Polynomial Identities in Ring Theory
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Polynomial Identities in Ring Theory
This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks
Editors: SAMUEL EILENBERG AND HYMAN BASS A list of recent titles in this series appears at the end of this volume.
Polynomial Identities in Ring Theory
Louis Hale Rowen Department of Mathematics and Computer Science Bar-llan University Ramat-Can. Israel
1980
ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers
New York London Toronto Sydney San Francisco
COPYRIGHT @ 1980, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF TMIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.
ACADEMIC PRESS, INC.
111 Fifth Avenue, New York, New York 10003
United Kingdom Edition published by ACADEMIC PRESS. I N C . (LONDON) LTD. 14/18 OvalRoad. London NW1
7DX
Library of Congress Cataloging in Publication Data
Rowen, Louis Halle. Polynomial identities in ring theory. Bibliography: p. 1 . Polynomial rings. I . Title. QA251.3.R68 51.Y.4 79-12923 ISBN 0-12-599850- 3 AMS (MOS) Classification Numbers: Primary 16A28,16A38,16A40 Secondary 16A46, 16A48
PRINTED IN THE UNITED STATES OF AMERICA
80 81 82 83
9 8 7 6 5 4 3 2 1
This book is written to honor the memory of Seymour M. Rowen. September 3, 1917-October 7, 1976
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CONTENTS ...
x111 xix
PREFACE PREREQUISITES
CHAPTER 1
The Structure of PI-Rings 1.1.
I .2. 1.3.
1.4.
1.5.
Basic Concepts and Examples The Free Monoid A ( X ) The Free Algebra $\X} Identities Multilinearization Normal Polynomials Examples of PI-Rings Facts about Normal Polynomials Capelli Polynomials and Standard Polynomials Matrix Algebras Matrices and Algebras of Endomorphisms . The Trace The Algebra & { Y } of Generic Matrices Modifring the Algebra of Generic Matrices The General Cayley-Hamilton Theorem and Newton’s Formulas The Regular Representation Nilpotent Subsets Identities and Central Polynomials for Matrix Algebras, and Their Applications to Arbitrary PI-Algebras The Amitsur-Levitzki Theorems The Roie of the Capelli Polynomial Central Polynomials, Featuring g , Properties oj nz-Normal. Central Polynomials of Arbitrary Rings Linear Dependence through the Capelli Polynomial Primitive Rings, Kaplansky’s Theorem, and Semiprimitive Rings Density Minimal Left Ideals The Closure (or Splitting) of a Primitive Ring Kaplansky’s Theorem: Two Proofs and Applications to Simple Algebras Primitive Ideals and the Jacobson Radical Semiprimitive Algebras vii
2 2 3 4 6 7 9 II 12 14 14 15 15 16 18 19 19 20 21 23 24 21 31 31 32 34 35
35 38 39
...
Vlll
CONTENTS
Injections of Algebras, Featuring Various Nil Radicals Admissible Algebras and the Injection Problem PI-Rings with Nilradical 0 The Sum of Nil Lejt Ideals of Bounded Degree (NIR)) Semiprime PI-Rings Nil Subsets of PI-Rings Amitsur’s Method of Obtaining Canonical Identities 1.7. Central Localization of PI-Algebras The Algebra of Central Quotients Localizing at a Set Intersecting g,(R)R Nontrivially Semiprime PI-Rings l . 8 . Tensor Products and the Artin-Procesi Theorem Tensor Products of Modules Tensor Products of Algebras Examples of Tensor Products Tensor Products over a Field The Artin-Procesi Theorem--“ Dificult Direction” Proper Maximally Central Algebras A Strengthened Artin-Procesi Theorem and Its Applications The Tensor Product Question: An Introduction The Brauer Group 1.9. The Prime Spectrum Comparing Prime Ideals of Related Rings Ranks of Prime Ideals Spec. ( R ) Prime Ideals Minimal over g,(R) Integral Extensions‘ of Commutative Rings I . 10. Valuation Rings, Idempotent Lifting, and Their Applications Valuation Rings The Transcendence Degree, and Its Application to Rank Ranks of Afline PI-Rings Affine Rings i f Generic Matrices L i j h g Idempotenis The A-adic Completion The Completion of a Valuation Ring Valuation Rings and the Integral Closure Algebras over Valuation Rings The “Little” Bergman-Small Theorem The “Big” Bergman-Small Theorem 1.11. Identities of Rings without I Exercises 1.6.
40 41 43 45 46 47 48 51 52 53 54 59 59
60 62 63 65 68 69 71 72 72 73 74 75 77 78 80 80 82 84 86 87 89
90 91
92 95 96 100 103
CHAPTER 2 The General Theory of Identities, and Related Theories 2.1.
Basic Concepts Generalized Identities Identities and Generalized Identities of Rings with Involution
109 111 114
CONTENTS
2.2. 2.3.
2.4.
2.5.
2.6.
Special Rings with Involution Generalized Monomials Degree and Related Concepts PI-Rings Which Have an Involution Sets of Identities of Related Rings (with Involution) Stability of Identities The Vandermonde Argument and Its Applications Relatively Free PI-Rings and T-Ideals Building T-Ideals The Jacobson Radical of a Relatively Free PI-Ring Is Nil Relatively Free PI-Rings of Prime and Semiprime Rings Relatively Free Prime Rings with Involution Specht’s Conjecture and Related Considerations Introduction to Trace Identities Identities of Matrix Rings with Involution Minimal Polynomials of Symmetric and Antisymmetric Matrices Some Identities of (M,,(C),1 ) and ( M J C J ,s) Minimal Identities Elementary Sentences of Algebraic Systems Exercises
ix 117 I I7 i19 119 123 125 129 133 133 134 135 136 137 139 140 141 143 144 145 148
CHAPTER 3 Central Simple Algebras 3.1.
