear map. On the other hand characteristic polynomials are the key tool to study the stabilization of linear systems, and the so-called pole-shifting property (that is, the ability to shift some roots of characteristic polynomial) gives rise to interesting algebraic properties of ringR of scalars. Finally, the algebraic equivalence (change of basis) of linear systems and the more general feedback equivalence are studied in terms of some invariant (up to isomorphism) R-modules associated to the given system. Some decomposition results involving these modules will gives us (in some cases) a way to set up canonical forms for linear systems. Here [4] and [30] are good references in matrix equivalence problems related with subjects in this chapter. This chapter is organized as follows: First we review some well (perhaps very well) known facts on linear algebra in order to give us some reference facts. This is done in the next paragraphs of the Introduction. Then in section 2 we study the reachability notion, which is a central subject of control systems, in terms of surjectivity of some linear maps. Section 3 is devoted to review the algebraic invariants associated to both the algebraic and feedback equivalence of linear systems while section 4 deals with the stabilization of linear systems in terms of the shifting, by feedback actions, the characteristic polynomial of the state transition linear map. Finally we include a section 5 as a conclusion of the chapter.

1.1. Some Linear Algebra Tools A Natural Block Structure for Matrices Let k be any field and let V1 , V2 be k-vector spaces of finite dimensions n1 and n2 respectively; and let f be a fixed linear map f : V1 → V2 . Every choice of basis B1 of V1 and B2 of V2 gives rise to isomorphisms V1 ∼ = k n1 and V2 ∼ = k n2 which assign to any vector in V1 (respectively V2 ) its coordinates re-

Linear Algebra over Commutative Rings Applied to Control Theory

67

ferred to the basis B1 (resp. B2 ) and consequently gives the expression of the matrix A = M at(f ; B1 , B2 ) for the linear map f in these bases. Choice of bases is crucial: In fact only one invariant (the rank r of f , which is the dimension of vector space Im(f ) = {f (v) : v ∈ V1 }) characterizes the linear map f up to change of bases. Rank of linear map f can be obtained from every matrix A of f by determinantal calculus on matrix A (as the greater integer r such that there exists some nonzero r × r minor of A) or directly for some“clever” choice of bases:Take a basis {f (v1 ), ..., f (vr )} of Im(f ) and complete it to a basis C2 = {f (v1 ), ..., f (vr )} ∪ {wr+1 , ..., wn2 }of V2 . Now the set {v1 , ..., vr } is a linearly independent subset of V1 and hence it may be completed to a basis C1 = {v1 , ..., vr } ∪ {vr+1 , ..., vn1 } of V1 . Then the matrix of f in bases C1 and C2 is 

   M = Mat(f ; C1 , C2 ) =    



1 ..

. 1 0 ..

.

      

We will denote this matrix by giving a block expression M=

1r×r

0r×(n1 −r)

0(n2 −r)×r

0(n2 −r)×(n1 −r)

or simply by M=

1 0 0 0

when the size of blocks are clear or can be easily obtained from the context. The above natural block structure (or normal form) for a matrix with entries in a field is of course a very well known fact in linear algebra. Our motivation for reviewing it here is that the iteration of this procedure will gives us a main tool to study in the sequel linear control systems over a field. For the study of linear systems over a commutative ring R with unit element 1 6= 0 we need to introduce the ”cast of characters” of linear algebra over commutative rings: We assume that the reader knows the definition and main properties of R-modules and, in particular, free R-modules. A linear map f : M1 → M2 between free R-modules of finite rank is defined by some matrix once we have fixed bases for M1 and M2 . But the above natural block structure cannot be reached in general ifR is not a field: Just take R = M1 = M2 = Z and f = multiplication by 2. The first gap is that Im(f ) must be free to obtain a basis; then supplement of Im(f ) to M2 must be also free to complete the basis to a basis of M2 ; the anti images of elements in the chosen basis of Im(f ) must also be completable to a basis of M1 . All these restrictions rarely are verified. In some cases we may give an analogous to the above natural block structure. In these cases we will be able to achieve similar results for linear systems over commutative rings that of obtained for the classical case of fields. For these cases we need to work with projective modules.

68

Miguel V. Carriegos

Projective Modules Definition 1.1. A R-module P is called projective if there exists a R-module Q such that the direct sum P ⊕ Q is free. This is equivalent to saying that P satisfies the ”projective lifting property”: For every surjective linear map s : M → N and every linear map f : P → N there exists a map g : P → M such that s ◦ g = f :

M

.∃g →s

P ↓f N

→ 0

Note that the projective lifting forces that if P is projective then every exact sequence 0 → N

→i M

→s P

→ 0

splits; that is to say, one has the decomposition M∼ =N ⊕P and there exist s0 and i0 , one-sided inverses for s and i respectively, such that the sequence may be reverted 0 ← N ←i0 M ←s0 P ← 0 Note that a direct summand of a projective R-module is straightforward projective itself. Lemma 1.2. Let f : P1 → P2 be a linear map where P1 and P2 are finitely generated projective R-modules. Suppose that the R-module Coker(f ) = P2 /Im(f ) is projective. Then: (i) There exists decompositions P1 = N ⊕K, and P2 =Im(f )⊕C, where N is isomorphic to Im(f ), K is isomorphic to ker(f ), and C is isomorphic to Coker(f ). (ii) According to decompositions in (i), the linear map f is defined by the matrix of linear maps f˜ 0 : N ⊕ K → Im (f ) ⊕ C 0 0 where fe : N →Im(f ) is an isomorphism. Proof. Since Coker(f ) is projective it follows that the following natural exact sequence splits 0 → Im (f ) →inclusion P2 →quotient P2 /Im (f ) = Coker (f ) → 0 , thus P2 =Im(f ) ⊕ C where C is isomorphic to Coker (f ). Denote by π1 : P2 →Im(f ) the projection of P2 onto its first direct summand, then the following exact sequence 0 → Ker (f ) →inclusion P1 →π1 ◦f

Im (f ) → 0 ,


69

splits because Im(f ) is projective. Then P1 = N ⊕Ker(f ) where N is isomorphic to Im(f) ˜ via (π1 ◦ f ) = f. With respect to the above decompositions, the linear map f : P1 = N ⊕ Ker(f ) → Im (f ) ⊕ C = P2 , f˜ 0 is given by the matrix of homomorphisms 0 0 The reader is referred to the first chapter of [37] as a wonderful concise introduction to projective modules. A-invariant Modules Definition 1.3. Let R be a commutative ring and let X be a finitely generated projective R-module of constant rank n. Let A : X → X be a linear map. We say that the submodule M ⊆ X is A-invariant if A(M ) ⊆ M . In the case of X = Rn being a free R-module then it is usual to state the A-invariance in terms of a matrix A ∈ Rn×n . Note that the trivial submodules 0 and Rn are A-invariant for all n × n matrix A. On the other hand it’s worth asking about the least A-invariant submodule containing M : From the easily checked fact that the character of A-invariance passes through intersections it follows that the intersection of all A-invariant modules containing M is in fact the least A-invariant module containing M . But we can do anything more: Lemma 1.4. Let X be a finitely generated projective R-module X of constant rank n. Let A : X → X be a linear map. The least A-invariant submodule containing M ⊆ X is given by the finite sum module A? (M ) = M + A(M ) + A2 (M ) + · · · + An−1 (M ) Proof.- First suppose that X = Rn . It is clear that A? (M ) ⊇ M . On the other hand is A-invariant containing M because, by Cayley-Hamiton Theorem, An can be written as a ”linear combination”of 1, A, ..., An−1 hence A? (M )

A(A? (M )) = A(M + A(M ) + A2 (M ) + · · · + An−1 (M )) = = A(M ) + A2 (M ) + · · · + An−1 (M ) + An (M ) ⊆ A? (M ) On the other hand, for the case of X being a finitelly generated R-module of constant rank n, the same result is valid because in this case there is a theory of determinant of endomorphisms that generalizes the usual one on matrices (see [25]). The characteristic polynomial and Cayley-Hamilton Theorem are valid just as in the free case (see [19], [22], [23] or [25]). Along this chapter we will use least A-invariant modules. If R-module M ⊆ Rn is finitelly generated by columns {b1 , ..., bs } then the columns of the block matrix B, AB, ..., An−1 B ,

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where B = (b1 , ...,bs ) , generate A? (M ). We will also denote by A? (B) the matrix B, AB, ..., An−1 B . Note 1.5. Let A : X → X and let M ⊆ X be a finitely generated submodule. Suppose that the quotient module X/A? (M ) is projective. Then one has the direct sum decomposition X ∼ = A? (M ) ⊕ X/A? (M ) and, according to this decomposition of X,linear map defined by A is given by a matrix of linear maps on a11 a12 : A? (M ) ⊕ X/A? (M ) → A? (M ) ⊕ X/A? (M ). the form A = a21 a22 Since A? (M ) is A-invariant then it follows that a21 = 0 is the zero linear map. Then a11 a12 A= 0 a22 then this gives a decomposition of a matrix A into the block matrix If R is a field A11 A12 . This block decomposition of A will be the key for results as Kalman’s 0 A22 reduction in section 3§6 Determinantal Ideals To conclude this introductory section we need to review some facts about determinantal calculations on a matrix we will use sometimes in the sequel. Interested readers are referred to [27] as main reference in the subject. Let A be a (n × m)-matrix over a commutative ring R. Let A : Rm → Rn the homomorphism of free R-modules defined by left multiplication u → Au. Put M = Coker(A) = Rn /Im(A) and consider the following exact sequence A

Rm → Rn → Coker (A) → 0 We denote by Uj (A) the j-th determinantal ideal of the matrix A (i.e. the ideal of R generated by all the (j × j)-minors of A). The following properties hold. (See [27]): det1 R = U0 (A) ⊇ U1 (A) ⊇ · · · ⊇ Uj (A) ⊇ · · · det2 Let f : R → R0 be a ring homomorphism, then we have the equality of ideals of ring R0 Uj (f (A)) = Uj (A) · R0 where f (A) is the (n × m)-matrix over R0 obtained by extension of A via f (that is to say, if A = (aij ) then f (A) = f (aij )). det3 If R = k is a field then dimk (Coker (A)) = n − sup {j : Uj (A) 6= (0)}


71

det4 M = Coker (A) is a projective R-module if and only if every determinantal ideal Uj (A) is principal generated by an idempotent element of R. In particular, if R is a domain (no idempotents 6= 0, 1) then M = Coker (A) is a projective finitely generated R-module of constant rank r if and only if the sequence of determinantal ideals is R = U0 (A) = U1 (A) = · · ·

1.2.

= Un−r (A) 6= Un−r+1 (A) = · · · q 0

Some Control Theory Topics

Let X be a projective finitely generated R-module of constant rank n and let A : X → X be a linear map. For a fixed element x0 ∈ X we obtain the sequence {xi }i≥0 generated by the initial element x0 and the map A xi = Axi−1 A control for this sequential linear dynamical system is a subspace B ⊆ X called the control subspace: One is allowed to add a vector of B to the current state xt at time t. For instance, suppose that we apply the control sequence u = {ui }i≥1 of elements of B to the above system ruled by A with initial state x0 . Then the sequence of states of controlled system is the following: = x0 ΦΣ (0, x0 , {ui }) ΦΣ (1, x0 , {ui })x1 = Ax0 + u1 ΦΣ (2, x0 , {ui }) = x2 = Ax1 + u2 = A2 x0 + Au1 + u2 .. .. .. . . . P t i = xt = A x0 + t−1 ΦΣ (t, x0 , {ui }) i=0 A ut−i .. .. .. . . . where ΦΣ (t, x0 , {ui }) denotes the state of system Σ = (A, B) at “time”= t when it is “initialized” at state x0 and the control sequence ui is applied. In general, a initialized linear system is a triple (A, B, x0 ) where A is the transition map, B are the control subspace, and x0 is the initial state. We usually study systems without a fixed initial state, thus our main definition is the following. Definition 1.6. Let R be a commutative ring and let X be a projective finitely generated Rmodule of constant rank n. A linear system over X is just a pair (A, B) where A : X → X is an endomorphism and B is a finitelly generated R-module of X. A linear system may be depicted by Σ:

B ⊆ X →A X

A linear system over X is just an endomorphism of some state module together with a submodule of controls. However, since B is a finitely generated R-module, we can describe

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B by a finite set of (say) m generators. Thus we can give a linear map B : Rm → X such that Im(B) = B. Then the linear system Σ = (A, B) is defined by a pair of linear maps Σ = (A, B) and the picture is now Rm Σ:

&B X

→A X

Does linear system depend on the set of generators we use to describe B? This is an interesting question we deal in section 3§2. The behavior ΦΣ (t, x0 , u) of system Σ = (A, B) with initial state x0 and control sequence u : N → Rm is given by: ΦΣ (t, x0 , u) = At x0 +

t−1 X

Ai But−i ;

i=0

In the case of X = Rn being free of rank n, system Σ = (A, B) is defined by a pair of matrices that we also denote by A and B. Note that if finitely generated projective R-modules are free then every linear system is given by a pair of matrices. Note 1.7. The class of commutative rings where all finitely generated R-modules are free is very wide containing fields, principal ideal domains, local rings and, among other properties (see [25]), passes through polynomial extensions; thus if finitely generated projective R-modules are free then finitely generated R[x]-modules are free (x an indeterminate). In this chapter we deal with linear algebra involved in the study of several problems related with linear systems theory. Most of these problems are directly related with practical applications we do not treat here. The main motivation of this chapter is to present a few techniques to attack a wide class of problems in control theory. Our techniques are inspired by the celebrated Kalman’s Reduction, with some linear algebra over commutative rings we will be able to attack some interesting control theoretic problems from an unified algebraic point of view. Some of the problems we’ll touch are the following: Problem 1. Reachable states.- Given a linear system Σ, to describe the set of reachable states of Σ from a given initial state x0 ∈ X, that is to say, to describe the image of the map ΦΣ (−, x0 , −). Problem 2. Reachability.- Given a linear system, is it possible to reach any state by some suitable control sequence? Problem 3. Equivalence.- The expression of a linear system as a pair of matrices depends on the chosen bases. Hence it is natural to define two systems Σ and Σ0 as algebraically equivalent if there exists bases change bringing system Σ to system Σ0 . But the key in control theory is the so-called Feedback Action which gives rise to a more general equivalence known as Feedback Equivalence. A central question in control theory is the classification of linear systems up to feedback equivalence.


73

Problem 4. Control of behavior.- Is it possible to design a control sequence u = {ui } in order to obtain some desired behavior ΦΣ (−, x0 , u)? For a given linear system Σ = (A, B), the study of characteristic polynomial χ (A, z) of matrix A is the key to describe the behavior of system Σ. Shifting χ (A, z) by feedback action is perhaps the main tool in linear control theory. We will deal with Problems 1 and 2 in section 2, with Problem 3 in section 3 and with Problem 4 in section 4. More general linear systems can be given for special applications we do not treat here. The reader is referred to [8] [14], [15] and [38] as some references dealing with special types of linear systems.

2.

Reachability

Consider the linear system Σ = (A, B) over the ring of integers Z given by the 1 × 1 matrices A = (1), B = (2). Given an initial state x0 ∈ Z we are asked to say if some target state ω ∈ Z can be reached by the behavior of system Σ for some control sequence u1 , u2 , .... Of course, this is a very easy exercise: Just write down the behavior of system for an indeterminate control sequence: Φ((1),(2)) (0, x0 , ui ) = x0 Φ((1),(2)) (1, x0 , ui ) = x1 = x0 + 2u1 Φ((1),(2)) (2, x0 , ui ) = x2 = x0 + 2u1 + 2u2 Φ((1),(2)) (3, x0 , ui ) = x3 = x0 + 2u1 + 2u2 + 2u3 .. . and then solve for ui to obtain that ω can be reached from x0 if and only if ω and x0 have the same parity (residue modulo 2). Notice that in fact we can reach any reachable state 0 and ui = 0, i ≥ 1; this is not the case in ω from x0 at time t = 1 by setting u1 = ω−x 2 general. But however we can obtain the ”time needed” to reach a fixed target by studying the partial reachability maps (see 2§1). In general, reachability questions are very important in systems theory. Next we review the main definitions and the usual approach to reachability in terms of linear equations. It’s worthy of remark that reachability is closely related to another central topic in control theory that is controllability (for instance, both properties are equivalent for linear systems over C, but they are not the same notion in general, see [8]). Let Σ = (A, B) be an linear system over X. Let x0 , ω be elements of X, following Sontag [34] we say that the state ω is reachable from the initial condition x0 if there exists a control function u : {0, 1, ...} → Rm such that ΦΣ (t, x0 , u) = ω for some t ≥ 0. In ω, or simply by x0 ω, when the expression of the this case we denote the fact by x0 u control function is not needed. Definition 2.1. An m-input linear system Σ over X is reachable if for each x0 ∈ X and ω. each ω ∈ X there exists a control function u such that x0 u

Using the expression of behavior of the system, and by Cayley-Hamilton Theorem for finitely generated projective R-modules of constant rank n, we have

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Theorem 2.2. Let Σ = (A, B) be a linear system over X and let x0 , ω ∈ X be respectively ω if and only if initial and target states. Then x0 n (ω − A x0 ) ∈ Im (B, AB, ..., An−1 B) : (Rm )⊕n → X Proof.- It is straightforward from the calculation of the behaviorof systemΣ = (A, B) given in section 1§2 If X = Rn is free, the result can be stated in terms of linear equations overR: Corollary 2.3. Let Σ = (A, B) be a linear system over Rn and let x0 , ω ∈ Rn be respecω if and only if the following system of nlinear tively initial and target states. Then x0 equations in n · m indeterminates (the entries of the ui ’s) over R has a solution   un−1  ..    (B, AB, ..., An−1 B)  .  = ω − An x0  u1  u0 Now the notion of reachability of system Σ is easily characterized: By definition, Σ is ω for all x0 , ω. This is actually equivalent, by means of the above reachable if one has x0 Theorem, to the linear map given by matrix of linear maps (B, AB, ..., An−1 B) being onto. Thus we have Proposition 2.4. Let Σ = (A, B) be a linear system over X. The following statements are equivalent: 1. System Σ is reachable. 2. Linear map (B, AB, ..., An−1 B) : (Rm )⊕n → X is onto 3. The least A-invariant module containing Im(B) equals X. If X = Rn is free then the above are also equivalent to: 4. Determinantal ideal Un (B, AB, ..., An−1 B) =

n × n-minors of (B, AB, ..., An−1 B)

generated by all n × n minors of (B, AB, ..., An−1 B) equals R. To conclude this review of reachability properties, we note that if system Σ = (A, B) over X is reachable then the mapping A ⊕ B = (A, B) : X ⊕ Rm → X need to be onto because (by definition of reachability) every target state is reachable from x0 = 0. Consequently every ω can be expressed as Bun−1 + ABun−2 + · · · + An−2 Bu1 n−1 Bu1 = (A, B) ω = Bun +ABun−1 +· · ·+A un Thus we have proved: Proposition 2.5. Let X be a finitely generated R-module of constant rank n and let Σ = (A, B) be a m-input linear system over X then the linear mapping X A ⊕ B = (A, B) : X ⊕ Rm → (x, u) → Ax + Bu is onto.


2.1.

75

Partial Reachability Maps

Let Σ = (A, B) be a linear system over X. For all i ≥ 1 denote by ϕΣ i the partial reachability map: (Rm )⊕i → X ϕΣ i : i P j−1 A Buj (u1 , ..., ui ) 7→ Note that the submodule Im ϕΣ is just the set of states that are reachable ”at time i t ≤ i” when linear system is initialized at x0 = 0. Thus, if X = Rn is free, the criterion to decide if some state is reachable at time t can be stated in terms of linear equations.

2.2.

Weak Reachability

Let R be a domain (that is, ab = 0 ⇒ a = 0 or b = 0 in R). Then there exists a field K (R) called “field of fractions of R” containing R that is constructed (see [1]) in the same way that Q arises from Z; that is, K (R) is the set of “formal” expressions pq such that q 6= 0 with the equivalence r p = ⇔ ps − qr = 0 q s pr Sum and product are the natural internal composition laws (pq + rs = ps+rq qs and q s = pr qs ). Consequently R can be immersed into its field of fractions by the canonical injective linear map i : R ,→ K(R) (sending r 7→ r1 ) which allows to consider every linear system Σ = (A, B) defined over X as a linear system Σ (R) = (i (A) , i (B)) over the K (R)vector space X ⊗R K (R).

Definition 2.6. Let R be a domain. Linear system Σ = (A, B) over X is weakly reachable if Σ (R) is reachable. Note that if k is any field then K (k) = k thus reachability = weak reachability for fields. Note also that if X = Rn then the pair of matrices (A, B) is weakly reachable if and only if there exists a nonzero n×n minor in (B, AB, ..., An−1 B), thus (by Proposition 2.4) reachability ⇒ weak reachability for the free case. For proving reachability ⇒ weak reachability, in the projective case, we only need to note that tensor product is right-exact functor thus surjectivity (and a fortiori reachability) passes from R to K (R), see [1]. On the other hand the converse doesn’t hold over commutative rings: Example 2.7. An example of a weakly reachable system that is not reachable is easily given in Z: System Σ = ((1) , (2)) is not reachable (because U1 (2) = 2Z 6= Z) but Σ = ((1) , (2)) is reachable because its reachability matrix (2) has full rank.

2.3.

Pointwise Reachability

Suppose that R is the ring of real-valued continuous functions C(Λ, R) defined on the compact topological space Λ. Let Σ = (A, B) a linear system over R = C(Λ, R). For every λ ∈ Λ we consider the linear system over R given by Σ(λ) = (A(λ), B(λ)) where

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A(λ) = (aij (λ)) and B(λ) = (bij (λ)) are the evaluations at λ ∈ Λ. If Σ is reachable then every Σ (λ) is reachable (note that if Σ is reachable then for each ω ∈ Rn there exists a sequence {ui } such that 0 {ui } ω. Therefore for each ω = ω (λ) ∈ Rn one has that 0 {ui (λ)} ω (λ)). This is clear but, does the converse result hold? That is to say: Does the statement ”if every Σ (λ) is reachable for each λ then Σ is reachable” hold? The same question can be stated for rings of polynomials over a field: Is the reachability property for a system Σ (t) = (A (t) , B (t)) over k[t] characterized by all evaluations Σ (t0 ) over k? Next we will prove that answer is ”yes if and only if k is algebraically closed field” (see Theorem 2.11 bellow). The hypothesis of algebraically closed ground field is necessary: Example 2.8. Let R = R[t] and consider the system Σ=

0 0 0 0

t −1 , ; 1 t

first we have that system Σ is reachable if and only if linear map defined by matrix B is onto or equivalently if B is an invertible matrix. This is not the case because det(B) = t2 + 1 is not an unit of R[t]. But on the other hand det(B (t0 )) is nonzero for each t0 ∈ R. Consequently every evaluation of system Σ is reachable but system Σ itself is not reachable. √ Obviously if we consider the above√linear because system in C[t] then, √ of i = −1 is −1 = 0 and system Σ −1 is not reachable. a root of det(B) it follows that det B The gap is that reachability can be stated from a pointwise point of view by checking at all maximal ideals of ring R. In the case of R = C[t] then the set of maximal ideals is M ax (C[t]) = {(t − a) : a ∈ C} and we only need to check the evaluations at any complex number. Next we review the general theory gives us the algebraic background to study the pointwise reachability over an arbitrary ring of scalars R. Definition 2.9. Let R be a commutative ring with unit and let Σ = (A, B) be a linear system over Rn . Let f : R → R0 be a ring homomorphism and denote by f (Σ) the linear system over (R0 )n given by f (Σ) = (f (A), f (B)) where f (A) = (f (aij )) and f (B) = (f (bij )). The linear system f (Σ) will be called the extension of Σ by change of scalars from R to R0 via f . Let m be a maximal ideal of R. We put Σ(m) = (A(m), B(m)) = π (Σ) where π : R → R/m is the natural quotient homomorphism. Theorem 2.10. Let R be a commutative ring and let Σ = (A, B) be a linear system over Rn . System Σ is reachable if and only if system Σ(m) is reachable for each maximal ideal m of R.


77

Proof.- Reachability is given in terms of the surjectivity of linear map A? B = (B, AB, ..., An−1 B) or equivalently if the determinantal ideal Un (A? B) generated by all n × n minors of A? B equals R. Thus if Σ is reachable then Un (A? B) = R and therefore Un (A(m)? B(m)) = R/m and Σ(m) is reachable for each maximal ideal m of R. Conversely if Σ(m) is reachable for each maximal ideal m of R then Un (A(m)? B(m)) 6= 0 and Un (A? B) is not contained in any maximal ideal of R. This is actually equivalent to Un (A? B) = R and to that Σ is reachable Next we rewrite the result in particular terms of polynomial rings. Theorem 2.11. Let k be al algebraically closed field. Let Σ = (A, B) be a linear system over k[t]n . Then the following are equivalent: 1. System Σ = (A, B) is reachable over k[t] 2. System Σ (t0 ) is reachable over k for each t0 ∈ k. Proof.- Because of k is algebraically closed it follows that every polynomial in k[t] splits as a product of linear (degree=1) polynomials, then M ax(k[t]) = {(t − a) : a ∈ k}. Therefore, for each maximal ideal m = (t − a) one has that Σ(m) = Σ(t = a) is the evaluation at t = a Note 2.12. The result is not true if k is not algebraically closed: As an example, put R = R[t] then x2 + bx + c is a maximal ideal for each b, c such that b2 − 4c < 0 and it is not sufficient to check the evaluations at any real number. To conclude we rewrite the above Theorem in terms of rings of continuous functions defined on a compact topological space. Theorem 2.13. Let Λ be a compact topological space. Let Σ = (A, B) be a linear system over C(Λ, R)n (i.e. entries of A and B are continuous real valued functions defined on Λ). Then the following are equivalent: 1. System Σ is reachable. 2. System Σ (λ) = (A (λ) , B (λ)) is reachable over R for each point λ ∈ Λ Proof.- Since Λ is a compact topological space, the map Λ → M ax (C(Λ, R)) λ → mλ = {f ∈ C(Λ, R) : f (λ) = 0} is an homeomorphism (see [1] or [16]) where Max(R) is the topological space of all maximal ideals of R equipped with the Zariski topology. Thus in the case of rings of continuous real valued functions defined in a compact topological space, pointwise reachability is in fact given by the reachability of evaluations at any point

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3.

Feedback Classification

It is natural to consider two linear systems over Rn Rm &B

Σ:

Rn →A Rn and Rm 0

&B

Σ :

0 0

Rn →A

Rn

as equivalent if we can bring one of them to the another by change of basesP in Rn and T in Rm Rm ↓T &B Rm Rn →A Rn 0 &B ↓P ↓P 0 n A R → Rn This is the algebraic equivalence of linear systems

3.1.

Algebraic Equivalence

Let Σ = (A, B) be a linear system over X Rm &B X →A X Algebraic equivalence of system Σ is given by the following two elementary actions ( I ) A 7−→ A0 = P AP −1 ; B 7−→ B 0 = P B for some isomorphism P : X → X 0 . of the state modules. ( II ) A 7−→ A0 = A ; B 7−→ B 0 = BT −1 for some automorphism T of Rm . Thus we say that system Σ = (A, B) over X and system Σ0 = (A0 , B 0 ) are algebraically equivalent via P, T −1 if the following diagram is commutative: Rm ↓T Rm

&B 0 &B

X →A ↓P 0 X 0 →A

X ↓P X0


79

It is an elementary exercise to prove that algebraic equivalence is an equivalence relation. We denote by Σ ≈ Σ0 the fact that systems Σ and Σ0 are algebraically equivalent. For historical reasons we denoteT −1 = Q. Thus a general algebraic action on system (A, B) is given by P AP −1 , P BQ . Theorem 3.1. (K ALMAN ’ S R EDUCTION ) Let R be a commutative ring and X a finitely generated R-module of constant rank n. Let Σ = (A, B) be a linear system over X. Then there exist an algebraic equivalence (P, Q) such that system is given by the following matrices of linear maps A11 A12 B1 −1 , (P AP, P BQ) = 0 A22 0 Proof.- Apply Note 1.5 in order to obtain P such that A11 A12 B1 −1 , , (P AP, P B) = 0 A22 B2 m ? ? 1 now note that the image of linear B = B B2 : R → A (Im B) ⊕ X/A (B) is forced to lie in A? (Im B). Therefore B2 = 0 is the zero linear map and we are done In the case of R = k is a field (the classical Kalman’s Reduction), or if all involved R-modules are free, the above Theorem can be stated in terms of block matrices: Corollary 3.2. (C LASSICAL K ALMAN ’ S R EDUCTION ) Let k be a field and Σ = (A, B) ∈ k n×n × k n×m be a linear system. There exist change of basis P of state space k n and a change of basis Q in the control space k m such that

(P −1 AP, P BQ) =

a11 .. . ar1 0 .. . 0

··· ··· ··· ···

a1r .. . arr 0 .. . 0

a1,r+1 .. . ar,r+1 ar+1,r+1 .. . an,r+1

··· ··· ··· ···

a1n .. . arn ar+1,n .. . ann

,

b11

···

b1m

br1 0

··· ···

brm 0

0

···

0

where r = rank B, AB, ..., An−1 B . Note that, in terms of algebraic equivalence, we have proved that A? (Im(B)) is an invariant, up to isomorphism, for the algebraic equivalence of linear systems. In the particular case of fields, the invariant is the dimension of the vector spaceA? (Im(B)) or, equivalently, the rank of block matrix (B, AB, A2 B, ..., An−1 B) Σ Now we are going to prove that in fact every partial reachability map ϕi = i−1 B, AB, ..., A B gives rise an invariant for the algebraic equivalence class of Σ = (A, B). First we need some notations.

Definition 3.3. Let Σ = (A, B) be a linear system over X. The image of ϕΣ i is denoted by Σ Σ Σ Σ Σ Ni , we define Mi = Coker ϕi = X/Ni , and finally we set N0 = 0 and M0Σ = X.

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Lemma 3.4. Suppose that the systems Σ = (A, B) and Σ0 = (A0 , B 0 ) are algebraically 0 equivalent then the R-modules NiΣ and NiΣ are isomorphic and the R-modules MiΣ and 0 MiΣ are isomorphic for all i ≥ 0. Proof. If Σ and Σ0 are feedback equivalent via (P, Q) then it is easy to see that the following equality of matrices of linear maps   Q 0   .. P ϕΣ = ϕΣ   . i i Q is verified for each i ≥ 1. Therefore the resultfollows straightforward from [25] because  Q   .. matrices of linear maps P and   are invertible . Q Theorem 3.5. Let Σ = (A, B) be a linear system over X. Then the characteristic polynomial χ (A, z) is an invariant for the algebraic equivalence associated to Σ. Proof.- Let Σ0 = (A0 , B 0 ) over X 0 be (algebraically) equivalent to Σ. It follows that there exists an isomorphism P : X → X 0 such that A0 = P AP −1 . Therefore χ (A0 , z) = χ (A, z) (see [19] or [25]) There exists some relations between invariants associated to a linear system: The first relation is given by the natural exact sequence 0 → NiΣ →inclusion X

→quotient MiΣ → 0

Σ we have the exact sequence From the natural inclusion NiΣ ⊆ Ni+1 Σ Σ /N Σ → 0 →quotient Ni+1 0 → NiΣ →inclusion Ni+1 i Σ → M Σ sending x+N Σ → x+N Σ whose and from the canonical quotient map Mi−1 i i−1 i Σ Σ kernel is Ni+1 /Ni , we obtain the exact sequence Σ /N Σ → Σ → Σ 0 → Ni+1 quotient Mi+1 → 0 inclusion Mi i

Moreover we also have the surjective linear maps Σ Σ /N Σ → 0 → Ni+1 NiΣ /Ni−1 i Σ x + Ni−1 → Ax + NiΣ

and Σ → Mi+1 → 0 MiΣ Σ Σ x + Ni → Ax + Ni+1

On the other hand, because of X is projective finitely generated of constant rank n, the Cayley-Hamilton Theorem applies and linear map An is in fact a linear combination of IdX , A, ..., An−1 . Thus there exists an index s ≤ n such that


0

N1Σ

···

Σ Ns−1

81

Σ NsΣ = Ns+1 = ···

Σ Σ 6= 0 for all i = 1, ..., s and NiΣ /Ni−1 = 0 for all i ≥ s + 1. A fortiori NiΣ /Ni−1 Σ Σ Σ = N Σ and Moreover we have that Ms = Ms+1 = · · · . For this reason we denote by N∞ s Σ Σ Σ Σ M∞ = Ms . Note that Σ is reachable if and only if N∞ = X or M∞ = 0. Then we’ve stated the reachability property in terms of nullity of some module. In the case of fields, invariants are vector spaces and thus characterized by their dimensions which are the invariants in that case. We introduce some notation:

• τiΣ = dim NiΣ = rank B, AB, ..., Ai−1 B • σiΣ = dim MiΣ = n − τiΣ Σ Σ • ξiΣ = dim NiΣ /Ni−1 = τiΣ − τi−1 Note that the above invariants are related by the following numerical properties: Proposition 3.6. Let Σ be a linear system over a vector space. Then we have: Σ for all i = 0, 1, ... (i) τiΣ ≤ τi+1 Σ then τ Σ = τ Σ for all j ≥ i (ii) If τiΣ = τi+1 j i Σ Σ Σ < τsΣ = τs+1 = τs+2 = · · · Thus we denote (iii) There exists some s such that τs−1 Σ Σ τ∞ = τs Σ =n (iv) Σ is reachable if and only if τ∞ Σ for all i = 0, 1, ... (v) σiΣ ≥ σi+1 Σ then σ Σ = σ Σ for all j ≥ i (vi) If σiΣ = σi+1 j i Σ Σ Σ > σsΣ = σs+1 = σs+2 = · · · Thus we denote (vii) There exists some j such that σs−1 Σ Σ σ∞ = σs Σ =0 (viii) Σ is reachable if and only if σ∞ Σ (ix) ξiΣ ≥ ξi+1 Σ = ξΣ = · · · = 0 (x) There exists some s such that ξsΣ 6= 0 and ξs+1 s+2

The above numerical relations also hold for the case of commutative rings when all involved R-modules are free, (otherwise the equalities are nonsense; think in the example Σ = ((1) , (2)) over Z which verifies that M1Σ = Z/2Z is not Z-free).

82

3.2.