3.2.
3.3.
Fundamental Results The Skolem-Noether Theorem and Wedderburn’s Theorem Splitting Fields The Index of a Simple PI-Ring Crossed Products Exponents of Central Simple Algebras Cyclic Aigebras The Reduced Trace Involutions of Central Simple Algebras Involutions and Maximal Subfields Characterization of Involutions of the First Kind Positive General Results about Maximal Subfields of Division Rings The Generic Division Rings Wedderburn’s Method The Cases n =3 and n =4 Generic Matrix Rings with Involution Simple PI-Rings with Involution of First Kind The Generic Division Rings Skew Polynomial Rings and Their Rings of Central Quotients Noncrossed Product Theorems Division Algebras over Fields Having a Complete. Discrete Valuation More Noncrossed Product Theorems The Center of the Ring of Generic Matrices Exercises
151 152 153 155 157 163 164 166 167 171 173 174 175 178 180 183 185 187 187 191 191 196 196 198
CONTENTS
X
CHAPTER 4 Extensions of PI-Rings 4.1. 4.2. 4.3. 4.4.
4.5.
Integral and Algebraic Extensions of PI-Rings Integral PI-E.\ tensions Formal Words and Shirshov’s Solution to the Kurosch Problem The Characteristic Closure of a Prime PI-Ring Finitely Generated PI-Extensions Hilbert‘s Nullstellensatz Generalized, and Related Results ACC (Ideals) and Related Notions Nilpotence of the Jacobson Radical A Small Counterexample Going Down Afine PI-Rings Are Catenary Generalizing the Razmyslov-Schelter Construction Exercises
202 203 204 208 210 210 212 214 215 216 217 218 220
CHAPTER 5 Noetherian PI-Rings 5.1.
5.2.
Sufficient Conditions for a PI-Ring to Be Noetherian Formanek’s Theorem Cauchon’s Theorem The Eahin-Formanek Theorem Noetherian Counterexamples The Theory of Noetherian P1-Rings The Principal Ideal Theorem Rank of Prime Ideals The Intersection of Powers of the Jacobson Radical Intersection oj Powers of Ideals Exercises
224 224 225 226 228 229 229 232 233 236 237
CHAPTER 6 The Theory of the Free Ring, Applied to Polynomial Identities 6. I . 6.2. 6.3.
The Solution of the Tensor Product Question Relation of Codimensions to Spechr’s Problem Representations of Sym(n) Obtaining Polynomial Identities from Young Diagrams Finite Generation of Certain T-ldeals Higman’s Theorem Application of the Combinatoric Method to Polynomials Exercises
239 242 243 246 248 249 250 252
CONTENTS
xi
CHAPTER 7 The Theory of Generalized Identities 7.1. 7.2.
7.3.
7.4. 7.5. 7.6.
Semiprime Rings with Socle The Basic Theorem of Generalized Polynomials and Its Consequences Amitsur‘s Theorem Improper Generalized Identities Strong GIs The Modified Density Theorem and Its Consequences Primitive Rings with Involution A(*)-Analogue of the Density Theorem Matrix Algebras with Involution Identities and Generalized Identities of Rings with Involution Ultraproducts and Their Application to GI-Theory Injecting Prime [and (*)-Prime]Rings into Nicer Rings Martindale’s Central Closure Exercises
254 257 26 1 261 263 264 265 266 270 271 276 279 282 286
CHAPTER 8 Rational Identities, Generalized Rational Identities, and Their Applications 8.1. 8.2.
8.3.
8.4.
Definitions and Examples Generalized Rational Identities of Division Rings Principal Left Ideal Domains and the Ore Condition The Division Ring Structure D on D(h) with the Division Ring of Laurent Series over D Identihing Lifting Generalized Rationalized Identities A Change of Division Rings The First Fundamental Theorem The “Generic” Division Ring W D ( X ) Rational Identities of Division Rings of Finite Degree Rational Equivalence of Simple Algebras of the Same Degree The Second Fundamental Theorem Applications of the Theory of Rational Identities Group Identities Desarguian Geometry Intersection Theorems Exercises
289 29 1 292 293 294 295 296 296 297 299 299 30 1 302 302 304 306 312
APPENDIX A Central Polynomials of Formanek Exercises
315 319
APPENDIX B The Theory of V3 Elementary Conditions on Rings Exercises
320 326
xii
CONTENTS
APPENDIX C Nonassociative PI-Theory NPI-Rings Kaplansky Classes of Algebras Application: Alternative Rings Exercises
327 328 331 332 338
POSTSCRIPT Some Aspects of the History BIBLIOGRAPHY MAJOR THEOREMS CON(ERNING IDENTITIES MAJOR COUNTEREXAMPLES NOTATION LISTOF PRINCIPAL INDEX
339
34 1 355 358 359 361
PREFACE One of the main goals of algebraists is to find large, natural classes of rings which can be analyzed in depth. An early example was M,(F), the algebra of n x n matrices over a field F , for varying n and F ; by the beginning of this century, the structure of M , ( F ) was well known. Then, much important work was done on finite dimensional algebras over a field ; Albert [61B] (written in 1939 and dealing exclusively with finite dimensional algebras) is still authoritative in many aspects. When studying the class of finite dimensional algebras over a field, one encounters the following difficulty: Suppose A is a finite dimensional Falgebra. Obviously every F-subalgebra of A is also finite dimensional. However, what can be said of the subrings of A ? Conversely, there is no clear-cut way to determine when a given ring is a subring of a finite dimensional algebra over a suitable field. To overcome this obstacle, an obvious strategy is to build a more general structure theory, based on properties common to all subrings of finite dimensional algebras. One such property turns out to be the most natural imaginable. Given a ring R and a polynomial , f ( X , ,.. . , X , ) in noncommutative indeterminates X , , .. ,, X , (having integral coefficients), callfan identity o f R iff(r,, . . ., rI) = 0 for all substitutions r l , . . . , rr of R. Iff is an identity of R, we also say R satisfies f . A PI-ring (polynomial identity ring) is a ring satisfying an identity whose coefficients are all & 1. Every commutative ring satisfies the identity X , X , - X , X , , and is thus a PI-ring; as we shall see, each finite dimensional algebra over a commutative ring is also a PI-ring. Moreover, all subrings. homomorphic images, and direct products of rings satisfying f ’ also satisfy j : Thus, the class of rings satisfying a given identity is quite large. Amazingly enough, many properties of finite dimensional algebras also pertain to PI-rings, yielding a broad theory called PI-theory. A straightforward exposition of PI-theory is possible because the major gap was recently filled by Formanek [72] and, independently, by Razmyslov [73a]. Their contribution was the construction, for each n, of central polynomials for M,(F), independent of the field F ; these are polynomials taking on all scalar values of M,(F), and no other values. Central polynomials provide a
...