Miguel V. Carriegos

A Fundamental Question

Let Σ = (A, B) be a linear system in the fundamental sense of picture B ⊆ X →A X

Σ:

that is to say, as an endomorphism of some state module X together with a submodule B ⊆ X of controls. Suppose that we choose m generators to present B and to obtain system Rm &B

Σ:

X

→A X

Does linear system depend on the set of generators we use to describe B? Unfortunately the answer is, in general, yes. Next we provide an example based on a very well known in Topology: The Hairy-BallTheorem. Example 3.7. Consider the ring R of continuous functions defined on the real 2-sphere S2 immersed in R3 in the usual way (that is to say, as the set of zeros of x2 + y 2 + z 2 − 1). Let B = R be the image of (1, 0, 0) : R3 → R and on the other hand choose (x, y, z) : R3 → R to generate the same control submodule B = R. Both linear maps are surjective thus one have the following decompositions of R3 as direct sums: R3 ∼ = R ⊕ ker (1, 0, 0) = R ⊕ R2 and on the other hand R3 ∼ = R ⊕ ker (x, y, z) but the Hairy-Ball Theorem assures that row (x, y, z) is not completable to an invertible 3 × 3 matrix (otherwise you could “comb the hair on a coconut”, see [27] for details). Thus linear systems R3 0

&(1,0,0)

Σ :

R →(0) R and R3 Σ00 :

&(x,y,z) R →(0) R

are two different representations of the same linear system Σ = (A = 0, B = R) that are not algebraically equivalent 3 × 3 matrix Q such that because there is no invertible 1 0 0 Q = x y z ; otherwise x y z is the first row of Q and hence completable to an invertible 3 × 3 matrix.


3.3.

83

Feedback Equivalence

Now we introduce the feedback equivalence allowing the feedback action of spaces onto the inputs. This action is at the very heart of control theory and is an usual way to determine the inputs as linear functions of the current “internal” state of the system. Linear system Σ = (A, B) is feedback equivalent to linear system Σ0 = (A0 , B 0 ), if Σ can be transformed to Σ0 by one element of the feedback group Fm,n (R), which is the group generated by algebraic equivalence actions (I) and (II), introduced in 3§1, together with feedback actions on the form: ( III ) A 7−→ A0 = A + BF ; B 7−→ B 0 = B for some linear map F : X → Rm , which is called the feedback or closed loop of system. Note 3.8. Let us denote by Σ ∼ Σ0 the feedback equivalence of Σ and Σ0 . Of course we have Σ ≈ Σ0 ⇒ Σ ∼ Σ0 Algebraic equivalence invariants introduced in the previous paragraph are also feedback invariants. The key is the following result: Lemma 3.9. Let Σ = (A, B) be a linear system over X. Then for any feedback map F : X → Rm we have the following equality (not only isomorphism) Im(B, AB, ..., Ai−1 B) = Im(B, (A + BF )B, ..., (A + BF )i−1 B) Proof.- The equality follows directly from the fact that: i−1 (B, (A + BF )B, ..., (A + BF 2) B) = 1 F B (F B) · · · (F B)i−1  .. ..  0 1 . FB .   .. . . i−1 . . .. .. = (B, AB, ..., A B)  . . (F B)2   .. .. ..  . . . FB 0 ··· ··· 0 1

        

which is easily checked by induction (see [7] for details) As a consequence of the previous result we have that algebraic equivalence invariants are in fact feedback invariants. Theorem 3.10. Let Rm Σ:

&B X

→A X

be a linear system over X. Then: 1. Modules NiΣ = Im(B, AB, ..., Ai−1 B) are invariant, up to isomorphism for the feedback action on Σ.

84

Miguel V. Carriegos 2. Quotient modules MiΣ = Rn /NiΣ are invariant, up to isomorphism for the feedback action on Σ. Σ are invariant for the feedback action on Σ. 3. Quotient modules NiΣ /Ni−1 0

That is to say, if Σ is feedback equivalent to Σ0 then MiΣ is isomorphic to MiΣ , NiΣ 0 Σ is isomorphic to N Σ0 /N Σ0 , for each i = 0, 1, 2, ... equals NiΣ , and NiΣ /Ni−1 i i−1 The characteristic polynomial of A is an algebraic equivalence invariant but it is not a feedback invariant. Perhaps the easiest example is system Σ = (A, B) = ((0) , (1)) over any field k: The characteristic polynomial of A is χ (A, z) = z. But a feedback action of matrix F = (f ) transforms Σ into system Σ0 = ((f ) , (1)), which is of course feedback equivalent to Σ but on the other hand χ ((A + BF ) , z) = z − f .

3.4.

Brunovsky’s Theorem

The first complete feedback classification result was given by P.A. Brunovsky in [5] for the case of reachable linear systems over a field. The set of invariants introduced is the Kronecker indices and the canonical form is known as Brunovsky Canonical Form. The classical result is stated as follows: Theorem 3.11. Let Σ be an m-input, n-dimensional reachable linear system P over a field k. Then there exist a finite set of positive integers k1 ≥ k2 ≥ ... ≥ ks > 0, with si=1 ki = n, uniquely determined by system Σ such that Σ is feedback equivalent to Σc = (Ac , Bc ) where Ac and Bc are described bellow:   E1 0 · · · 0  0 E2 · · · 0    Ac =  . .. ..  . . .  . . . .  0 ··· Es where Ei is the ki × ki matrix 

    Ei =     and



  Bc =  

0 1

0

0 0 .. .. . .

1 .. .

··· .. . .. . .. . 0

0 0 ··· 0 0 ···

e1 0 · · · 0 e2 · · · .. .. . . . . . 0 0 ···

0 0 .. . es

0

       1  0

0 .. .

0 ··· 0 ··· .. . . . . 0 ···

where ei is the ki × 1 matrix ei =

0 ···

0 1



t

0 0 .. . 0

    


85

The integers {ki }si=1 are called the Kronecker indices of Σ, and Σc = (Ac , Bc ) described above is called the Brunovsky canonical form associated to the Kronecker indices {ki }si=1 . Proof.- See [5] or [20] As an example we put the Brunovsky canonical P form with Kronecker invariants k1 = 4, k2 = 3, k3 = 1 and m = 4. It is clear that n = ki = 8            

0 0 0 0

1 0 0 0

0 1 0 0

 

0 0 1 0 0 1 0 0 0 1 0 0 0 0

          ,          



0 0 0 1 0 0 1 1

          

What are the relation between Kronecker invariants of Σ and τiΣ , σiΣ and ξiΣ obtained from invariant vector spaces NiΣ and MiΣ ? Theorem 3.12. If the sequence of Kronecker indices associated to reachable linear system Σ is {k1 , ..., ks } where: ks = · · · = ks1 +1 < ks1 = · · · = ks2 +1 < ks2 = · · · < kst = · · · = k1 , then the sequence τ1Σ , ..., τsΣ0 is s 2s 3s .. . ks s ks s + s1 ks s + s2 .. . ks s + (ks1 − ks ) s1 .. . ks s + · · · + kst−1 − kst−2 st−1 + st .. . ks s + · · · + kst−1 − kst−2 st−1 + kst − kst−1 st Proof.- The reader can see [10] for a complete proof Now, we conclude that we can obtain the Kronecker indices {ki }1≤i≤s from the invari ants dim NiΣ 1≤i≤n and since σiΣ = dimk (MiΣ ) = n − dim NiΣ , it follows that the set

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Miguel V. Carriegos

of Kronecker indices {ki }1≤i≤s of system Σ = (A, B) can be obtained from the indices Σ σi 1≤i≤n . This proves that the set of indices σiΣ 1≤i≤n are equivalent data to the set of Kronecker indices of the system Σ. But there is another way to relate the Kronecker invariants of a linear system with Σ by using Ferrers diagram (see [24]) of the partition the invariants ξiΣ = dim NiΣ /Ni−1 of integer n: Let Σ be a reachable linear system over k n (k a field). Then the partition Σ of n given by the Kronecker indices of Σ and the partition of n given by indices ξi = Σ Σ dim Ni /Ni−1 are dual in the sense of their Ferrers diagrams are transposed. We put the above example of n = 8 with ξ1 = 3 ξ2 = 2 ξ3 = 2 ξ4 = 1 k1 = 4 k2 = 3 k3 = 1 The Ferrers diagram is k1 ↓ ξ1 ξ2 ξ3 ξ4

k2 ↓

k3 ↓

→ → → →

Thus it is very easy to pass from the Kronecker indices to the invariants obtained from partially reachability maps. In the particular case of m = 1 there exists only one feedback class called the Canonical Controller Form which corresponds to the Ferrers diagram k1 ↓ ξ1 ξ2 .. .

→ → .. .

ξn

→

.. .

and invariants k1 = n, ξ1 = ξ2 = · · · = ξn = 1, τi = i, σi = n − i, for 1 ≤ i ≤ n. Corollary 3.13. (C ANONICAL CONTROLLER FORM ) Let R be a commutative ring and let Σ = (A, b) ∈ Rn×n × Rn×1 be a single input (m = 1) reachable linear system. Then Σ is feedback equivalent to the Canonical Controller Form:      0 1 0 ··· 0 0   ..  .. ..  0 0   0  . . .         Σ[ =  ... . . . . . . . . . 0  ,  ...       ..   0  .. ..  .  . . 1  1 0 ··· ··· 0 0 Unfortunately, the Brunovsky result is a complete classification only over fields:


87

Definition 3.14. A Brunovsky ring as a commutative ring R such that every reachable linear system Σ over Rn is feedback equivalent to a Brunovsky canonical form. Then we have the following characterization: Theorem 3.15. Let R be a commutative ring. The following are equivalent: 1. R is a Brunovsky ring. 2. R is a field. Proof.- Note that (2) ⇒ (1) is a consequence of the Brunovsky’s Theorem. Conversely assume that R is a Brunovsky ring. First, we claim that R is absolutely flat ring (that is, every finitely generated ideal of R is principal generated by a single idempotent e, with e2 = e, see [1] or [2]): Let a be a finitely generated ideal of R. a = (a1 , ..., am ). Consider the linear system Σ over R2 given by 0 0 1 0 0 ... 0 . Σ = (A, B) = , 1 0 0 a1 a2 . . . am Note that the reachability matrix of Σ 1 0 0 ... 0 ∗ A B= 0 a1 a2 . . . am

0 0 0 ... 0 1 0 0 ... 0 ,

verifies that the determinantal ideal U2 (A∗ B) is the whole ring R. Consequently Σ is reachable and therefore a Brunovsky system by hypothesis. Then, the R-module M1Σ = R2 /Im(B) is free: Put the finite free resolution of the R-module M1Σ B

Rm+1 → R2 → M1Σ → 0. Since M1Σ is free, it follows that the determinantal ideals associated to the matrix B are principal ideals generated by an idempotent element of R. In particular, the ideal U2 (B) = (a1 , a2 , ..., am ) = a, is a principal ideal generated by an idempotent and we have proven that every finitely generated ideal of R is principal generated by an idempotent. This is equivalent to R being an absolutely flat ring (see [1]). Now let us prove that R is a local ring (i.e. R has only one maximal ideal). Suppose, by contradiction, that there exist m and m0 two maximal ideals of R such that m 6= m0 . Let x ∈ m and x ∈ / m0 and consider the linear system over R2 given by 0 0 1 0 , . Σx = (Ax , Bx ) = 1 0 0 x

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Miguel V. Carriegos

The system Σx is reachable, so Σx is feedback equivalent to a Brunovsky canonical form. But note that the following two systems are the only Brunovsky canonical forms over R2 :The form Σ{1,1} associated to the sequence of Kronecker indices k1 = k2 = 1, k3 = k4 = · · · = 0, with the following Ferrers diagram:

and given by the canonical form Σ{1,1} =

0 0 0 0

1 0 , . 0 1

And the form Σ{2} associated to the sequence of Kronecker indices k1 = 2, k2 = k3 = · · · = 0, and Ferrers diagram

and given by Σ{2} =

0 1 0 0

0 0 , . 1 0

An easy calculation shows that the matrix

is not equivalent to the matrix

neither to the matrix

1 0 0 x

1 0 0 1

0 0 1 0

,

,

.

Then the system Σx is neither feedback equivalent to the system Σ{1,1} nor to the system Σ{2} . This is a contradiction with R being a Brunovsky ring and therefore m = m0 . Consequently ring R is local. Collect the properties of R; that is, R is local and absolutely flat: This is equivalent to R being a field (see [1])


3.5.

89

The Iterative Procedure (Free invariants)

Let R be a commutative ring such that finitely generated projective R-modules are free. Then for reachable linear systems such that invariant R-modules MiΣ are free one has essentially the same Brunovsky canonical form (see [20]). Next we give give an iterative procedure that obtain a canonical form in this case. Note that this procedure is not the same given in [20] but on the other hand our method will allow us to generalize the classification results for the case of projective invariants. Theorem 3.16. Let R be a commutative ring such that finitely generated R-modules are free. Let Σ = (A, B) be a linear system over Rn Rm &B Rn →A Rn If the R-module M1Σ = Rn /Im (B) is free then: 1. System Σ is feedback equivalent to a system on the form 1 0 0 0 , 0 0 B1 A1 Σ

Σ

Σ

Σ

for some matrices B1 ∈ Rσ1 ×ξ1 and A ∈ Rσ1 ×σ1

0

2. If Σ0 is another linear system over Rn such that M1Σ is free and if (A01 , B10 ) are the matrices obtained in the previous item for Σ0 then system Σ is feedback equivalent to system Σ0 if and only if (A1 , B1 ) is feedback equivalent to (A01 , B10 ). 3. Feedback invarinats of system δ (Σ) = (A1 , B1 ) Rξ1 &B1 R σ1

→A1

R σ1

are related to feedback invariants of Σ by the isomorphisms δ(Σ) NiΣ ∼ = N1Σ ⊕ Ni−1 δ(Σ) MiΣ ∼ = Mi−1

Proof.First matrix Bis on the note that by Lemma 1.2 we can assume that 1 0 −A11 −A12 brings matrix A = form . Now the feedback action 0 0 0 0 0 0 A11 A12 to matrix . Denote B1 = A21 and A1 = A22 . This proves A21 A22 A21 A22 (1).

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Miguel V. Carriegos

0 To prove (2) first assume that is to say: There Σ arefeedback equivalent, that Σ and Q11 Q12 F11 F12 P11 P12 ,Q= , and F = ; exist block matrices P = P21 P22 Q21 Q22 F21 F22 1 0 1 0 such that P Q= . Therefore P21 = 0. 0 0 0 0 On the other hand, one has the equality 0 0 0 0 P11 P12 1 0 F11 F12 P11 P12 = + 0 P22 0 P22 F21 F22 B1 A1 B10 A01 0 0

Consequently P22 B1 = B10 P11 and P22 A1 = A01 P22 + B10 P12 . Hence subsystems (A1 , B1 ) and (A01 , B10 ) are feedback equivalent via (P22 , P11 , P12 ). Conversely assume that δ (Σ) = (A1 , B1 ) is feedback equivalent to δ (Σ0 ) = (A01 , B10 ), that is to say P A1 P −1 + P B1 F

= A01

P B1 Q = B10 then one has that 0 0 Q Q−1 −QF P −1 0 P B1 A1 0 −1 1 0 T1 −QF P −1 Q + 0 P 0 0 0 −1 Q T1 ∗ ∗∗ + −1 0 P B1 Q P A1 P + P B1 F

F P T2 0

+

=

Q−1 T2 0

=

and, by setting T1 = −Q (∗) and T2 = −Q (∗∗), the above equals: 0 0 . = B10 A01 On the other hand, −1 1 0 Q 0 1 0 −QF P −1 Q = 0 P 0 0 0 1 0 0 therefore Σ is feedback equivalent to Σ0 via thefeedback action −1 −1 Q 0 Q T1 Q−1 T2 −QF P −1 Q , , . 0 P 0 1 0 0 To prove (3) it is only needed to consider the partial reachability matrices ofΣ 1 0 0 0 0 0 0 0 ··· 0 0 Σ ϕi = 0 0 B1 0 A1 B1 0 A21 B1 0 · · · Ai−2 1 B1 0 and of δ (Σ)

δ(Σ)

ϕi−1

= B1 , A1 B1 , ..., Ai−2 1 B1


91

The hypothesis of all finitely generated R-modules are free is necessary to state the classification theorem and the canonical formin termsof matrices. In particular, the iterative 1 0 procedure in this case first obtain a form for B and then obtain the same form 0 0 for B1 , ... and so on. Each step is a feedback action in the original systemΣ (Theorem 3.16 (2)) and the procedure is terminated because the rank of state space decreases in each step. Consider a single input reachable linear system Σ. By applying the iterative we have that Σ is feedback equivalent to Σ\ bellow, which is an alternate Canonical Controller Form: Corollary 3.17. (C ANONICAL CONTROLLER FORM II) Let R be a commutative ring and let Σ = (A, b) ∈ Rn×n × Rn×1 be a single input (m = 1) reachable linear system. Then Σ is feedback equivalent to the Canonical Controller Form (compare to the one obtained on page 85:     0 0 ··· ··· 0 1     . . . . 0   1 0 . .      ..     . \ . . .. . . ..  ,  .  Σ =  0 1     ..   .. . .    .. ..  .   . . . . 0  0 0 ··· 0 1 0 Example 3.18. In the case of invariants m = 4 ξ1 ξ2 ξ3 ξ4

=3 =2 =2 =1

, , , , τ4

τ1 = 3 τ2 = 5 τ3 = 7 = 8 = τ∞

, , , , σ4

σ1 = 5 σ2 = 3 σ3 = 1 = 0 = σ∞

the iterative procedure generates the following canonical form:            

3.6.

0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0

            ,          

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 0

           

The Iterative Procedure (Projective invariants)

If the base ring R is not projectively trivial decompositions as before can be performed for linear systems Σ such that M1Σ is projective, but the reduced system δ (Σ) is defined on a projective R-module and the input space becomes also a projective (in general non-free R-module). The situation may be described as follows:

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Miguel V. Carriegos

Let R be any commutative ring and let Σ = (A, B) be a linear system over the X (a finitely generated projective R-module of constant rank n) Rm &B X →A X If M1Σ = X/Im (B) is projective of constant rank σ1 then linear system δ (Σ) characterizing the feedback class of Σ may be constructed as follows: Lemma 3.19. Let Σ = (A, B) be a m-input system over X and suppose that the R-module M1Σ is projective. Then: (i) There exists decompositions Rm = N ⊕K, and X = N1Σ ⊕X1 , where N is isomorphic to N1Σ , K is isomorphic to ker (B), and X1 is isomorphic to M1Σ . (ii) Σ is feedback equivalent to system b 0 0 0 , , 0 0 a21 a22 where

and

0 0 a21 a22

b 0 0 0

: N1Σ ⊕ X1 → N1Σ ⊕ X1

: N ⊕ K → N1Σ ⊕ X1

are matrices of linear maps and b is an isomorphism. Proof. The form of matrix of linear maps B is obtained by Lemma 1.2 Now since b is an isomorphism it follows that feedback given by matrix of linear maps −b−1 a11 −b−1 a12 0 0 Let brings matrix of linear maps A to the form of the statement b 0 0 0 , be a m-input system over X in the conditions of the Σ= 0 0 a21 a22 above Lemma. We denote by δ (Σ) the linear system over X1 given by N1Σ

&B1 X1 ∼ = M1Σ →A1

X1 ∼ = M1Σ

where δ (Σ) = (A1 , B1 ) = (a22 , a21 b)

Linear Algebra over Commutative Rings Applied to Control Theory 93 0 0 b 0 Lemma 3.20. Consider the m-input system Σ = , over X. a21 a22 0 0 δ(Σ)

Suppose that M1Σ is projective. Then Mi

Σ for all i ≥ 0. is isomorphic to Mi+1

Proof.- Direct calculation on the partial reachability maps of linear systemΣ Theorem 3.21. Let Σ = (A, B) be an m-input system over X and let Σ0 = (A0 , B 0 ) be a 0 m -input system over X 0 . If M1Σ and M1Σ are projective, then the following statements are equivalent: (i) Σ and Σ0 are feedback equivalent. 0

(ii) The R-modules K Σ = ker (B) and K Σ = ker (B 0 ) are isomorphic and the systems δ (Σ) and δ (Σ0 ) are feedback equivalent. Note that the system δ (Σ) obtained from Σ is not unique in general, but by the above Theorem it is clear that if δ (Σ) and δ 0 (Σ) are obtained by system Σ as in then δ (Σ) and δ 0 (Σ) are feedback equivalent.

3.7.

Non-Reachable Systems over a field

Kalman’s Reduction needs only change of basis (algebraic equivalence) to work. If we allow feedback actions we have a more powerful reduction result. Theorem 3.22. (K ALMAN ’ S F EEDBACK R EDUCTION OVER A FIELD ) Let k be any field. Let Σ = (A, B) ∈ k n×n × k n×m be a linear system. Then there exist change of basis P of state space k n and a change of basis Q in the generator space k m of control subspace B = Im(B) and a feedback law F : k n → k m such that the system is on the form A11 0 B1 −1 , (P (A + BF )P, P BQ) = 0 0 A22 In this case, we say that system Σ splits as a reachable subsystem (A11 , B1 ) together with a subsystem (A22 , 0) with no controls. Σ = A? (ImB) is an A-invariant vector space. Thus Classical Proof.- First note that N∞ Kalman’s Reduction Theorem 3.2 applies and there exists an algebraic equivalence (change of bases) bringing Σ to the system A11 A12 B1 , Σ0 = A0 , B 0 = (P −1 AP, P BQ) = 0 A22 0

Now put the reachability map of Σ0 B1 A11 B1 A211 B1 · · · 0 ? 0 A B = 0 0 0 ···

n−1 A11 B1 0

n−1 Σ ∼ N Σ0 we have that Because of N∞ B1 generB1 A11 B1 A211 B1 · · · A11 = ∞ Σ . By Cayley-Hamilton, and due to A ates a vector subspace of dimension τ∞ 11 is a square

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Miguel V. Carriegos

Σ × τ Σ -matrix, it follows that subsystem (A , B ) is reachable. Now since (A , B ) τ∞ 11 1 11 1 ∞ τn ⊕ k m → k τn is onto, consequently it has a , B ) : k is reachable it follows that (A 11 1 H1 such that A11 H1 + B1 H2 = 1. Apply to system Σ0 the change of right inverse H2 1 −H1 and then the feedback matrix F = 0 H2 to conclude the bases P = 0 1 proof

Corollary 3.23. The feedback classification problem over any field k is solved by giving the solution of the feedback classification problem for reachable systems together with the classification of endomorphisms of k-vector spaces.

3.8.

Non-Reachable Systems (general case)

It is possible a generalization of above theorem to the projective case. Bellow we summarize the main steps we propose for a formal proof: Because of A (A? Im (B)) ⊆ A? Im(B) one has that the quotient linear map A¯ :

X/A? Im (B) X/A? Im (B) → x + A? Im (B) → Ax + A? Im (B)

Σ is projective then the state space X splits as X ∼ is well defined. Hence if M∞ = ? n ? A Im(B) ⊕ R /A Im(B) and, linear system Σ = (A, B) is given (with regard to the given decomposition of state space) by matrices of linear maps on the form B1 A11 A12 , 0 A22 0

because subsystem (A11 , B1 ) over A? Im(B) is reachable and A? Im(B) is projective then ? m ? it follows thatsurjective linear map (A11 , B1 ) = A11 ⊕ B1 : A B ⊕ R → A B has a H1 and we continue as in the above theorem to conclude that systemΣ right inverse H2 is in fact equivalent to a system on the form 0 B1 A11 , 0 0 A22

3.9.

Weakly Reachable Linear Systems

Weak reachability is sometimes used when we deal with linear systems over the polynomial ring k[t] (see[28]). A single input (m = 1) weakly reachable linear system Σ = (A, b) verifies (see Definiton 2.6) that the determinant det b, Ab, ..., An−1 b 6= 0 is nonzero. Thus weakly reachable system Σ is not in general feedback equivalent to a Canonical Controller Form; but changing scalars to the quotient field K(k[t]) = k(t) turns extended


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system Σk(t) into reachable and hence we can design feedback actions to control system Σk(t) “as a Canonical Controller Form”. Thought very useful in practice, the change of scalars R → K (R) looses too much information. In fact, every weakly reachable linear system over domain R is equivalent to the Canonical Controller Form over field K (R). But it is clear that is not the case over R. Example 3.24. Let R = Z; there are infinite feedback classes of weakly reachable linear systems over Z1 on the form ((a) , (b)) because ((a) , (b)) is feedback equivalent to ((a0 ) , (b0 )) if and only if b0 = ±b and b is a divisor of (a − a0 ). All these feedback classes over Z collapse in a single feedback class over Q, which is ((0) , (1)), the Canonical Controller Form for n = 1. Note 3.25. In general, it is shown in [9] that every single input weakly reachable system Σ over Zn (reapectively k[t]n ) is equivalent to the canonical form     a11 a12 · · · a1,n−1 a1n d1  d2 a22 · · ·  a2,n−1 a2n  0          .   .. .. .. .. .. ∆ ∆ ∆    ..  . . . . . Σ = A , b =  . ,      .  0  .. a  0 0 n−1,n−1 an−1,n  0 0 0 ··· dn ann where di > 0 (resp. nonzero monic polynomials) and 0 ≤ aij < di (resp. deg(aij ) ≤ deg(aij ) ) for all j. On the other hand it is also proved in [10] that the correspondence Σ → Σ∆ is computable (an algorithm is given there) and that Σ∆ is canonical in the sense of ∆ ∆ ∆ Σ∆ 1 ∼ Σ2 ⇔ Σ1 = Σ2 Note that this result implies each election onNn of invariants d1 , ..., dn there do Q that for n−1 n−i n 2 exist d1 d2 · · · dn−1 dn = 1≤i≤n di different feedback classes of single input weakly reachable systems over Zn .

4.

Pole Shifting

The characteristic polynomial is an invariant of a matrix up to semejance, that is to say, if A is a square n × n matrix with entries in the commutative ring R, and χ (A, z) = det (A − z1) is its characteristic polynomial, then for any invertible matrix P one has that −1 χ (A, z) = χ P AP, z . Thus characteristic polynomial of state matrix A is an invariant associated to system Σ = (A, B), for the algebraic equivalence. But this is not the case for the feedback action because in general, χ (A, z) 6= χ (A + BF, z). Hence it is possible to change the characteristic polynomial of state matrix of a system by a suitable feedback action. This is one of the main tools in stabilization of linear systems. For example one can ask if, given a linear system Σ = (A, B) there exists a deadbeat control which is a feedback

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matrix F such that χ (A + BF, z) = z n (this deadbeat control change all eigenvalues ofA to zero and stabilize the behavior of system Σ in the origin (0, ..., 0) t ). Note that the Canonical Controller Form     0 ··· ··· ··· 0 1  ..   0   1 0  .      ..    ..  ,  ..   . Σ\ = A\ , b\ =  0 1  . .       .. . . .   ..  .. ..  . . . . ..   .  0 0 ··· 0 1 0 is a deadbeat linear system because χ A\ , z = z n . Hence every reachable single input linear system can be brought to a deadbeat system by a feedback action. On the other hand, every monic polynomial z n +an−1 z n−1 +· · · +a1 z +a0 of degree n can be assigned to Σ\ just by applying the feedback matrix F = (−an−1 , ..., −a0 ). Thus the Canonical Controller Form is ”easy to control” in this sense. Consequently every sigle input reachable linear system Σ can be brought to a system with any characteristic polynomial (first take Σ to Σ\ by feedback and then change the characteristic polynomial as before). Definition 4.1. Let Σ = (A, B) be a linear system over R. Let p (z) be a monic polynomial of degree n. 1. We say that p (z) is assignable to Σ if there exists a feedback matrix F such that χ (A + BF, z) = p (z). 2. We say that Σ is coefficient assignable if every monic polynomial of degree n is assignable to Σ. In terms of the poles of system Σ (roots of χ (A, z)) we give the following Definition 4.2. Let Σ = (A, B) be a linear system over R. Let λ1 , ..., λs ∈ R 1. We say that poles λ1 , ..., λs can be assigned to Σ if there exists a feedback matrix F such that (z − λ1 ) · · · (z − λs ) | χ (A + BF, z). 2. We say that s poles can be assigned to Σ if poles µ1 , ..., µs can be assigned to Σ for all µi ∈ R. 3. We denote by ρ (Σ) the greater integer ssuch that s poles can be assigned to Σ. 4. We say that Σ is pole assignable if ρ (Σ) = n. Note 4.3. If poles λ1 , ..., λs can be assigned to system Σ by the feedback action F1 , and system Σ0 is feedback equivalent to Σ by using the feedback action F2 then it is easy to check that poles λ1 , ..., λs can be assigned to Σ0 by using the feedback action F1 + F2 . Thus the integer ρ (Σ) is a feedback invariant associated to Σ. Lemma 4.4. Let Σ = (A, B) be a linear system over a field k then ρ (Σ) ≥ ξ1Σ + ρ (δ (Σ))


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Proof.- System Σ is feedback equivalent to 1 0 0 0 , 0 0 B1 A1 Assume that poles λ1 , ..., λρ(δ(Σ)) are already assigned to (A1 , B1 ) . Remaining ξ1Σ F11 0 where F11 is the compoles can be assigned by the feedback action F = 0 0 Q panion matrix of polynomial 1≤j≤ξ Σ (z − µj ) 1

Theorem 4.5. (P OLE -S HIFTING OVER A FIELD ) Let Σ = (A, B) be a linear system over a field k then Σ ρ (Σ) ≥ τ∞ Proof.- Note that we have the chain of equalities δ(Σ)

ρ (Σ) ≥ ξ1Σ + ρ (δ (Σ)) ≥ ξ1Σ + ξ1 ≥

X

δi (Σ)

ξ1

=

X

+ ρ δ 2 (Σ) ≥ · · ·

Σ ξiΣ = τ∞ .

If system is reachable then the number of poles is exactly n (consequence of the above result) and hence any splitting polynomial can be assigned: Corollary 4.6. (C LASSICAL P OLE -A SSIGNABILITY ) Let Σ = (A, B) be a reachable linear system over a field. Then Σ is pole-assignable. Σ and n is not true in general. In fact If Σ is not reachable then the equality between τ∞ the equality is true only for infinite fields (see [11]). An example of that the equality doesn’t hold on finite fields is easily given in F2 , the field of integers modulo 2:

Example 4.7. 2-Input linear system over F22 given by: Σ=

0 0 0 1

0 0 , 0 0

Σ = 0 and however ρ (Σ) = 1. verifies τ∞

4.1.

Pole-Shifting over Commutative Rings

Now suppose that Σ = (A, B) is a linear system over a commutative ring R. For every maximal ideal m of R let Σ (m) = (A (m) , B (m)) be the extension of Σ to the residual field R/m. Then it is clear that ρ (Σ) ≤ ρ (Σ (m)) for each m. We use this property in the following classical result (see [3]). Theorem 4.8. Let Σ = (A, B) be a linear system over Rn . Then if Σ is pole assignable then Σ is reachable.

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Proof.- If ρ (Σ) = n then ρ (Σ (m)) = n for all maximal ideal m of R. By classical Pole-Shifting one has that Σ (m) is reachable for all m, which is equivalent to Σ being reachable When does the converse of the above result hold? It depends on the commutative ringR: For instance we have proven that if R = k is a field then the answer is “yes”, but in general the answer is “no”. The seminal papers [6], [17], [32] and [33] deal with the problem of a characterization of PA-rings, which are those rings with the property that every reachable linear system is pole assignable. Definition 4.9. Let R be a commutative ring. We say that R is a PA ring if every reachable linear system over X is pole-assignable. A commutative ring is PAF if every reachable linear system over a free module Rn is pole-assignable. The characterization of PA-rings is not given yet. It is not known even if PA=PAF. However the least known class of rings containing PA-rings were given in [36] and it is the class of BCS-rings. Definition 4.10. A commutative ring R has the BCS property (or R is a BCS ring) if every basic submodule of a finitely generated projective R-module X contains a rank 1 summand of X. Recall that a submodule B of a finitely generated projective R-module M is basic if locally B contains a non-trivial summand of X, or equivalently, if the image of B in X/mX is nonzero for each maximal ideal m of R. If X = Rn is free and B is an n × m matrix whose columns span B then the submodule B is basic if and only if U1 (B) = R. The class of BCS-rings gives has more applications: In a BCS ring it is possible to estimate the number of poles we can assign to a linear system. This account is performed by the residual rank of a system. Definition 4.11. Let R be a commutative ring and let Σ = (A, B) be a linear system over finitely generated R-module X of constant rank n. For each maximal ideal m ∈ M ax (R) we consider the residual system Σ (m) = (A (m) , B (m)) of Σ at m (see 2§3). Then we define the residual rank of Σ as the integer: o n res.rk. (Σ) = min rank B (m) |A (m) B (m) | · · · |A (m)n−1 B (m) : m ∈ M ax (R) . Now we give the class of rings where the residual rank estimates the number of poles one can assign to a given system. Definition 4.12. We say that R is a PS ring if ρ (Σ) ≥ res.rk. (Σ) for every linear system Σ. Since system Σ is reachable if and only if res.rk. (Σ) = n then it follows that every PS Σ. ring is a PA ring. Note that in a PS ring, res.rk. (Σ) plays the rôle of τ∞ Note 4.13. In the case of Σ being a free system (X = Rn ) then res.rk. (Σ) = max ν ∈ Z+ : Uj (A? B) = R , where Uj (A ∗ B) denotes the ideal generated by all the j × j−minors of the matrix A? B (see 1§1).


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The residual rank of a linear system is also a feedback invariant associated to the system. Proposition 4.14. Suppose that Σ = (A, B) is feedback equivalent to Σ0 = (A0 , B 0 ). Then res.rk. (Σ) = res.rk. (Σ0 ). Proof. It is sufficient to note that for every maximal ideal m of R we have the equality Σ( ) Σ0 ( ) = τ∞ = rank A0 (m) ∗ B 0 (m) rank (A (m)? B (m)) = τ∞

Lemma 4.15. Let Σ = (A, B) be a linear system over X. Then res.rk. (Σ) ≥ 1 if and only if the image of B is a basic submodule of X. Proof. res.rk. (Σ) = 0 if and only if there exists a maximal ideal m of R such that A (m) ∗ B (m) = 0, or equivalently, B (m) = 0 Theorem 4.16. Let R be a BCS ring. Then R is a PS ring and hence a PA ring. Proof. Let R be a BCS ring and let Σ = (A, B) be a linear system over a finitely generated projective R-module X of rank n. Suppose that res.rk. (Σ) ≥ 1. Then Im (B) is basic and hence there exists a rank one direct summand P1 of X such that P1 ⊆ Im (B). Put X = P1 ⊕ P2 and let πi be the canonical projection of X onto Pi for i = 1, 2. Since P1 ⊆ Im (B) it follows that π1 B : Rm → P1 is onto and consequently there exists a rank one direct summand P10 of Rm isomorphic via π1 B to P1 . Put Rm = P10 ⊕ P20 where P20 = ker (π1 B). Respect to the decompositions Rm = P10 ⊕ P20 and X = P1 ⊕ P2 the system Σ is defined by b11 b12 a11 a12 , B= , Σ= A= a21 a22 b21 b22 where bij = (πi B) |Pj0 is the restriction to Pj0 of πi B and aij = (πi A) |Pj is the restriction to Pj of πi A. By construction b11 is an isomorphism and b12 is zero. Using a suitable feedback action we have that Σ is feedback equivalent to the system 0 0 b11 0 0 0 , B= . Σ = A = a021 a022 0 b22 Now we prove the result by induction on res.rk. (Σ). Suppose that res.rk. (Σ) = 1 and let fλ be the endomorphism of P1 such that χ (fλ ) = (x − λ). Since Σ0 is feedback equivalent to fλ 0 b11 0 00 00 ,B = , Σ = A = a021 a022 0 b22 it follows that χ (A00 ) = (x − λ) · χ (a022 ) and hence one pole can be assigned to Σ. Suppose that res.rk. (Σ) > 1. Let Γ be the linear system over P2 given by Γ = (a22 , (b22 |a021 b11 )). Since A0 ∗ B = P1 ⊕ a022 ∗ b22 |a021 b11 , then res.rk. (Γ) = res.rk. (Σ) − 1 and therefore the result follows by induction

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Corollary 4.17. The following classes of rings are BCS rings and hence PS rings: (i) Elementary divisor rings. (ii) Semilocal rings. (iii) Both the rings C (X, R) of continuous and C ∞ (X, R) of differentiable real valued functions over the connected manifold X, where dim (X) ≤ 1. (iv) 1-dimensional rings.