XI11
xiv
PREFACE
link between PI-theory and commutative ring theory, and have led to a complete revolution in the subject through the application of classical methods of commutative ring theory. Results considered deep ten years ago have been reduced virtually to trivialities, clearing the way for applications to other subjects, and also for further insights into PI-theory itself. The most startling illustration of this phenomenon is the major theorem of Artin [69] characterizing Azumaya algebras. The initial work was 3 2 pages of difficult reading; Procesi [72a] improved Artin’s theorem and gave a simpler proof of nine pages. Shorter proofs were found by Amitsur [73] and Rowen [74a], and a new proof, about five pages long, of a much stronger result was discovered independently by Amitsur [75] and Rowen [75P]. Their proof is given here, as shortened even further by Schelter [77b]. Most of the “best” proofs of the main PI-theoretical results have now apparently been found, many new fundamental results have recently been proved, and several outstanding problems in related subjects have been solved. At the same time, new ideas are developing into theories branching from PI-theory. Hence, it seems a good time to write a new, comprehensive book on the subject. The reader ma) find useful the following brief survey of the book’s contents. The text falls naturally into three parts. The first part, comprising Chapters 1-3, is the general PI-structure theory which, in my view, is the descendent of the theory of finite dimensional algebras as given in Albert [61B]. The material in 341.1-1.8 is crucial for an understanding of the proper use of polynomial identities, and could be used for an introduction to the subject. Included are the famous theorems of Amitsur-Levitzki [on identities of M J F ) of minimal degree] and Kaplansky (that all primitive PIrings are simple). However, the main focus is on certain other identities and central polynomials of M , ( F ) ; building up sufficient information in these polynomials, one can transfer this information to all rings satisfying the identities of M,(F). This yields immediately much of the structure theory of prime and semiprime PI-rings, as well as of Azumaya algebras. In $1.9 we investigate the prime ideals of a PI-ring, exploiting the “center.” 41.10 introduces more sophisticated techniques, leading to fundamental results about finitely generated PI-rings, as well as to the difficult Bergman-Small theorem. Several theorems can be proved more easily for rings without 1, which are discussed in $1.1 1. In Chapter 2 we introduce two related theories, “identities of rings with involution” and “generalized identities,” in order to treat them together with the usual PI-theory, in a unified framework. At the same time, a study is made of the set of all identities of a given ring R . (These sets correspond naturally to the “7Lideals” of the free ring.) This study is very important because it leads to the notion of a “relatively free” PI-ring ~ ( R )#.( R )
PREFACE
xv
accumulates properties from each ring satisfying the same set of identities as R , highlighting the importance of the following fact: If R is an algebra over an infinite field, all central extensions of R satisfy the same set of identities as R . (This is proved in the course of $2.3, but can be obtained directly; cf. Exercise 2.3.4.) Chapter 3 contains a detailed discussion of simple PI-rings, largely inspired by Albert [61], but in a more modern setting. It turns out that if R is simple, & ( R ) is a (noncommutative) domain which yields a division algebra when we formally invert its central elements. This leads to a simplification of many classical proofs about finite dimensional simple algebras, for they can be obtained “generically,” obviating the painful caseby-case analysis often previously required. More spectacularly, as discovered by Amitsur [72a], many times these “generic” division rings turn out not to be crossed products. The second part (Chapters 4-6) contains the theory of specific classes of PI-rings, whose development depends mainly on Chapter 1. In Chapter 4 the theory of finitely generated PI-algebras is given. This is the proper setting for a noncommutative “algebraic geometry,” which we look at from a strictly ring-theoretical point of view. The entire subject was motivated by a problem of Kurosch, on whether every finitely generated algebra algebraic over a field is necessarily finite dimensional as a vector space. Although the answer is no in general, the answer is yes for PI-algebras; generalizations of this fact have led to a theory of integral PI-extensions paralleling the commutative theory. A similar generalization from commutative theory can be found for Noetherian PI-rings, in Chapter 5, although the proofs are somewhat trickier and less satisfying. The reader should be aware that many of the theorems (including Jategaonkar’s “principal ideal theorem”) hold in more general settings (without PI), although the proofs are considerably more difficult. The theory of Chapter 6 stems from Regev’s theorem, that the tensor product of two PI-rings is necessarily PI. The proof involves a close look at 7’-ideals of the free ring, leading to a study of T-ideals through representations of the symmetric group (cf. 46.2). T-ideals have also been studied by the Russian school, and a representative result of Latyshev is given in $6.3. The third part (Chapters 7, 8, and Appendix B) concerns related theories on (associative) rings which usually are riot PI-rings; these theories are included because they throw more light on PI-theory and are also important in the history of PI-theory. Chapter 7 gives a fairly complete account (based on $92.1-2.3) of the theory of generalized identities, including the (*)-theory, as initiated by Amitsur and developed by