4.2.

A Word on BCS Rings

To conclude this section devoted to pole-shifting we focus some properties of BCS rings. The interested reader is referred to the excellent paper [36] where this class of rings is studied in detail. Let R be a BCS ring and let a be an ideal of R. Then The quotient ring R/a is BCS. On the other hand, it is also proved in [36] that: Finitelly generated projective modules over a BCS ring split in rank one direct summands. Moreover we have that If R is BCS then the group Pic(R) of finitely generated rank one projective modules over R maps onto Pic(R/a) (by L → L/aL) for all ideal a of R. From the point of view of linear systems over rings of real valued continuous functions these are important facts: Let D2 be the unit closed disk in R2 and S1 be the unit circle in R2 . Of course ∂D2 = S1 . On the other hand, since D2 is a normal topologicalspace it follows (see [16]) that C S1 , R is a quotient ring of C D2 , R . Therefore C D2 ,R is not a BCS ring because Pic(C D2 , R ) = 1 (D2 retracts to a point) while Pic(C S1 , R ) = Z/2Z and it is not possible to map the trivial group {1} onto Z/2Z. The same argument shows that C (Λ, R) is not a BCS ring for any compact topological space Λ containing a copy of the unit disk D2 of R2 . The case of polynomial rings or affine algebras can be studied by the same argument: Note that the polynomial ring R = R[x1 , ..., xn ] is not BCS for n ≥ 2 because Pic(R) = 1 and R = R[x, y]/(x2 + y 2 − 1) is a quotient of R verifying Pic(R) ∼ = Z/2Z. The case n = 1 is solved by another argument: R[x] is a BCS ring because it is a principal ideal domain (see [6] or [17]).

5.

Conclusion

Systems Theory over commutative rings is an active subject of research since the 1980’s. It is a generalization of systems theory over a field. Linear algebra over commutative rings is used instead of usual linear algebra of vector spaces. Some cornerstones of the theory have been developed but some other remains unsolved: We point out four of them related to this chapter:


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• The fundamental question of §3.2; that is the relationship between the description of a system as (A, B) (endomorphism + control subspace) and the description as (A, B) (endomorphism + generators of the control subspace). Linear algebra over commutative rings would help to solve this question in the near future. • The Once it is stated that known feedback invariants for feedback Σ invariants: Σquest /NiΣ do not characterize (in general) the feedback class of a Ni , MiΣ , Ni+1 linear system, it is needed to introduce new invariants. • The calculation of assignable polynomials to a given linear system: Over a field we know what are the feedback assignable polynomials to a given system Σ: Just put Σ in the iterative canonical form and assignable polynomials are all monic polynomials of degree n on the form λ (z) · χ (A22 , z) (see [9]). A similar result holds for systems over rings where all feedback invariants are projective. But in the general case we know that we only can shift res.rk.(Σ) poles. For weakly reachable systems we are in general not allowed to assign any pole, but it is obvious that the characteristic polynomial may be shifted by feedback actions. What are the polynomials we can reach by feedback? In the case of single input weakly reachable linear systems over Z (or over k[t]) a canonical form is provided in [9] (see Note 3.25 above). So it would be possible to perform this calculation. • Locally Brunovsky systems: Define locally ΣBrunovsky Σ linear system as a system aΣ Σ Σ such that all feedback invariants Ni , Mi , Ni+1 /Ni are projective. Then one may perform iterative procedures in order to relate those invariants an to construct a ”canonical form” for linear system. In this problem the class of BCS rings is very suitable because every finitely generated projective R-module split as a direct sum of rank one projective modules. Thus if we know the set Pic(R) of finitely generated rank one projective R-modules we would perform arithmetic calculations and Ferrers diagrams in order to give a classification of locally Brunovsky linear systems over the BCS ring R.

References ´ [1] Atiyah, MF; Macdonald, IG. Introducción al Algebra Conmutativa. Barcelona: Reverté; 1989. [2] Bourbaki, N. Algèbre Commutative, Chapitres 5 a` 7. Paris: Masson; 1985. [3] Brewer, JW; Bunce, JW; Van Vleck, FS. Linear Systems over Commutative Rings; Lecture notes in pure and applied mathematics 104. New York: Marcel Dekker; 1986. [4] Brewer, JW; Klingler, L. On feedback invariants for linear dynamical systems. Linear Algebra and Its Applications, 2001, 325, 209-220. [5] Brunovsky, PA. A classification of linear controllable systems. Kybernetika, 1970, 3, 173-187.

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[6] Bumby, R; Sontag, ED; Sussmann, HJ;Vasconcelos, WV. Remarks on the PoleShifting problem over rings. Journal of Pure and Applied Algebra, 1981, 20, 113-127. [7] Carriegos, MV. On the local-global decomposition of linear control systems.Communications in Nonlinear Science and Numerical Simulation, 2004, 9(2), 149-156. [8] Carriegos, MV; Diez-Mach´ıo, H; Garc´ıa-Planas, MaI. On Higher order linear systems: Reachability and feedback invariants. Linear Algebra and Its Applications, 2006, 413, 285-296. [9] Carriegos, M; Hermida-Alonso, JA. Canonical forms for single input linear systems. Systems & Control Letters, 2003, 49, 99-110. [10] Carriegos, M; Hermida-Alonso, JA; Sánchez-Giralda, T. The pointwise feedback relation for linear dynamical systems. Linear Algebra and its Applications, 1998, 279, 119-134. [11] Carriegos, M; Hermida-Alonso, JA; Sánchez-Giralda, T. Pole-shifting for linear systems over commutative rings. Linear Algebra and its Applications, 2002, 346, 97-107. [12] Casti, JL. Linear Dynamical Systems. Academic Press; 1987. [13] Garc´ıa-Planas, MaI; Magret, MaD. Miniversal deformations of linear systems under the full group action. Systems & Control Letters, 1998, 35, 279-286. [14] Garc´ıa-Planas, MaI; Magret, MaD. An alternative system of structural invariants of quadruples of matrices. Linear Algebra and Its Applications, 1999, 291, 83-102. [15] Garc´ıa-Planas, MaI; Magret, MaD. Associating matrix pencils to generalized linear multi variable systems Linear Algebra and Its Applications, 2001, 332-334, 235-256. [16] Gillman, L; Jerison, M. Rings of continuous functions. New York: Springer-Verlag; 1976. [17] Hautus, ML; Sontag, ED. New results on pole-shifting for parametrized families of systems. Journal of Pure and Applied Algebra, 1986, 40, 229-244. [18] Heij, C; Ran, A; Van Schagen, F. Introduction to Mathematical Control Theory. Berlin: Birkhäuser-Verlag; 2007. [19] Hermida-Alonso, JA. Linear Algebra over commutative rings. In: Hazewinkel M, editor. Handbook of Algebra III. Elsevier; 2003; 3-61. [20] Hermida-Alonso, JA; Pérez, MP; Sánchez-Giralda, T. Brunovsky canonical form for linear dynamical systems over commutative rings. Linear Algebra and its Applications, 1996, 233, 131-147. [21] Hermida-Alonso, JA; Pérez, MP; Sánchez-Giralda, T. Feedback invariants for linear dynamical systems over a principal ideal domain. Linear Algebra and its Applications, 1995, 218, 29-45.


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[22] Hermida-Alonso, JA; Pisonero, M. Invariant factors of an endomorphism and finite free resolutions. Linear Algebra and its Applications, 1993, 187, 201-206. [23] Hermida-Alonso, JA; Pisonero, M. What polynomials satisfies a given endomorphism?. Linear Algebra and its Applications, 1995, 216, 125-138. [24] Knuth, DE. The art of computer programming. Prefascicle 3B: Sections 7.2.1.4-5. ”Generating all partitions”. Addison-Wesley; 2004. [25] McDonald, BR. Linear Algebra over Commutative Rings. New York: MarcellDekker; 1984. [26] Morse, AS. Ring models for delay differential systems. Automatica, 1976, 12, 529531. [27] Northcott, DG. Finite free resolutions. London: Cambridge University Press; 1976. [28] Richard, J-P. Time-delay systems: an overview of some recent advances and open problems. Automatica, 2003, 39, 1667-1694. ´ [29] Sánchez-Giralda, T. Algebra y Teor´ıa de Sistemas. In: Homenaje al Profesor Hayyek. Tenerife: Universidad de La Laguna; 1990; 299-308. [30] Sergeichuk, VV. Canonical matrices for linear matrix problems. Linear Algebra and Its Applications, 2000, 317, 53-102. [31] Serre, D. Matrices. Theory and Applications. New York: Springer; 2002. [32] Sharma, PK. On pole assignment problems in polynomial rings. Systems & Control Letters, 1984, 5(3), 49-54. [33] Sharma, PK. Some results on pole-placement and reachability. Systems & Control Letters, 1986, 6(5), 325-328. [34] Sontag, ED. Mathematical Control Theory. New York: Springer; 1990. [35] Strang, G. Linear Algebra and Its Applications. 4th edition. Belmont, CA, USA: Thompson Brooks/Cole; 2006. [36] Vasconcelos, WV; Weibel, CA. BCS rings. Journal of Pure and Applied Algebra, 1988, 52, 173-185. [37] Weibel, CA. An Introduction to algebraic K-theory. [2007-03-03]. Available from: http://www.math.rutgers.edu/∼weibel/Kbook.html [38] Willems, JC. Paradigms and puzzles in the theory of dynamical systems. IEEE Transactions on Automatic Control, 1991, AC-36, 259-294.



Chapter 7

A C HARACTERIZATION OF C OMMUTATIVE C LEAN R INGS∗† Warren Wm. McGovern‡ Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403, USA

Abstract A commutative ring A is said to be clean if every element of A can be written as a sum of a unit and an idempotent. This definition dates back to 1977 where it was introduced by W. K. Nicholson [7]. In 2002, V. P. Camillo and D. D. Anderson [1] investigated commutative clean rings and obtained several important results. In [4] Han and Nicholson show that if A is a semiperfect ring, then A[Z2 ] is a clean ring. In this paper we generalize this argument (for commutative rings) and show thatA[Z2 ] is clean if and only A is clean. We also show that if the group ring A[G] is a commutative clean ring, then G must be a torsion group. Our investigations lead us to introduce the class of 2-clean rings.

Key Words : commutative clean ring, commutative group ring

1. On 2-Clean Rings We start with the following definition: The element a ∈ A is said to be clean if there exists an idempotent e ∈ A such that a − e is invertible. If every element of A is clean, then A is said to be a clean ring. ∗ A version of this chapter was also published in Mathematics, Game Theory and Algebra Compendium edited by Jacob H. Mathias, published by Nova Science Publishers. It was submitted for appropriate modifications in an effort to encourage wider dissemination of research. † Communicated by Daneil Anderson ‡ E-mail address: [email protected]

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The set of units and idempotents of A shall be denoted by U(A) and Id(A), respectively. As usual, M ax(A) denotes the set of maximal ideals of A, and it is equipped with the hull-kernel topology. This means that the collection of sets of the form V (a) = {M ∈ M ax(A) : a ∈ M } for a ∈ A form a base for the closed sets. The complement of V (a) is denoted by U (a). It is well known that von Neumann regular rings, and more generally zero-dimensional rings, are clean. Also, local rings are clean, and, in fact, are precisely the indecomposable clean rings. Thus, if A is an integral domain, then A is clean if and only if A is local. Some fundamental facts regarding clean rings are that every homomorphic image of a clean ring is a clean ring, and that a ring A is clean if and only if A/n(A) is clean where n(A) denotes the nilradical of A. Also, a direct product of rings is clean if and only if each factor is clean. Most of these facts can be found in [4] or [1]. We urge the interested reader to check there for more information and selected proofs.

2. All Rings Are Commutative and with Identity In [6] a list of several characterizations of commutative clean rings is given. Included in the list is one given by Johnstone [5] which we presently state. Theorem 2.1 (Johnstone). A is a clean ring if and only if M ax(A) is zero-dimensional and every prime ideal is contained in a unique maximal ideal. A topological space is said to be zero-dimensional if it has a base of clopen sets. It is known that if e ∈ Id(A), then U (e) is a clopen subset of M ax(A), but the reverse is not true in general. A clopen subset of M ax(A) of the form U (e) for some e ∈ Id(A) will be called an idempotent clopen. In [6], it is shown that M ax(A) is zero-dimensional and every clopen is an idempotent clopen precisely when A is a clean ring. Theorem 2.2. The following are equivalent for a ring A: (i) A is a clean ring. (ii) For each a ∈ A there exists an idempotent e such that V (a) ⊆ U (e) and V (a − 1) ⊆ V (e). (iii) The collection of idempotent clopen subsets of M ax(A) is base for the topology on M ax(A). (iv) M ax(A) is zero-dimensional and every clopen subset is of the form U (e) for some e ∈ Id(A). Here is another characterization of a clean ring.

A Characterization of Commutative Clean Rings

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Proposition 2.3. A is clean if and only if for every a ∈ A, there is an idempotent e ∈ A such that both a + e and a − e are invertible. Proof. The sufficiency is clear. As for the necessity, let a ∈ A. By Theorem 2.2, choose an idempotent e ∈ A such that V (a2 ) ⊆ U (e) and V (a2 − 1) ⊆ V (e). Let M be any maximal / M . It follows that a + e, a − e ∈ / M . If ideal of A. If a ∈ M , then so is a2 and hence e ∈ a, e ∈ / M , then neither a + 1 nor a − 1 belong to M . Thus, a + e + M = a + 1 + M 6= M and a − e + M = a − 1 + M 6= M , whence a + e, a − e are invertible. We call a commutative ring A 2-clean if for every pair of elements a, b ∈ A there exists an idempotent e ∈ A such that a + b − e and a − b − e are both invertible. Observe that letting b = 0 we obtain that a 2-clean ring is clean. Also, if charA = 2, then A is clean if and only if A is 2-clean. The proof of our next proposition is straightforward. Proposition 2.4. 1) A homomorphic image of a 2-clean ring is 2-clean. 2) A direct product A = Πi∈I Ai is 2-clean if and only if each factor Ai is 2-clean. Proposition 2.5. The following are equivalent for a ring A. (i) A is 2-clean. (ii) A/n(A) is 2-clean. (iii) A is clean and A/J(A) is 2-clean. (iv) A/J(A) is 2-clean and idempotents lift modulo the Jacobson radical. Proof. The proof is exactly the same as the one used to show that a ring A is clean if and only if A/n(A) is clean. We leave the verification to the interested reader. Our next result gives an easier way of checking whether a clean ring is 2-clean. Proposition 2.6. A is 2-clean if and only if A is clean and 2 ∈ J(A). Proof. Necessity. If A is 2-clean, then it is clean. Let a ∈ A be an arbitrary element. Using the 2-clean condition on the pair a, a there is an idempotent e ∈ A such that 2a − e and −e (and hence e) are invertible. The only idempotent that is invertible ise = 1. Therefore, 2a − 1 is invertible, whence 2 ∈ J(A). Sufficiency. Let a, b ∈ A. Choose an idempotent e ∈ A such that a + b − e is invertible. We claim that a − b − e is also invertible. Let M be a maximal ideal of A. By hypothesis, 2 ∈ M so that a − b − e + M = a + b − e + M 6= M . Therefore, a − b − e ∈ / M for any maximal ideal of A. It follows that both a − b − e and a + b − e are invertible.

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Theorem 2.7. Let A be a zero-dimensional ring. Then the following are equivalent: (i) A is 2-clean. (ii) charA = 2k for some natural k. (iii) 2 is nilpotent. (iv) For each maximal ideal M of A, charA/M = 2. Proposition 2.8. Suppose A is a local ring. Then A is 2-clean if and only if charA/M = 2. Proof. The necessity is clear, so suppose that charA/M = 2. This means that a+b+M = a − b + M for all a, b ∈ A. Therefore, a + b ∈ M if and only if a − b ∈ M . Thus, if both a + b, a − b ∈ M , then a + b − 1, a − b − 1 ∈ / M and hence both are invertible. Otherwise, a + b, a − b ∈ / M means that they are both invertible.

3.

Commutative Clean Group Rings

In this section we address the question of when a group ring is clean. Some work on this question was done in [4] where they showed that if A is a boolean ring and G is a torsion group, then A[G] is a clean ring. We will only consider the case where the group in question is abelian. When a group ring is zero-dimensional or local has been characterized, and therefore we have some sufficient conditions for a group ring to be a clean ring. We list [3] as our main reference on commutative group rings, though we warn the casual reader that most of the information in [3] is done for semi-group rings. All groups are abelian. Let A be a ring and (G, +) be a group. The group ring is the set of formal sums of the form Σni=1 ai X gi under pointwise addition and multiplication defined by X g X h = X g+h on monomials and extending this in the obvious way. Theorem 3.1. A[G] is zero-dimensional if and only if A is zero-dimensional and G is a torsion group. For commutative rings and abelian groups we obtain the following corollary. Corollary 3.2. If A is a boolean ring and G is a torsion group, then A[G] is a clean ring. Theorem 3.3. A[G] is local if and only if A is local, charA/M = p, and G is a p-group. In particular, if A[G] is local, then G is torsion. Our aim is to show that for A[G] to be a clean ring it is necessary that A be clean and G be torsion. The first part is clear as A is a homomorphic image of A[G]. To obtain the second part we need a series of lemmas. The first is well known.


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Lemma 3.4. A[G] is an integral domain if and only if A is an integral domain and G is torsion free. Lemma 3.5. For any ring A and group G, A[G] is clean if and only if A/n(A)[G] is clean. Proof. The necessity is obvious since A/n(A)[G] is a homomorphic image of A[G]. As for the sufficiency it is known that n(A)[G] ≤ n(A[G]), which means that A[G]/n(A[G]) is a homomorphic image of A/n(A)[G] and therefore A[G] is clean. Proposition 3.6. If A[G] is a clean ring, then A is clean and G is a torsion group. Proof. Clearly, A is a clean ring. Let t(G) denote the torsion subgroup of G. Let P be a prime ideal of A. Then (A/P )[G/t(G)] is a homomorphic image of A[G] and hence is clean. Since G/t(G) is a torsion-free group and A/P is an integral domain, it follows by Lemma 3.4 that (A/P )[G/t(G)] is a clean integral domain and hence is local. By Theorem 3.3 it follows that G/t(G) is torsion, whence G = t(G). Remark 1. The converse of the previous proposition is certainly not true. In [4] it is shown that if A is the localization of Z at the prime 7Z (A is clean because it is local) and G = Z3 (clearly torsion), then A[G] is not clean. In general, discovering when A[G] is clean has eluded us except in certain cases. In the next section we consider the specific case when G = Z2 . Proposition 3.7. Suppose A is zero-dimensional. Then the following are equivalent: (i) A[G] is clean. (ii) A[G] is zero-dimensional. (iii) G is a torsion group. Proof. This is in fact a corollary to Proposition 3.6 and Theorem 3.1. Corollary 3.8. Let F be a field. Then F [G] is clean if and only if G is torsion.

4.

When G = Z2

In this section we investigate the cleanliness of the group ringA[Z2 ]. Let e be an idempotent of A. The element eX 0 ∈ A[G] is an idempotent of A[G]. Any idempotent of A[G] of this form is said to be an idempotent in A. If every idempotent of A[G] is in A, then we say that the idempotents of A[G] are in A. Lemma 4.1. The element rX 0 +sX 1 ∈ A[Z2 ] is invertible if and only if r+s, r−s ∈ U(A). Proof. Suppose rX 0 + sX 1 ∈ U(A[Z2 ]). Then there are f, g ∈ A such that (rX 0 + sX 1 )(f X 0 + gX 1 ) = 1. This means that rf + sg = 1 and rg + sf = 0. Adding these two equations gives us that (r + s)(f + g) = 1. Subtracting yields (r − s)(f − g) = 1. Therefore, r + s, r − s ∈ U(A). r −s Conversely, if r + s, r − s ∈ U(A), then so is r 2 − s2 . Let f = r2 −s 2 and g = r 2 −s2 . A quick check shows that rf + sg = 1 and rg + sf = 0. It follows that rX 0 + sX 1 is an invertible element of A[Z2 ].

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Lemma 4.2. The element cX 0 + dX 1 ∈ A[Z2 ] is an idempotent if and only if c2 + d2 = c and 2cd = d. In this case, c + d, c − d ∈ Id(A). Proof. The first statement follows from the equation cX 0 + dX 1 = (cX 0 + dX 1 )2 = (c2 + d2 )X 0 + 2cdX 1 . The second is obtained by adding (and subtracting) the equations. Proposition 4.3. The following are equivalent for the ring A: (i) A is clean. (ii) For every a ∈ A, the element aX 0 ∈ A[Z2 ] can be written as the sum of a unit and an idempotent in A. (iii) For every a ∈ A, the element aX 1 ∈ A[Z2 ] can be written as the sum of a unit and an idempotent in A. Proof. (i) ⇒ (ii) If A is clean, then for each a ∈ A let u ∈ U(A) and e ∈ Id(A) such that a = u + e. Then aX 0 = uX 0 + eX 0 is a representation of aX 0 as a sum of a unit of and an idempotent in A. (ii) ⇒ (i) Let a ∈ A. If aX 0 = (rX 0 + sX 1 ) + eX 0 is a representation of aX 0 as a sum of a unit and an idempotent in A, then s = 0 and thus r = r + s ∈ U(A). Therefore, A is a clean ring. (iii) ⇒ (i) Let a ∈ A. Suppose that aX 1 is the sum of a unit and an idempotent in A, say aX 1 = (rX 0 + aX 1 ) + eX 0 . for some e ∈ Id(A). By Lemma 4.1 r + a, a − r ∈ U(A). Since r = −e it follows that a − e and a + e are both invertible and so by Proposition 2.3 A is clean. (i) ⇒ (iii) Let a ∈ A. By Proposition 2.3 there exists an e ∈ Id(A) such that a−e, a+e are invertible. Set r = −e so that aX 1 = (rX 0 + aX 1 ) + eX 0 . Since r + a = a − e and r − a = −(a + e) are invertible it follows that aX 1 is the sum of a unit and an idempotent in A. Proposition 4.4. For every a ∈ A, the element aX 0 + aX 1 ∈ A[Z2 ] can be written as the sum of a unit and an idempotent in A if and only if 2 ∈ J(A). In this case every idempotent of A[Z2 ] belongs to A.


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Proof. Suppose aX 0 + aX 1 ∈ A[Z2 ] can be written as the sum of a unit and an idempotent in A for every a ∈ A, say ax0 + aX 1 = (rX 0 + sX 1 ) + eX 0 . Since s = a it follows that r + a, r − a ∈ U(A) by Lemma 4.1. Notice that e = a − r which is invertible and hence e = 1 and r = a − 1. Thus r + a = 2a − 1 ∈ U(A) for every a ∈ A, whence 2 ∈ J(A). Conversely, let a ∈ A and 2 ∈ J(A). Since aX 0 + aX 1 = ((a − 1)X 0 + aX 1 ) + X 0 and because (a − 1) + a = 2a − 1 and (a − 1) − a = −1 are both invertible, the result follows from Lemma 4.1. Finally, suppose 2 ∈ J(A) and let aX 0 +bX 1 be idempotent. This means that a2 +b2 = a and 2ab = b. The latter equality implies that (2a − 1)b = 0. By hypothesis, 2 ∈ J(A) and therefore 2a − 1 is invertible. It follows that b = 0 and so the idempotents of A[Z2 ] belong to A. Proposition 4.5. The following are equivalent. (a) For each a ∈ A, the element aX 0 + aX 1 is clean. (b) For each a ∈ A, there exists an e ∈ Id(A) and t, u ∈ A such that (2a−1)t = (1−e) and 2au = e. (c) For each a ∈ A, there exists an e ∈ Id(A) such that V (2a) ⊆ V (e) and V (2a−1) ⊆ U (e). Proof. (b)⇒(a). Let a ∈ A and choose e ∈ Id(A) and t, u ∈ A satisfying (b). Let f = 1−e so that ef = 0 and e + f = 1. Observe that since both e and f are idempotent we may, without loss of generality, assume that ue = u, and tf = t. Let x = aue and observe that 2x = e and ex = x. Let y = (a − 1)f , z = (a − x)e, and r = y + z. Next let d = −x and c = d + 1 = 1 − x. Now, c2 + d2 = d2 + 2d + 1 + d2 = 2d2 + 2d + 1 = 2x2 − 2x + 1 = x + f. Notice that 2x = e = 1 − f so that c = 1 − x = x + f and therefore c2 + d2 = c. Next, 2cd = 2c(−x) = −ce = −(1 − x)e = ex − e = x − e = x − 2x = −x = d and so cX 0 + dX 1 is an idempotent. Now, r+c = y+z +c = (a − 1)f + (a − x)e + 1 − x = af − f + ae − ex + 1 − x = a − f − ex + 1 − x = a + e − 2x = a

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Warren Wm. McGovern so that aX 0 + aX 1 = (rX 0 + (r + 1)X 1 ) + (cX 0 + dX 1 ).

What is left to be shown is that (rX 0 + (r + 1)X 1 ) is invertible. To that end, first notice that 2r + 1 = 2y + 2z + 1 = 2(a − 1)f + 2(a − x)e + e + f = (2a − 1)f + 2ae. Next, (2r + 1)(t + u) = ((2a − 1)f + 2ae)(t + u) = (2a − 1)t + 2aue = f +e = 1. Therefore rX 0 + (r + 1)X 1 is invertible by Lemma 4.1, whence aX 0 + aX 1 is clean. (a)⇒(c). Suppose aX 0 + aX 1 is clean and write it as a sum of a unit and idempotent, say aX 0 + aX 1 = (rX 0 + sX 1 ) + (cX 0 + dX 1 ). Note that a = r + c = s + d. Since (rX 0 + sX 1 ) is invertible, r + s ∈ U(A), and since (cX 0 + dX 1 ) is an idempotent, it follows that c + d ∈ Id(A). Now 2a = a + a = r + c + s + d = (r + s) + (c + d) and so 2a is a clean element. It follows that there exists an idempotent e such that V (2a) ⊆ V (e) and V (2a − 1) ⊆ U (e). The equivalence of (b) and (c) is straighforward. Theorem 4.6. For a ring A, the following are equivalent: (i) A is 2-clean. (ii) A[Z2 ] is clean and the idempotents of A[Z2 ] belong to A. (iii) A[Z2 ] is clean and 2 ∈ J(A). (iv) Every element of A[Z2 ] can be written as the sum of a unit and an idempotent in A. (v) A is clean and for every a, b ∈ A, V (a + b) ∩ V (a − b − 1) = ∅. Proof. (i)⇒ (ii) If A is 2-clean, then by Proposition 2.6 and Proposition 4.4 every idempotent of A[Z2 ] is in A. Let aX 0 + bX 1 ∈ A[G]. By the hypothesis there is an idempotent e such that a + b − e and a − b − e are both invertible. Set r = a − e so that r + b and r − b are invertible. It follows that aX 0 + bX 1 = (rX 0 + bX 1 ) + eX 0 is a clean representation,whence A[Z2 ] is a clean ring.


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(ii) ⇒ (iii) Suppose that A[Z2 ] is clean and that idempotent elements are in A. Let a ∈ A and write aX 0 + aX 1 = (rX 0 + aX 1 ) + eX 0 where e2 = e, r + e = a, and rX 0 + aX 0 is invertible. As previously noted it follows that r + a, r − a are both invertible. Since r − a = −e is invertible it follows that e = 1. Therefore, the invertibility of 2a − 1 = r + a and arbitrariness of a imply that 2 ∈ J(A). (iii) ⇒ (ii) This is patent. (ii) ⇒ (i) Let a, b ∈ A. Then the element aX 0 + bX 1 is clean and thus we can write it as aX 0 + bX 1 = (rX 0 + bX 1 ) + eX 0 where e ∈ Id(A) and (rX 0 + bX 1 ) ∈ U(A[Z2 ]). Observe that r + e = a and r + b, r − b ∈ U(A). Since a + b − e = r + b and a − b − e = r − b we are done. (ii) ⇒ (iv). This is clear. (iv) ⇒ (i). Let a, b ∈ A. By hypothesis there is a unit rX 0 + sX 1 ∈ A[Z2 ] and an idempotent e ∈ A such that aX 0 + bX 1 = (rX 0 + sX 1 ) + eX 0 . It follows that s = b and that a − e = r. It is straighforward from here to show that A is 2-clean. (i) ⇒ (v). If A is 2-clean, then V (a − b − 1) = V (a + b − 1) which is always disjoint from V (a + b). (v) ⇒ (1). For any a ∈ A we have ∅ = V (a − a) ∩ V (a + a − 1) = V (0) ∩ V (2a − 1) = V (2a − 1) and therefore 2a − 1 is invertible. It follows that 2 ∈ J(A). Corollary 4.7. Let charA = 2. Then A[Z2 ] is clean if and only if A is clean. The next result was shown in [4]. We include a sketch of proof for completeness sake. Proposition 4.8. Let 2 ∈ U(A). Then A[Z2 ] ∼ = A × A. In this case A is clean if and only if A[G] is clean. Proof. Define ψ : A[G] → A × A by ψ(aX 0 + bX 1 ) = (a + b, a − b). Then ψ is a ring homomorphism. When 2 is a regular element this homomorphism is injective, and when 2 is invertible the map is a surjection. Remark 2. If V (2) is a clopen subset of M ax(A), then there is a decomposition of A = A1 ⊕ A2 where the rings A1 and A2 satisfy that 2 ∈ U(A1 ) and 2 ∈ J(A2 ). It follows that A[Z2 ] is a clean ring, but not every element of A[Z2 ] is the sum of a unit and an idemptotent in A.

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Lemma 4.9. Suppose cX 0 + dX 1 ∈ Id(A[Z2 ]). Then both V (c), V (d) r V (c) are idempotent clopen subsets of M ax(A). Moreover, V (d) is an idempotent clopen subset of M ax(A). Proof. The second statement clearly follows from the first. By hypothesis we know that c2 + d2 = c, 2cd = d, and c + d, c − d ∈ Id(A). The second equation implies that V (2), V (c) ⊆ V (d). In particular, we obtain that V (c) ⊆ V (c + d) ∩ V (c − d). Now, if M ∈ V (c + d) ∩ V (c − d), then both 2c ∈ M and c2 − d2 ∈ M . If 2 ∈ M , then the residue field has characteristic 2, and c + M = c2 + d2 + M = c2 − d2 + M = M so that c ∈ M . It follows that V (c) = V (c+d)∩V (c−d), and hence V (c) is an idempotent clopen subset of M ax(A). Next we show that V (c + d) ∪ V (c − d) = U (d) ∪ V (c) from which it will follow that V (d) r V (c) is an idempotent clopen subset of M ax(A). Suppose that d ∈ / M . Then 2 2 2c + M = 1 + M . From this and c + d = c we conclude that 2d + M = 1 + M or 2d + M = −1 + M . (Observe that in any field of characteristic different from 2 we have that 1/2 − (1/2)2 = (1/2)2 ). Therefore, c + M = d + M or c + M = −d + M in which case M ∈ V (c − d) or M ∈ V (c + d), whence U (d) ∪ V (c) ⊆ V (c + d) ∩ V (c − d). The reverse is the easy case. Theorem 4.10. For a ring A, A[Z2 ] is clean if and only if for each a, b ∈ A there exist idempotents e1 , e2 , e3 , e4 ∈ Id(A) such that U (ei ) ∩ U (ej ) = ∅, i 6= j, 1 = e1 + e2 + e3 + e4

(1)

V (2) ⊆ U (e3 ) ∪ U (e4 )

(2)

V (a + b) ⊆ U (e3 ) ∪ U (e1 )

(3)

V (a + b − 1) ⊆ U (e4 ) ∪ U (e2 )

(4)

V (a − b) ⊆ U (e3 ) ∪ U (e2 )

(5)

V (a − b − 1) ⊆ U (e4 ) ∪ U (e1 )

(6)

Proof. Necessity. Suppose A[Z2 ] is clean and let a, b ∈ A. Then we can write aX 0 + bX 1 = (rX 0 + sX 1 ) + (cX 0 + dX 1 ) where rX 0 +sX 1 ∈ U (A[Z2 ]) and (cX 0 +dX 1 ) ∈ Id(A[Z2 ]). This means that c2 +d2 = c, 2cd = d, c + d, c − d ∈ Id(A) and both r + s and r − s are invertible. By Lemma 4.9 it follows that there are idempotents e1 , e2 , e3 , e4 such that U (e4 ) = V (c), U (e3 ) = V (d) r V (c), U (e1 ) = V (c − d) r V (c) and SU (e2 ) = V (c + d) r V (c). Observe that U (ei ) ∩ U (ej ) = ∅ for each i 6= j and that 1≤i≤4 U (ei ) = M ax(A). Without loss of generality we can assume that ei ej = 0 and thus e1 +e2 +e3 +e4 is an invertible idempotent and hence (1) is satisfied. That (2) is satisfied follows from the fact that V (2) ⊆ V (d) = U (e3 ) ∪ U (e4 ).