xvi
PREFACE
Martindale, Jain, arid Rowen. This theory is very close to PI-theory in spirit and yields interesting PI-theorems as applications, especially in the determination of when a ring is PI. Chapter 8 deals with the theory of rational identities, which is tied in with the origins of PI-theory through a paper of Dehn [22]. Dehn raised the question as to which nontrivial intersection theorems are possible in a Desarguian projective plane whose underlying division ring is infinite dimensional over the center; he invented PI-theory to frame his question. Amitsur [66a] showed that the answer is none, and invented the theory of rational identities for the proof. Now there is a simpler proof, using part of the generalized identity theory; and the ensuing theory is (in my opinion) very beautiful, with several important applications. In the book we use a “second-generation” central polynomial and, for historical interest, include the original central polynomials of Formanek in Appendix A. (See Appendix A for other reasons why these polynomials are interesting.) In Appendix B a theory is developed permitting one further step in the ambitious program of classifying rings through identitylike conditions. Finally, in Appendix C we glance at a PI-theory for nonassociative rings, where it turns out that an identity-oriented treatment of alternative rings yields striking results. [Note that an associative ring is merely a nonassociative ring satisfying the additional identity (X , X 2 ) X 3 - X I(x2x3).1 I hope module-oriented readers will not be irritated by the way modules are sometimes slighted ; for example, projective modules are not mentioned, even in the treatment of Azumaya algebras! It just so happens that PItheory is rarely enhanced by the use of modules, except in the theory of Noetherian PI-rings (Chapter 5 ) , where the PI plays a subsidiary role as mentioned above. The reason perhaps is that (as far as the literature has developed, at least) the PI is defined on the ring, not on its modules. There are three main aims in this book: to give some people an understandable entry into PI-theory through the first eight or nine sections of Chapter 1; to supply others with a complete account of the “state of the art”; and to point others to directions for further research. (Actually, I think further research will mostly involve the use of PI-theory in related areas.) These three aims are not always consistent, and have led to the following general guidelines: (1) Little prior knowledge is assumed (cf. the prerequisites), although it is certainly useful. ( 2 ) The point of view is not particularly modern. (3) Proofs of important results are given in detail. (4) A few areas itre relegated to exercises (such as the maximal quotient ,
PREFACE
xvii
ring of a semiprime PI-ring, in $1.1 I ) . The "exercises" are often sophisticated pieces of research, and hints are provided in abundance. Nevertheless, I feel little compunction in relegating them to exercises because their proofs have become so much easier in light of the new PItheoretic techniques. There are many mathematicians to whom I am indebted, foremost among whom are N. Jacobson and S. A. Amitsur. There were stimulating and enlightening discussions with E. Formanek, A. Regev, and L. Small; helpful suggestions also came from R. Snider and M. Smith, and G. Bergman generously sent a wealth of related preprints. I am deeply grateful to M. Cohen, A. Regev, and S. Dahari for their careful reading and criticism, respectively, of Chapters 7, 6, and 4. Rachel Rowen translated several key Russian articles. Finally, the staff of Academic Press has been courteous and competent throughout the production of the book.
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PREREQUISITES Formally, the prerequisite to this book is a thorough knowledge of undergraduate abstract algebra, as exemplified in Herstein [64B], which is used as a standard reference. For us, “ring” means “associative ring with I,” and fields are taken to be commutative. Z is the ring of integers and Q is the field of rational numbers. “Algebra” means “associative unital algebra over a commutative ring.” (On rare occasions, when explicitly indicated, we use algebras without I , to facilitate discussions involving subsets without 1.) Every ring is a Z-algebra in the natural way, and usually may be considered thus, at the reader’s pleasure. For a ring R, define the center Z(R) = ( r , ~ R l r r= , r l r for all r in R}. Obviously R is also a Z(R)-algebra. In the book, 4 is fixed and “algebra” means “algebra over 4.” Note that there is a homomorphism 4 + 4 . 1 G R. Often we shall be interested in the case
4 = 62. “Ideal” means “2-sided ideal”; “module” means “unital left module,” i.e., for all y in an R-module M , we have l y = v . Note that for M a module over an algebra R, 4 acts on M by ay = (crl)y for all a €4, Y E M (1 E R). For any module homomorphism or ring homomorphism $, ker $ denotes the kernel of $, the preimage of 0 ; $ is an itljectiott if ker$ = 0, and an onto injection is called an isomorphism. Contrary perhaps to normal usage, the and then $2,” i.e., ( I ~ ~ $ ~ ) ( - Y ) “composition” $,4b2 denotes, “first = $2($l(.Y)).