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Next, notice that a+b−(c+d) = r+s and so (3) V (a+b) ⊆ U (c+d) = U (e1 )∪U (e3 ). Similarly, a − b − (c − d) = r − s and so (5) V (a − b) ⊆ U (c − d) = U (e2 ) ∪ U (e3 ). Let / M (otherwise 2c ∈ M M ∈ V (a + b − 1). If M ∈ U (e1 ) = V (c − d) r V (c), then c + d ∈ and 2 ∈ / M would imply c ∈ M .) Since c + d is idempotent, c + d + M = 1 + M . Hence M = a + b − 1 + M = a + b − (c + d) + M contradicting that r + s is invertible. Similarly, if M ∈ U (e3 ) = V (d) r V (c) then d + M = M and so c + M = 1 + M . Therefore M = a + b − 1 + M = a + b − (c + d) + M. Therefore, (4) is satisfied. A similar argument yields (6). Sufficiency. Let aX 0 + bX 1 ∈ A[Z2 ] and suppose that e1 , e2 , e3 , e4 are idempotents satisfying the hypothesis. Observe that ei ej = 0 for all i 6= j. We shall produce elements r, s, c, d ∈ A such that c2 + d2 = c, 2cd = d, and both r + s = a + b − (c + d) and r − s = a − b − (c − d) are invertible. Thus, it follows that aX 0 + bX 1 = (rX 0 + sX 1 ) + (cX 0 + dX 1 ) is a clean representation. To that end observe that (1) implies there is an x ∈ (e1 + e2 )A such that 2x = e1 + e2 . We further suppose that x(e1 + e2 ) = x. Let y1 = xe1 and y2 = −xe2 . Set c = x + e3 and d = y1 + y2 . Observe that V (c) = U (e4 ) and V (d) = U (e4 ) ∪ U (e3 ). Since c2 + d2 = (x + e3 )2 + (y1 + y2 )2 = x2 + e3 + x2 e1 + x2 e2 = 2x2 + e3 = x + e3 = c and 2cd = 2(x + e3 )(xe1 − xe2 ) = 2(x2 e1 − x2 e2 ) = 2x2 (e1 − e2 ) = x(e1 + e2 )(e1 − e2 ) = x(e1 − e2 ) = d it follows that cX 0 + dX 1 ∈ Id(A[Z2 ]). By construction c + d = 2xe1 + e3 so that V (c + d) = U (e2 ) ∪ U (e4 ).

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Let r = a − c and s = b − d. Then r + s = a + b − (c + d) and r − s = a + b − (c − d). If r + s is not invertible, then it lies in some maximal ideal M . If M ∈ V (a + b), then M ∈ V (c + d) contradicting (3). Therefore, M ∈ U (a + b) and hence M ∈ U (c + d). So a + b + M = c + d + M = 1 + M , i.e. M ∈ V (a + b − 1). By (4) M ∈ U (e2 ) (since M∈ / V (c) = U (e4 )). Our contradiction is 1 + M = c + d + M = xe1 + xe2 + e3 + xe1 − xe2 + M = 2xe1 + xe3 + M = M since M ∈ V (e1 ) ∩ V (e3 ). Therefore, r + s is invertible. A similar argument shows that r − s is invertible. Theorem 4.11. A[Z2 ] is clean if and only if A is clean. Proof. As we have pointed so many times already, if A[Z2 ] is clean, then so is A. So suppose that A is clean and let a, b ∈ A. Choose idempotents f1 , f2 ∈ A such that V (a + b) ⊆ U (f1 ), V (a + b − 1) ⊆ V (f1 ) and V (a − b) ⊆ U (f2 ), V (a − b − 1) ⊆ V (f2 ). Let e1 , e2 , e3 , e4 be idempotents such that U (e1 ) = U (f1 ) ∩ V (f2 ), U (e2 ) = U (f2 ) ∩ V (f1 ), U (e3 ) = U (f1 ) ∩ U (f2 ), U (e4 ) = M ax(A) r (U (e1 ) ∪ U (e2 ) ∪ U (e3 )). Observe that U (ei ) ∩ U (ej ) = ∅ whenever i 6= j. Without loss of generality ei ej = 0 whenever i 6= j. Therefore, all the conditions of Theorem 4.10 are satisfied except for possibly (2). We claim that V (2) ∩ U (e1 ) and V (a + b) ∪ V (a − b − 1) are disjoint closed sets. To see this let M ∈ V (2) ∩ U (e1 ). Suppose that M ∈ V (a + b). Since M ∈ V (2), M ∈ V (a − b) and hence M ∈ U (f2 ), contradicting that M ∈ U (e1 ). If M ∈ V (a − b − 1), then M ∈ V (a + b − 1) which implies that M ∈ V (f1 ), again contradicting that M ∈ U (e1 ). Similarly, V (2) ∩ U (e2 ) and V (a − b) ∩ V (a + b − 1) are disjoint closed sets. Since A is clean it follows that the collection of idempotent clopen sets form a base for the topology. It follows that there is an idempotent clopen subset K1 ⊆ U (e1 ) such that V (2) ∩ U (e1 ) ⊆ K1 and (V (a + b) ∪ V (a − b − 1)) ∩ K1 = ∅. Similarly, there is an idempotent clopen subset K2 ⊆ U (e2 ) such that V (2) ∩ U (e2 ) ⊆ K2 and (V (a − b) ∪ V (a + b − 1)) ∩ K2 = ∅. By combining U (e4 ) together with K1 and K2 and shrinking down U (e1 ) and U (e2 ) in the appropriate manner, we obtain idempotent clopen subsets which satisfy conditions (1)-(6) of Theorem 4.10. Therefore, A[Z2 ] is clean. Corollary 4.12. Suppose G is an elementary 2-group. Then A[G] is clean if and only if A is clean. Proof. Since G is an elementary 2-group, for each f ∈ A[G] there exists a finite subgroup H of G such that f ∈ A[H] and H is a finite direct product of copies of Z2 . It follows that A[H] is a finite product of copies of A[Z2 ] which is clean. Therefore, f ∈ A[H] can be written as a s um of a unit and an idempotent in A[H], and hence can be written as a sum of a unit and an idempotent in A[G].


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I. Connell, On the group ring, Can. J. Math. 15, 650-685 (1963).

[3]

R. Gilmer, Commutative Semi-Group Rings, Univ. Chicago Press, Chicago (1984).

[4]

J. Han, W.K. Nicholson, Extensions of clean rings, Comm. Alg. 29 no. 6, 2589-2595 (2001).

[5]

P.T. Johnstone, Stone Spaces, Cambridge Stud. Adv. Math., 3, Cambridge Univ. Press, Cambridge (1982).

[6]

W.Wm. McGovern, Neat rings, preprint.

[7]

W.K. Nicholson, Lifting idempotents and exchange rings, Trans. A.M.S., 229, 269278 (1977). Received: September 2004 revised: November 2004



Chapter 8

J ORDAN AUTOMORPHISMS OF C ERTAIN J ORDAN M ATRIX A LGEBRA OVER C OMMUTATIVE R INGS ∗ Ruiping Yao†, Dengyin Wang and Yanxia Zhao Department of Mathematics, China University of Mining and Technology, Xuzhou, 221008, P.R.China

Abstract Let R be a commutative ring with identity 1 and unit 2, and let S (see section 1) be the 2m by 2m Jordan matrix algebra over R. In this article, we prove that any Jordan automorphism of S can be uniquely decomposed as a product of inner and extremal automorphisms, respectively.

Keywords: Jordan algebra; Jordan automorphism; Commutative rings MR(2000) Subject Classification: 17C

1. Introduction Let R be a commutative ring with identity 1 and unit 2, R∗ the subset of R consisting of all invertible elements in R. Mm (R) denotes the m × m matrix algebra over R. Let Tm (R) be the subalgebra of Mm (R) consisting of 0 all upper triangular matrices. For A ∈ Mm (R), A denotes the transpose of the A. ∗A

version of this chapter was also published in International Journal of Mathematics...Volume 16, Number 5, published by Nova Science Publishers. It was submitted for appropriate modifications in an effort to encourage wider dissemination of research. † E-mail address: [email protected]

120

Ruiping Yao, Dengyin Wang and Yanxia Zhao

In Lie algebra, we have studied the symplectic algebra. Now, we consider a new set which have the analogue construction with symplectic algebra. Let ! A B 0 0 S1 (2m, R) = 0 A, B,C ∈ Mm (R), B = B,C = C C A be the matrix set over a commutative ring R. Let ! A B 0 S(2m, R) = 0 A ∈ Tm (R), B = B ∈ Mm (R) , O A then S(2m, R) is a subset of S1 (2m, R). In this article, S(2m, R) is abbreviated to S. Let Γ be a set. We define the Jordan multiplication onΓ by x◦y = xy+yx for all x, y ∈ Γ. Recall that the set Γ is called a Jordan algebra if a, b ∈ Γ, r, s ∈ R implies that ra + sb ∈ R and a ◦ b = ab + ba ∈ Γ. Evidently, Mm (R), Tm (R), S1 (2m, R) and S are Jordan algebra. If an R− module automorphism ϕ of Γ satisfies ϕ(x ◦ y) = ϕ(x) ◦ ϕ(y) for all x, y ∈ Γ, then ϕ is called a Jordan automorphism. More recently, significant research has been done in studying the Jordan automorphisms or Jordan derivations of matrix algebra. Tang, Cao and Zhang [1] studied module automorphisms preserving idempotence and Jordan isomorphisms of triangular matrices over commutative rings. Beidar, Bresar and Chebotar [2] studied the Jordan isomorphisms of triangular matrix algebra over connected commutative rings. Huang, Ban, Li and Zhao [3] considerd Jordan isomorphisms on symmetric matrices over PID. Wang and You [4] proved that any Jordan automorphism ϕ of strictly upper triangular matrix algebra can be uniquely decomposed as a product of a graph, a diagonal, an inner and a central automorphisms, respectively. Benkovic [5] determined the Jordan derivation and anti-derivation on triangular matrices over commutative rings. The aim of those articles are matrix algebra and also R−algebra. We will study S which is Jordan matrix algebra but not R−algebra. In this article, we proved that any Jordan automorphism ϕ of S can be uniquely expressed as ϕ = φS,c · IntS X , where φS,c and IntS X are extremal and inner automorphisms, respectively.

2.

Preliminaries

In the following, we always suppose that m ≥ 2 and 2 ∈ R∗ . For 1 ≤ i ≤ j ≤ m, let Ei, j denotes the 2m by 2m matrix whose sole nonzero element 1 is in the (i, j) position; Ei,− j the 2m by 2m matrix whose sole nonzero element 1 is in the (i, j + m) position; E−i,− j the 2m by 2m matrix whose sole nonzero element 1 is in the ( j + m, i + m) position. Let E denotes the identity matrix in Mm (R). For a ∈ R, 1 ≤ i < j ≤ m, set Ti j (a) = a(Ei j + E− j,−i);

Jordan Automorphisms of Certain Jordan Matrix Algebra ...

121

Ti j = {Ti j (a)|a ∈ R}; Ti,− j (a) = a(Ei,− j + E j,−i ); Ti,− j = {Ti,− j (a)|a ∈ R}. For a ∈ R, 1 ≤ i ≤ m, set Ti,−i (a) = aEi,−i , Ti,−i = {Ti,−i (a)| a ∈ R}, Ti,i (a) = a(Ei,i + E−i,−i). If all diagonal entries of T ∈ Tm (R) are 0, we call T a strictly upper triangular matrix. The set {T | T ∈ Tm (R) be strictly upper triangular matrix } be denoted by Nm (R), which is a subalgebra of Mm (R). Let

! h= 0 Λ be diagonal matrix in Mm (R) O Λ ! A O t= 0 A ∈ Tm (R) , O A ! A O v= 0 A ∈ Nm (R) , O A ! O B 0 w= B = B ∈ Mm (R) . O O and let u = v + w, then u is an ideal of S. Aut(S) denotes the Jordan algebra automorphism group of S. Λ

O

Definition 2.1. An ideal L of S is called invariant in S if it keeps stable under each Jordan automorphism ϕ of S, that is, ϕ(L) = L. Lemma 2.2. The set {Ti,i+1 (1), Ti,− j (1)|1 ≤ i < j ≤ m} generates u by Jordan multiplication. The proof is trivial, we omit it. Before proving the next lemma and theorem, we first give some general comments on Jordan automorphisms. Let R be a commutative ring with 1 and unit 2, T be a Jordan algebra over R. For every Jordan automorphism ϕ : T → T clearly satisfies ϕ(A2 ) = [ϕ(A)]2 , ϕ(ABA) = ϕ(A)ϕ(B)ϕ(A) for all A, B ∈ T . Let E ∈ T be an idempotent, if AE = EA = 0 for some A ∈ T , then we have ϕ(E)ϕ(A) = ϕ(A)ϕ(E) = 0. Lemma 2.3. u is invariant in S. Proof. Let ϕ be any Jordan automorphism of S. Set ϕ(Tii (1)) =

m

∑ akk Tkk

k=1

(i)

u, where 1 ≤ i ≤ m.

122

Ruiping Yao, Dengyin Wang and Yanxia Zhao ϕ(Ti,i+1 (1)) =

m

∑ bkk Tkk (i)

u, where1 ≤ i ≤ m − 1.

k=1

ϕ(Ti,− j (1)) =

m

∑ ckk Tkk (i)

u, where1 ≤ i < j ≤ m.

k=1

For any x ∈ S, since 2 is a unit of R and ϕ is a Jordan automorphism of S, we have = [ϕ(x)]2 . By Tii2 (1) = Tii (1), i = 1, 2, · · · , m, we have

ϕ(x2 )

ϕ(Tii (1)) = ϕ(Tii2 (1)) = [ϕ(Tii (1))]2 .

(1)

By comparing the two side of (1), we obtain (i) (i)

(i)

akk akk = akk

(2)

for 1 ≤ i ≤ m and 1 ≤ k ≤ m. Moreover, for 1 ≤ i ≤ m − 1 by applying ϕ on Ti,i+1 (1) = Tii (1) ◦ Ti,i+1 (1), that is ϕ(Ti,i+1 (1)) = ϕ(Tii (1)) ◦ ϕ(Ti,i+1 (1)). Looking at the diagonal entries, we get (i)

(i) (i)

(i) (i)

bkk = akk bkk + bkk akk ,

(3) (i)

where 1 ≤ k ≤ m and 1 ≤ i ≤ m − 1. Multiplying this equality by akk from the left and the (i) (i) right respectively, we know by (2) that bkk akk = 0, where 1 ≤ k ≤ m, 1 ≤ i ≤ m − 1. By (i) formula (3), we know that bkk = 0, for 1 ≤ k ≤ m, 1 ≤ i ≤ m − 1. Similarly, by applying ϕ on Ti,− j (1) = Tii (1) ◦ Ti,− j (1), 1 ≤ i < j ≤ m, we get (i)

(i) (i)

(i) (i)

ckk = akk ckk + ckk akk .

(4)

(i)

Also, by (2) we have ckk = 0, where 1 ≤ k ≤ m. Thus by lemma 2.2, we have ϕ(u) ⊆ u. Considering ϕ−1 , we have ϕ(u) = u. So u is invariant in S. Let u(1) = u ◦ u, u(2) = u ◦ u(1) , · · · , u(k) = u ◦ u(k−1) , · · · . It is obvious that i+ j≤n+1

u(m−1) =

∑

Ti,− j ,

1≤i≤ j≤n

u(2m−3) = T1,−1 + T1,−2 , u(2m−2) = T1,−1 , and w is exactly the centralizer of u(m−1) in u. The following lemma is obvious. Lemma 2.4. w and T1,−1 are all invariant in S.

3.

The Standard Jordan Automorphisms of t

It is obvious that t is isomorphic to Tm (R). Tang, Cao and Zhang [1] has described the Jordan automorphism of Tm (R). We now thransfor them to t for using later. t has the following standard automorphisms.

Jordan Automorphisms of Certain Jordan Matrix Algebra ... (a) Inner automorphisms. For invertible A ∈ Tm (R), set T =

A

O

123

!

∈ t, and define Intt T : t → t, sending 0 O A −1 X ∈ t to T XT −1 , then Intt T is a Jordan automorphism of t, called the inner automorphism of t induced by T . (b) Graph automorphisms. Let ε = ε2 be an idempotent in R, J = E1m + E2,m−1 + · · · + Em−1,2 + Em1 . Define φt,ε : ! ! 0 O εA + (1 − ε)JA J A O , then ∈ t to t → t by sending any 0 0 O εA + (1 − ε)JAJ O A φt,ε is a Jordan automorphism of t, called the graph automorphism of t induced by ε. The main theorem in [1] is as follows. Theorem 3.1. Suppose R is a commutative ring with 1 and 2 ∈ R∗ , m ≥ 2. Then following two conditions are equivalent: (i) f is a Jordan isomorphism on Tm (R) over the ring R; (ii) There exist an invertible matrix P ∈ Tm (R) and an idempotence ε ∈ R such that 0 f (X ) = P(εX + (1 − ε)JX J)P−1 for any X ∈ Tm (R). That is, if R is a commutative ring with identity 1 and unit 2, then every Jordan automorphism ϕt of t can be written uniquely in the form: ϕt = Intt T · φt,ε , where Intt T and φt,ε are the inner and graph Jordan automorphisms of t just defined.

4.

The Standard Jordan Automorphisms of S

We now define some standard Jordan automorphisms of S in order to construct the Jordan automorphism of S. (a) Inner automorphisms. 0

For invertible A ∈ Tm (R) and B = −B ∈ Mm (R), set X =

A

AB

!

∈ S1 (2m, R) 0 O A −1 and define IntS X : S → S, sending Y ∈ S to XY X −1 , then IntS X is a Jordan automorphism of S, called the inner automorphism of S induced by X . Let ! A AB 0 G1 = IntS 0 A ∈ Tm (R) be invertible, B ∈ Mm (R) satisfy B = −B . O A −1 Then G1 forms a subgroup of Aut(S). (b) Extremal automorphisms. Let c ∈ R∗ and define φS,c : S → S, by sending φS,c

A

B

!

A cB

!

∈ S to . Then 0 0 O A O A is a Jordan automorphism of S, called the extremal automorphism of S induced by

124

Ruiping Yao, Dengyin Wang and Yanxia Zhao

c ∈ R∗ . Note that if c = r2 for ! some r ∈ R∗ , then φS,c is exactly the inner automorphism rE O . If c ∈ / (R∗ )2 , φS,c is not an inner automorphism. Let of S induced by −1 O r E G2 = {φS,c | c ∈ R∗ }. Then G2 forms a subgroup of Aut(S). It is easy to see that G1 ∩ G2 = {φS,c | c ∈ (R∗ )2 }.