If M , M , are R-modules, then Hom,(M, M I ) denotes the set of module homomorphisms from M to M , . Define End, M = Hom,(M, M ) . We return to End,M in s1.5, but note here that End, M is a ring, “multiplication” given by composition of functions. An R-module is Jiriite dimemiorid if it is spanned (over R) by a finite number of elements; the smallest possible such number is called the dimettsion of the module. (Likewise for algebras, viewed as modules.) Also, motloid means, “set with associative multiplication and unit element 1 .” As customary, the symbols and denote, respectively, sum and product; an “impossible” sum (such as E.,”=, k) will be taken to be 0. In general, write 0 for the empty set ; s will denote set inclusion, and c will denote proper (i.e., unequal) set inclusion. If :/ is a family of sets and
n
xix
xx
PREREQUISITES
S E .V such that S qL S’ for all S’ E .Y’, we say S is maximal in .Y. :f is Zorn if, SiE 9’. We frequently use for every chain S , c S , G ... in :/,’ we have the principle known as “Zorn’s lemma.” that every Zorn family has maximal sets. For example, suppose A is a proper ideal of R (i.e., 1414) and Y = (proper ideals of R containing A ] , .Then Y is Zorn, so A is contained in a maximal (proper) ideal of R . The same situation holds for left and right ideals, and we shall use these facts implicitly. A good treatment of Zorn’s lemma and equivalent statements is found in Kelley [55B]. Z + denotes the natural numbers { I, 2,. . .}. The reader should be familiar with the principle of mathematical induction, which says that every subset of Z + has a minimal element. Thus, to verify the assertion of a statement P ( t i ) for all 11 in b ’,it suffices to verify P(1) and then to prove, for all m in Z+, that if P ( j ) holds for every j < m, then P ( m ) holds. Similarly, we shall use “definition by induction,” in which way we define a property P(n) for all n in Z+by defining P(1) and then giving P(m)in terms of { P ( j ) l j < m ) . We shall also use other variants of induction. Ring theory usually requires the principle of “transfinite induction,” but one of the pleasant aspects of PItheory is the fact that this more complicated concept is unnecessary. A set S is countuble if it is in one-to-one correspondence with Z +,i.e., if its elements can be enumerated sl, s,, . . . . For any set S, the Cartesian product S x *.. x S taken t times is denoted P). Otherwise, the book is self-contained, with proofs provided for all the needed results, excepting a few places where we need a well-known theorem to get a further refinement of one of the results.
ui=
CHAPTER 1
THE STRUCTURE OF PI-RINGS In this chapter we present the basic structure theory of rings with polynomial identities, called PI-rings, thereby generalizing much of the theory of finite-dimensional algebras. The main technique will be to show that a given ring R satisfies the same multilinear identities as a matrix algebra; then we shall pile information into a few identities and central polynomials of matrix algebras, and transfer it to R . This program is carried out quite deliberately in order to lay the proper foundations for the remainder of the chapter (and the book). In $1.1 we give the basic definitions and examples. (See the Preface for a quick, “intuitive” notion of polynomial identity.) In $1.2 we focus on the most important class of polynomials, the ‘?-normal” polynomials. Basic properties of matrix algebras are reviewed in $1.3 from a relatively free standpoint taken throughout the rest of the book. Only in $1.4 are we really ready to start the program; besides the Amitsur-Levitzki theorem, this section discusses the Capelli polynomials, central polynomials, and structural properties linked with them. $1.5 features Kaplansky’s theorem and its applications; many people do not realize that Kaplansky’s theorem simplifies some important results about division rings. In $1.6 we present various injection theorems needed to apply the above program to semiprime PI-rings. In $1.7 we introduce the very important technique of central localization and some of its uses. Tensor products are introduced in $1.8, leading to an easy proof “from scratch” of the Artin-Procesi theorem. In $1.9 we examine extensions of PIrings, focusing on lifting prime (and semiprime) ideals, the main application being the relation between the prime ideal structures of Z ( R ) and R. Using valuation rings, we commence a deeper study of prime ideals in $1.10, leading to the finite rank of “affine” PI-rings as well as a very pretty (and difficult) theorem of Bergman-Small. Up to this point we have put off the discussion of rings without 1, but some results seem to require it, so in $1.11 we duly define PI-rings without 1 and show that the same underlying theory is thereby obtained.
I
2
THE STRUCTURE OF PI-RINGS
[Ch. 1
$1.l.Basic Concepts and Examples The Free Monoid A ( X )
Given a class of +-algebras that are “P,” where “ P denotes some property (such as “associative,” “commutative,” etc.), we shall often be interested in the free “P” algebra, generated (as +-algebra) by a countable set of elements. say y,, y 2 . .. ., that has the following property: For any R that is ‘&P” and for any countable subset { r , , r,, . . .} E R , there is a unique homomorphism J/ from the free “P’algebra to R such that + ( y k ) = rn for all k . This concept will become clearer as we go along; meanwhile, as examples, let us construct the “free (associative) algebra” and the “free commutative algebra” from the “free monoid.” Definition 1.1.l. A word is a formal string of natural numbers ( i l ... in);we call k the length of the word, and max{i,, . . . ,i k ) is the height of the word. A word of length k and height < t will often be called a ( k , t ) word. Define the product of two words by juxtaposition, i.e., (i, . . . ik)(jI. . .j,J= (il . . . id, . . If w 1 is a ( k , , t,)-word and w 2 is a (k2,r2)-word, then w 1 w 2 is a ( k , + k , , max{tl, t,})-word. Consequently, for each t , {words of height < t } is closed under products. Now, for convenience, we use formal symbols Xi in place of the number i, so that (i, ... in) IS replaced by (Xi,...Xi,) called a formal monomial of degree k. Using the corresponding multiplication (juxtaposition) and adjoining a formal element 1 such that by definition l h = h l = h for each formal monomial h, we obtain a monoid of all formal monomials, written . & ( X ) and called the free monoid. For ease of notation, we write (inductively) X l for X:-’Xi; e.g., X : means X , X , X , . Also, it is convenient to consider 1 as corresponding to the “blank” word ( ), which has length 0 ; thus 1 has degree 0. ej,).
Write deg(h) for the degree of a formal monomial h. As we saw above, deg(h,h,) = deg(h,)+deg(h,). Let deg,(h) be the number of times the indeterminate X ioccurs in h ; then clearly deg(h) = Zidegi(h). Recall that a partial order is a relation ( 6 )that is reflexive (s < x), transitive (x ,< y and y ,< 2 imply s ,< z), and antisymmetric (x < y and y ,< x imply x = J). We digress a bit to introduce a very important partial order. Definition 1.1.2. A word w1 is a subword of a word w if w = w ‘ w , ~ ‘ ’ for suitable words w’, W“ (possibly blank). For example, (132) is a subword of (651 32).
$1.1.1
3
Basic Concepts and Examples
Definition 1.1.3. Given words w, = ( i , ... ik) and w2 = (j, ...j,), which are not blank, we say w , < w 2 if either i, < j , or, inductively, i , = j l and (i2 . . . ik) < (j2. ..j J .
Note. This ordering is stronger than the usual ''lexicographic'' ordering, which also stipulates that a word is greater than each of its beginning subwords. The lexicographic ordering totally orders the words [and thus, correspondingly, .R;c(X)], whereas our ordering is not a total ordering [e.g., (12) and (123) are not comparable]. The point of this difference is the following remark, which does not hold for the lexicographic ordering. Remark 1.1.4. < w2wq.
If w , < w 2 then for any words w3, w4 we have w I w 3
Similarly, we could define another monoid A-(,we have .f= C n E G ( f ; r , a ) + f ; f , ( i j ) n ) ) = E : n s C ( . & t , n ) - ( i j ) ~ f ; ~ , , J ,implying f ( ..., Xi, ..., Xi, ...) = 0. Thus f is t-normal by definition. QED
CnsSym(f)f;r,n)r
Corollary 1.2.4.
Zf(ii+I)o,f= -fforall i d t - - , t h e n f i s t - n o r m a l .
Proof. The transpositions ( I 2), (23), (34), . . . generate all transpositions since, for i < j , ( i j + 1 ) = ( i j ) ( j j + l)(ij). QED
We are ready for a good criterion for t-normality. Theorem 1.2.5. (i) f is t-normal ij’for any 71 in Sym(t), f;,,,) (sg x)n of;,). (ii) f = ZnsSym(L) (sg n)n oAl)g f i s t-normal.
=
Proof. (i) Immediate from Proposition 1.2.3 because every permutation is a product of transpositions. (ii) This is now obvious. QED
Corollary 1.2.6.
f is t-normal if
. f ( X R l ? . . . .X n t r
X L + , , . . .)
=
(sg n ) f ( X *
1 - - .
, x,,x,+1, -..I
for all n in Sym(t). Corollary 1.2.7.
Suppose f ( X , , . . .,X , ) is t-normal. Then ,for any
12
[Ch. 1
THE STRUCTURE OF PI-RINGS
u 2 0 atid any po/.vnomialh ( X 1, . . .,Xu) (possibly constant), r+1
c
(- 1
- If
( X l , . . ., xi-1 7 xi+1 , ‘ . ., x,+ 1, x,+ 2.
‘ ’ ’ 7
X d + 1)
i= 1
. h(Xci + 23 . . ., Xu + d + is (r
1
)Xi
+ 1)-normal.
Proof.
Immediate from Corollary 1.2.4. QED
Thus we have a way of extending t-normal polynomials to polynomials. Definition 1.2.8. is 1.
+ 1)-normal
f is t-primitive if&,, is a monomial whose coefficient
Lemma 1.2.9. lf f is t-normal, then f t-primitive, t-normal fi. Proof.
(t
= x a i f i for
suitable aiin
I$
and
Immediate from Theorem 1.2.5. QED
Now we use the above results to describe two of the most important polynomials in PI-theory.
Capelli Polynomials and Standard Polynomials
Example 1.2.10. C2t- 1
>
...)X2r- 1 )
Define the Capelli polynomials =
C
(sg n)Xr,Xr+1Xn2Xr + 2 .. X r u - 1 )X2r- 1 Xnr,
nESyrn(r)
and C,
(C2t - 1 )(I)
=
C 2 , - 1X2t.C2,- and CZt are t-primitive and t-normal, with
= XI X t
+ 1 . * Xr - 1 Xi?-I X ,
Lemma 1.2.11.
and
(CZt)(r)= X1Xt + 1 .. . XtX2r.
Every t-primitive, t-normal polynomial has tke,form
h o C 2 r - 1 ( X I . . . ., X,, h , , . . ., h,-,)h, = hOC,,(X,, . . .. X , , h , , . . .)h,),
where ho, .. . ,h, E K(X). Proof.
Immediate from Theorem 1.2.5. QED
Proposition 1.2.12. 1 f C 2 , - (resp. C2,) is an identity of R , then every t-normal polynomial is an identity of R . Proof.
Combine Lemmas 1.2.9 and 1.2.11. QED
Thus, in some sense, C2,- “generates” the t-normal polynomials.
41.2.1
Facts about Normal Polynomials
13
Example 1.2.13. Thestaridur~pol~~riomialS, =~:-ts,,,,(f,(sgn)X . . .-X r , . (If t = 0, define S o = 1.) S, is t-normal, with (S,),,, = X,X,...X,, and also satisfies the following formulas:
(i) S , = E ~ = l ( - l ) i - l x i ,..., ~ , -xli (- ,~, xl i + ,,..., x,); (ii) S , = E;=,( - l ) l - i S f -, ( X I , .. . , X i - X i + . . ., x,)X,.
,,
The following fact follows trivially from 1.2.13(i): Remark 1.2.14. If S, is an identity of R , then S d + is also an identity of R . If .f is normal of degree t , then ,f = as, for some OL in Remark 1.2.15. 4. (Immediate from Theorem 1.2.5.)
This fact provides us with some nice equalities. Lemma 1.2.16.
1 n E Sym( 1 )
For t 2 3,
(sg n)St-Z(XlrlXnZXn3,x,,, ' . Xn,)= ( t - 2) !S,(Xl,.. . , X,). '
1
Proof. The left-hand side is obviously t-normal, and there are ( t - 2) ! t ! monomials with coefficients ? I, none of which cancel, so the result is immediate from Remark 1.2.15. QED
Proposition 1.2.17.
S,(Q{X})L S,-,(Q{X})+,for all r 3 3.
Proposition 1.2.17 should be viewed as an improvement of Remark 1.2.14 in characteristic 0. A connection between S,, and [X,, X,] ... [ X , , - , , X,,] is given in Proposition 1.2.18.