5. Jordan Automorphisms of S In this paper, we obtain the main theorem as follows. Theorem 5.1. Let R be a commutative ring with identity, 2 ∈ R∗ and m ≥ 2. S be the 2m × 2m Jordan matrix algebra over R. Then for any Jordan automorphism ϕ of S there exist inner and extremal automorphism IntS X and φS,c respectively, the ϕ can be written in the form: ϕ = φS,c · IntS X. In other words Aut(S) = G2 G1 and G1 ∩ G2 = {φS,c | c ∈ (R∗ )2 }. Proof: Let ϕ be a Jordan automorphism of S. Since w is stable under ϕ, then ϕ induces Jordan automorphism ϕ of S/w by ϕ(Y ) = ϕ(Y ), for Y ∈ S. Since S/w is isomorphic to t, we may directly view S/w as t. Thus ! by theorem 3.1, ϕ can be written in the form: ϕ = A O for certain invertible A ∈ Tm (R), ε is an idempotent Intt T · φt,ε , where T = 0 O A −1 in R. It is easy to see that Intt T = IntS T , so ϕ · IntS T −1 = φt,ε . Denote ϕ · IntS T −1 by ϕ1 . In the following, we shall give the proofs by steps. Step 1. ε = 1. Note that ϕ1 (T11 (1)) = T11 (ε)+Tmm (1−ε)+W for some W ∈ w. Since T1,−1 is invariant ideal of S, we may assume that ϕ1 (T1,−1 (a)) = T1,−1 (1), where a ∈ R. It is obvious that a ∈ R∗ . By applying ϕ1 on T11 (1) ◦ T1,−1 (a) = T1,−1 (2a), we have that (T11 (ε) + Tmm (1 − ε) +W ) ◦ T1,−1 (1) = T1,−1 (2), which shows that 2ε = 2, thus ε = 1 and ϕ1 fix each T + w for T ∈ t. ! E B 0 Step 2. There exist certain W = with B = −B ∈ Mm (R) such that IntSW · O E ϕ1 (H) = H for any H ∈ h. ! O B1 0 , where B1 = B1 ∈ Mm (R). Let ϕ1 (Tii (1)) = Let ϕ1 (T11 (1)) = T11 (1) + O O ! O Bi 0 , 2 ≤ i ≤ m, where Bi = Bi ∈ Mm (R). Since [T11 (1)]2 = T11 (1), we Tii (1) + O O have ϕ1 (T11 (1)) = ϕ1 ([T11 (1)]2 ) = [ϕ1 (T11 (1)]2 . That is ! ! O B1 O B1 = [T11 (1) + ]2 . T11 (1) + O O O O

Jordan Automorphisms of Certain Jordan Matrix Algebra ... ! O B1 , where We get ϕ1 (T11 (1)) = T11 (1) + O O 

0

 (1)  b  12 B1 =  .  ..  (1) b1m

b12

(1)

b13

(1)

···

b1m

0 .. .

0 .. .

··· .. .

0 .. .

0

0

···

0

0 ···  . ..  . .  .  (1)  Bi =  b1i · · ·  . ..  .. .  0 ···

!

O Bi

Similarly, we have ϕ1 (Tii (1)) = Tii (1) + 

(1)

O O

b1i .. .

(i)

··· .. .

0 .. .

··· .. .

(i)

···

bim



    ∈ Mm (R).  

, where

 0 ..   .   (i) bim   ∈ Mm (R). ..  .   0

Since [T11 (1)]2 = T11 (1), T11 (1)Tii (1) = 0, 2 ≤ i ≤ m. We get ! O B1 )(Tii (1)+ ϕ1 (T11 (1)Tii (1)) = ϕ1 (T11 (1))ϕ1 (Tii (1)) = (T11 (1)+ O O (1)

(2)

(1)

(3)

(1)

125

O Bi O O

!

) = 0.

(m)

So b12 + b12 = b13 + b13 = · · · = b1m + b1m = 0. Similarly, by [Tii (1)]2 = (2) (3) (2) (4) (m−1) Tii (1), Tii (1)T j j (1) = 0, 2 ≤ i 6= j ≤ m, we have b23 + b23 = b24 + b24 = · · · = bm−1,m + (m)

bm−1,m = 0. Let 

0

(1)

b12

(1)

b13

···

 (2)  −b(1) 0 b23 · · · 12  B= . .. .. ..  .. . . .  (1) (2) −b1m −b23 · · · · · ·



(1)

b1m

    ∈ Mm (R).  

(2)

b2m .. . 0

Then IntSW · ϕ1 (Tii (1)) = Tii (1). Moreover, T11 (1), · · · , Tmm (1) generate the h, so ϕ1 (H) = H for any H ∈ h. Denote IntSW · ϕ1 by ϕ2 . Step 3. ϕ2 (V ) = V for any V ∈ v. Let ϕ2 (T12 (1)) = T12 (1) +

O C

!

0

, where C = C ∈ Mm (R). By T12 (1) ◦ T11 (1) =

O O T12 (1), we have ϕ2 (T12 (1) ◦ T11 (1)) = ϕ2 (T12 (1)) ◦ ϕ2 (T11 (1)) = ϕ2 (T12 (1)), which shows

126

Ruiping Yao, Dengyin Wang and Yanxia Zhao ! O C1 , where that ϕ2 (T12 (1)) = T12 (1) + O O 

0

 (1)  c  12 C1 =  .  ..  (1) c1m

c12

(1)

c13

(1)

···

c1m

(1)

0 .. .

0 .. .

··· .. .

0 .. .

0

0

···

0



    ∈ Mm (R).  

Similarly, by T12 (1) ◦ T22 (1) = T12 (1), ϕ2 (T12 (1)) ◦ ϕ2 (T22 (1)) = ϕ2 (T12 (1)), we get (1) ϕ2 (T12 (1)) = T12 (1)+T1,−2 (c12 ). By applying ϕ2 on (T12 (1))2 = 0, we know that [T12 (1)+ (1) (1) (1) T1,−2 (c12 )]2 = 0, which shows that 2c12 = 0, thus c12 = 0. Hence ϕ2 fix T12 (1). Similarly, we have ϕ2 (Ti,i+1 (1)) = Ti,i+1 (1), 2 ≤ i ≤ m − 1. Since v can be generated by {Ti,i+1 (1)|1 ≤ i ≤ m − 1}, ϕ2 fix all V ∈ v. Step 4. ϕ2 (Ti,− j ) = Ti,− j for all 1 ≤ i < j ≤ m. Notice that w is stable under ϕ2 . For any 1 ≤ i < j ≤ m, suppose that ϕ2 (Ti,− j (1)) = ! O D 0 , where D = (di j )m×m ∈ Mm (R) satisfies D = D. By applying ϕ2 on Tii (1) ◦ O O ! ! O D O D = , which shows that all Ti,− j (1) = Ti,− j (1), we have that Tii (1) ◦ O O O O dkl = 0 except for the case that k = i or l = i. By applying ϕ2 on T j j (1) ◦ Ti,− j (1) = Ti,− j (1), ! ! O D O D = , which shows that dkl = 0, except for the we have that T j j (1) ◦ O O O O case that k = j or l = j. So ϕ2 (Ti,− j (1)) ∈ Ti,− j . It follows that ϕ2 (Ti,− j ) = Ti,− j for all 1 ≤ i < j ≤ m. Step 5. ϕ2 is an extremal automorphism of S. Notice that ϕ2 (T1,−1 ) = T1,−1 . Suppose that ϕ2 (T1,−1 (1)) = T1,−1 (c), then c ∈ R∗ and ϕ2 (T1,−1 (a)) = T1,−1 (ac) for any a ∈ R. For any 2 ≤ j ≤ m, by applying ϕ2 on T1 j (1) ◦ T1,− j (a) = T1,−1 (2a), we see that ϕ2 (T1,− j (a)) = T1,− j (ac) for all a ∈ R. For any 2 ≤ i < j ≤ m, a ∈ R, by applying ϕ2 on Ti1 (1) ◦ T1,− j (a) = Ti,− j (a), we have that ϕ2 (Ti,− j (a)) = Ti,− j (ac). In fact, ϕ2 is exactly the extremal automorphism φS,c induced by c. Now we see that ϕ = IntSW −1 · φS,c · IntS T . By this we can easily obtain the desired expression for ϕ. Now, the proof is complete.

References [1] Modular automorphisms preserving idempotence and Jordan isomorphisms of triangular matrices over commutative rings, Linear Algebra Appl. 2001, 338:145-152. Tang Xiaomin. Cao Chongguang. Zhang Xian.

Jordan Automorphisms of Certain Jordan Matrix Algebra ...

127

[2] Jordan auomorphisms of triangular matrix algebra over a connected commutative ring, Linear Algebra Appl.2000,312:197-201. K.I.Beidar.M.Bresar, M.A.Chebotar. [3] Jordan isomorphisms and additive rank preserving maps on symmetric matrices over PID, Linear Algebra Appl.2006,419:311-325. Huang Liping. Ban Tao. Li Deqiong. Zhao Kang. [4] Decomposition of Jordan automorphism of strictly triangular matrix algebra over local rings, Linear Algebra Appl. 2004, 392:183-193. X.T.Wang, H.You. [5] Jordan derivation and antiderivation on triangular matrices over commutative rings, Linear Algebra Appl. 2005, 397:235-244. Dominik Benkoviˇc.

INDEX A Aβ, 69, 75, 76, 77 Abelian, 2, 6 adaptation, 19, 45 algorithm, 95 alternative, 102 alternatives, 34 AMS, 1, 14 Amsterdam, 6 APP, 39, 65 appendix, 10 application, 6, 50, 64 applied mathematics, 101 argument, ix, 41, 42, 43, 47, 54, 100, 105, 115, 116 arithmetic, 101 assignment, viii, 39, 50, 51, 103 assumptions, 18, 22 asymptotic, vii, ix, 53, 54, 60, 63, 64

B behavior, viii, 19, 25, 28, 32, 34, 35, 36, 53, 54, 60, 64, 72, 73, 96 blocks, 42, 44, 45, 67 bounds, 4 building blocks, 19, 21

C cast, 67 China, 119 classes, viii, ix, 19, 31, 39, 40, 47, 65, 95, 100 classical, viii, 13, 19, 21, 30, 67, 79, 84, 97, 98 classification, vii, 13, 15, 16, 17, 18, 22, 41, 72, 84, 86, 89, 94, 101 closure, 25, 26, 32, 35, 63 coconut, 82 coding, 66

collaboration, 20, 23 community, 1 complement, 106 complex numbers, ix, 27, 65, 66 composites, viii, 13, 25 composition, 75 computation, vii, 7 conductor, 15, 23 conjecture, vii, 7, 12, 51 construction, 15, 16, 18, 27, 32, 44, 99, 115, 119 control, ix, 65, 66, 67, 71, 72, 73, 79, 82, 83, 93, 95, 96, 101, 102

D decomposition, 45, 51, 66, 68, 70, 94, 102, 113 definition, ix, 19, 20, 23, 32, 34, 67, 71, 74, 105 dichotomy, 15, 28, 29, 31, 34 dimensionality, 51 division, 63 duality, 58 dynamical system, 71, 101, 102, 103

E election, 95 equality, 40, 44, 70, 80, 83, 90, 97, 99, 111, 122 equilibrium, 66 exaggeration, 16 exercise, 2, 73, 79

F family, 59 feedback, viii, ix, 39, 40, 41, 42, 43, 45, 47, 48, 50, 51, 65, 66, 72, 73, 80, 83, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 99, 101, 102 field theory, 13, 28 flavor, 23 freedom, 9 FSP, 23, 24, 28, 29, 33, 36

130

Index G

gene, vii, ix, 13, 15, 36 generalization, viii, 6, 13, 15, 16, 17, 21, 22, 29, 30, 31, 36, 54, 59, 66, 94, 100 generalizations, vii, ix, 13, 15, 36 generators, 57, 72, 82, 101 globalization, 14 government, iv graph, 12, 120, 122 groups, vii, 1, 2, 5, 6, 108

H heart, 83 height, 20 Hilbert, ix, 53 Holland, 6 homomorphism, 14, 16, 18, 23, 69, 70, 76, 113 hypothesis, 3, 5, 6, 18, 30, 36, 76, 87, 91, 107, 111, 112, 113, 114, 115

I idealization, 13, 16, 17, 25 identity, vii, ix, x, 1, 6, 7, 8, 10, 12, 14, 23, 49, 54, 119, 120, 122, 123 Illinois, 62 images, 24, 67 inclusion, viii, 13, 14, 19, 20, 30, 32, 33, 34, 68, 80 indices, 33, 84, 85, 86, 88 induction, 43, 83, 99 infinite, 1, 2, 3, 4, 5, 8, 16, 21, 22, 35, 56, 62, 95, 97 initial state, 71, 72, 73 interface, viii, 53 invariants, 66, 80, 81, 83, 84, 85, 86, 89, 91, 95, 101, 102 Iran, 53 isomorphism, 14, 15, 17, 18, 19, 25, 30, 31, 35, 50, 66, 68, 78, 79, 80, 83, 84, 92, 99, 122 iteration, 67

83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103 linear function, 83 linear systems, vii, viii, ix, 39, 40, 50, 65, 66, 67, 72, 73, 78, 79, 82, 84, 89, 91, 94, 95, 100, 101, 102 localization, 109 London, 51, 63, 103 Louisiana, 12

M manifold, 100 mapping, 74 matrices, 66, 103 matrix, viii, x, 39, 40, 41, 42, 45, 46, 47, 48, 49, 50, 51, 66, 67, 68, 69, 70, 73, 74, 75, 76, 79, 82, 84, 87, 88, 89, 92, 94, 95, 96, 97, 98, 102, 103, 119, 120, 122, 123, 126 matrix algebra, x, 119, 120, 123, 126 minors, viii, 39, 40, 70, 74, 77, 98 models, 103 modules, v, vii, viii, ix, 1, 2, 3, 4, 5, 6, 14, 18, 25, 40, 47, 53, 54, 58, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 78, 79, 80, 81, 84, 89, 91, 93, 100, 101 monogenic, 36 motivation, 8, 25, 67, 72 multiplication, vii, 16, 47, 67, 70, 108, 120, 121

N natural, 5, 17, 19, 35, 67, 68, 72, 75, 76, 78, 80, 108 New York, 37, 38, 50, 51, 63, 64, 101, 102, 103 normal, 67, 100

O Ohio, 1, 6 Oman, 1, 2, 4, 6 operator, 3 overrings, viii, 13, 14, 15, 16, 17, 19, 20, 21, 22, 26, 27, 29, 35

P

J Jordan, v, x, 119, 120, 121, 122, 123, 125, 126 justice, 31

L law, 93 laws, 75 Lie algebra, 119 lift, 107 linear, vii, viii, ix, 39, 40, 50, 51, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82,

Pacific, 38, 63 PAF, 98 Paris, 101 partition, 49, 86 PBQ, 41, 43, 48, 49, 79, 93 pedigree, 21 play, 19 polynomial, ix, 22, 27, 33, 40, 41, 42, 50, 53, 65, 66, 69, 72, 73, 76, 77, 84, 94, 95, 96, 97, 100, 101, 103 power, 4, 28, 35, 50

131

Index powers, ix, 3, 11, 12, 53, 63, 64 programming, 103 property, viii, ix, 6, 14, 19, 21, 22, 23, 28, 30, 34, 36, 39, 40, 41, 42, 45, 46, 47, 48, 49, 50, 65, 66, 68, 76, 81, 97, 98 proposition, 2, 3, 5, 49, 57, 107, 109 pseudo, 15, 36

R range, viii, 10, 39, 40, 42, 45, 46, 50, 51 reading, 66 real numbers, ix, 65 reasoning, 33 recall, vii, 1, 2, 3, 5, 20, 30, 40, 45, 47 recalling, 4 regular, 13, 17, 18, 26, 29, 30, 31, 35, 36, 46, 55, 56, 57, 63, 106, 113 relationship, 46, 101 resolution, 87 returns, 10, 11 Rome, 63 routines, vii, 7, 8, 9, 10

S search, 10, 11 semigroup, 4, 5, 12 series, 4, 11, 50, 108 set theory, vii, 1 similarity, 41 Spain, 39, 65 speed, vii, 1, 11 stability, vii, ix, 40, 41, 45, 53, 55, 56, 62, 63 stabilization, viii, ix, 39, 43, 50, 65, 66, 95 stabilize, 50, 96

stages, vii, 7 strategies, vii, 1 subgroups, 1 subrings, viii, 6, 13, 14, 23, 27, 28 substitution, 27 symplectic, 119 systems, viii, ix, 39, 40, 41, 42, 43, 44, 48, 50, 51, 65, 66, 67, 71, 72, 73, 79, 80, 88, 93, 94, 95, 100, 101, 102, 103

T Tennessee, 12, 13, 37 tension, 22 time, 16, 17, 24, 57, 65, 71, 73, 75 title, 14 topological, 35, 36, 51, 66, 75, 77, 100, 106 topology, 77, 106, 116 transformations, 46, 48 transition, 66, 71 transpose, 47, 119

U unconditioned, 57

V values, 7, 8, 9, 10, 11, 54 vector, 21, 27, 33, 40, 42, 46, 47, 48, 49, 65, 66, 67, 71, 75, 79, 81, 85, 93, 94, 100 vein, 22, 29

Y yield, 45

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