Let
1'= L S ) m ( Z l , ( ~ g ~ " n x"21~~~[x"(21-1)1 lr Xr(2rJ' Theri f = 2'S,,. Proof. Clearly f is 2t-normal and so equals mS,, for some integer m. Since f is a sum of 2'(2t)! monomials, none of which cancel, we have m = 2'. QED
Corollary 1.2.19. S(Qa(X}1'.
[ f ' g = [ X l , X 2 ] ~ ~ . [ X 2 , - 1 , X z lthen ] Sz,(C9{X))L
Exercise 1 indicates that S , is the "simplest" t-normal polynomial, and could lead to the feeling that S , is the most worthwhile t-normal polynomial. (Indeed, S , was the only t-normal polynomial in the PI literature until 1973.) Actually, S , turns out to be too good to reflect typically the property "t-normal." In fact, for every matrix algebra R of
14
[Ch. 1
THE STRUCTURE OF PI-RINGS
dimension r z , S,, is an identity of R, as we shall see in Section 1.4. The Capelli polynomial is a far more effective tool than the standard polynomial for studying matrix algebras, and thus most PI-rings.
$1.3. Matrix Algebras
The fundamental building block in the structure of PI-algebras is the theory of matrices, as we shall see time and time again. I n this section some basic classical facts are brought together, with the emphasis in proofs on the matrix algebra over Z[r]. These facts are well known to algebraists; they are included to make the book self-sufficient and to indicate the point of view to be taken later. Matrices and Algebras of Endomorphisms
ForanyringR, .M,(R)denotestheringofn x nmatrices withcoefficients in R. Let 6 denote the “Kronecker de1ta”map: 6, = 0 unless i = j , and 6 , = 1. Then we can define the set ofmatric units ( e i j [1 d i , j d n), where eij is the matrix whose entry in the i j position is hii. Each element of M,(R) can be written uniquely in the form )3:tj= rijeijfor suitable rij in R ; we denote this matrix as (rij).Addition and multiplication then are given by the respective rules
,
Clearly eijeuI.= ‘bjueiv, so r,,,e,, = euu(rij)et,u, an obvious but useful fact which helps us pinpoint t heentries ofa matrix. Wecan identify Z ( R )with Z ( M , ( R))via the map c + ce,,. Similarly, any homomorphism 4 : R + R, extends naturallytoa homomorphism $: M,(R) M,(R,),given by $ ( ( r i j ) )= (I(/(rij)). Suppose M is a free n-dimensional R-module with basis y , , . . .,y,; define eij in End, M such that e i j ( x ; = ruy5)= riyj, and for each r in R define ruyu)= ruryU. Letting R = { ; I r e R}, we see R 2 R and End, M = Reij % M,(l?) % M,(R). Even when M is not free, there is a close connection between End, M and M,(R), which we give now for the commutative case. Let C be a commutative algebra.
x;=
-+
?(x:=l
x:zl xy,j=l
(Procesi-Small). Suppose M is an n-dimensional module over C . Then End,M < + M,(C). In,fact, as a C-algebru, End,M is a homomorphic image of a C-subalgebra of M,(C). Proposition 1.3.1
Proof. =
Suppose M
=
x;=C x i . For any p
E End,
M . we can write p ( x i )
xy,j=lpijsj for suitable P i j e C . Now let R = ( r = ( c i j ) eM,(C)lfor some
$1 3.1
15
Matrix Algebras
B,E End,. M , p , ( x i ) =
cijsj).Then R is a subalgebra of M,(C), and the natural map r b P r is an onto homomorphism of R onto End, M , as can be verified routinely. Thus End,M d R < M,(C). QED The Trace
Of course we can define the trace and determinant (written tr and det) for arbitrary matrices in M,(C), in a manner completely analogous to the special case when C is a field. Here are some useful facts. Remark 1.3.2. a11 i, j , then r = 0.
For r
= ( r i j ) EM,(C),
tr(rejj) = rji. So if tr(rejj) = 0 for
Remark 1.3.3. [ M , ( C ) , M,,(C)]= (Xj+jCejj+ZY:l C ( e , i - q + l . i + l ) ) {elements of trace 0}, a C-module of dimension n2 - 1. (Proof is an easy verification.) =
Remark 1.3.4. 2trS2,(al, . . . , a 2 , ) = 0 for all a,, . .., aZk in M,(C) with k , n arbitrary. [Indeed, by Example 1.3.13,
2trSZk(a,, . . .
7
a2k)
= tr
(-
1 ) i - 1 a i S 2 k -l ( a 1 , .
. .)a,- 1 , a,+
1 3 . .
.
3
a2k)
i= I
+tr 1 ( - 1 )2k-’S2,-
(
( a l , .. .,a,-
,,ai +
.. . ,
i =2k1
The Algebra & ( Y ) of Generic Matrices
In algebra it is often convenient to deal with a “generic” object, on which verifying a given question is often equivalent to verifying the question in general. Our foremost generic object is the algebra of generic matrices, defined as follows: In the free commutative algebra 4[4], label the indeterminates of 4 as {
4,,{ Y } 1. WelZ, take F 3 4[(] and b E M,(F) such that b-' Y, b is diagonal, and let Y'' = b - ' x b . If f ( Y;, Y,, . . . , Yd)= 0, then, by specializing suitably the
18
[Ch. 1
THE STRUCTURE OF PI-RINGS
0 = . f (Y,", Y;, . . . , &") = b-',f( Y , , . . ., Y,)b, implying f (Vl, . . . . Y,) = 0, so $ can indeed be defined. QED
invertible,
then
The General Cayley-Hamilton Theorem and Newton's Formulas
Definition 1.3.17.
Let ui,, be defined on commuting indeterminates = 1, ol,(l)= C;= l i , u2,(A)= Zi<jliAj,. .., ,Ai. We call bin the ith elementary symmetric function on
1,,...,A,,as follows: o0&) a&)
=
n;=
n variables.
. . ., (!,I), 0 d i 6 n. Then clearly x1 = tr Y; and a, Suppose ori = u1,,(t'i1/, Y;. Let p(1) = ( - ~ ) ' C L ~ A= ' ' -det(1~ Y ; ) ;we have p ( Y ; ) = 0. The power of the method is ready to be displayed.
x;=o
= det
Theorem 1.3.18 (Cayley-Hamilton theorem over arbitrary commutative rings). Ij a E M J C ) and p = det(A-a), then p ( a ) = 0. Proof. By Proposition 1.3.15 we may assume C = Z[(] and u = Y;, so we are done by the above observations. QED
Here is another famous application of this method, giving an inductive description of the xk. Theorem 1.3.19 (Newton's formulas). Let a E M J C ) and ( - 1 )kakbe the coefficient of in the characteristic polynomial of'a. Then or, = 1 and ka, = ( -- 1 ) i - ak_,tr(ai)forall k, 1 ,< k d 11.
x!=
'
n;=
Proof.
Again we may assume a = Y;. Let q(1) = (1 -A$)) = l ) k a k l k .Letting q' be the formal derivative of 9 with respect to 2, we take the logarithmic derivative to get q'q-' = Eni = 1 - S;("(l ii -1($')K'= -xP=,tr(a k f l )Ak. developed as a formal power series in 1. Thus
C;=o( -
q' = - q x ; :
tr(ak)Ak-', yielding
(-l)kka,lk-l = k= 1
(-
( - I ) k o r k j . k ) ( i , tr(ak)Ak-l k=O
Matching the coefficients of Ak-' formulas. QED
for
1 d k d n gives Newton's
91.3.1
Matrix Algebras
19
There is a moral here, one of the basic principles of the book: Given a problem, try to treat it in the “generic” setting, and often a technique will present itself that is not otherwise available. (In this instance the technique was diagonalization.) This procedure is sometimes complicated at the beginning and demands considerable patience, but almost always pays off at the end. The Regular Representation An interesting connection exists between finite-dimensional algebras and matrices, which we present more generally. Definition 1.3.20. For any ring R, we define Reg(R) to be the set of “right” multiplications, i.e., those maps !Ir:R --* R given by t,hI,(r’)= r’r, r fixed. Reg(R) is called the (right) regular representation of R. Remark 1.3.21. Viewed as functions, Reg(R) form a ring with respect to addition and composition of functions, and the map r H $r provides an isomorphism of R and Reg(R). [Indeed, let us show that if $ r = 0, then r = 0. If $, = 0, then 0 = 1//~(1) = lr = r.] Remark 1.3.22. If K is a subring of R, then, viewing R as K-module, we have Reg(R) c End, R ; hence there is an injection R + End, R. Proposition 1.3.23. If’ R is t-dimensional as a module over a commutative subring K , then R f M , ( K ) . Proof.
Apply Remark 1.3.22 to Proposition 1.3.1. QED
Along the same lines, we have Proposition 1.3.24. I f R is a t-dimensional algebra over a subring C c Z(R), then R is integral o w r C, of‘ bounded degree < t.
Proof.
Apply Remark 1.3.22 to the Hamilton-Cayley theorem. QED Ni lpotent Subsets
Let us conclude with a few useful remarks about nilpotence. Definition 1.3.25. A subset B of a ring is nil if every element is nilpotent; B is nilpotent ofindex t if B‘ = 0 and I 3 - l # 0. Remark 1.3.26. (This remark holds even for rings without 1.) If Bi are nilpotent left ideals of index ti, i = 1,2, then ( B , + B 2 ) f 1 + ‘ 2 = - 1 0. (Just check each term.) Hence any finite sum of nilpotent left ideals is nilpotent.
20
[Ch. 1
THE STRUCTURE OF PI-RINGS
Definition 1.3.27. If A , B are sets with a formal operation (written as multiplication) A . B -+ S , where S is some set with an element 0, write Ann,B for { a ~ A l a B = 0 ) and AnnbA for { b e B J A b= 0). The subscript is deleted if there is no ambiguity. If bEB, write Ann,b for Ann,{b), and similarly for Annba. "Ann" is short for "annihilator." Remark I .3.28. Suppose R is a ring, M is an R-module, and '4 c R. Then Ann',, A' c' Ann:, A' for all j 1 i ; if Ann',, A' = Ann',, A''' then Ann; A' = Ann', A' for all j > i. (For the second assertion, let B = Ann:, A'. Then AB G Ann',, . 4 j - ' = Ann',, A', by induction, so A ' + ' B = 0, implying B c Ann',, ,4'.) Lemma 1.3.29. Euery riilpotenr subset q f M , ( F ) has irides s
I?.
Proof. View l ' = F(")as M,(F)-module, and let A be a nilpotent subset of M , , ( F ) of index u. For each i, Ann;, ('4') is an F-subspace of V ; Ann',, A" = I/ and Ann; 4 " - - ' # I/. Thus by Remark 1.3.28, Ann; A'" 2 Ann; .4' for all i < u. Hence Ann;. .4" has dimension 3 t i , implying A"C' = 0, so A " = O . QED Proposition 1.3.30. Suppose A is a nil, niultipliratiaely closed subset of M,(F). Then A" = 0.
Proof. Let A' = F A c M,(F), the F-algebra (without 1 ) spanned by A . We argue by induction on the F-dimension of A'. Well, any U E A induces an F-module homomorphism A' + A'u whose kernel is Ann',. a # 0; since A'a = FAa, the F-algebra spanned by A a , we have by induction (Aa)" = 0. Hence each element of A generates a nilpotent left ideal of A', implying A' is nilpotent; by Lemma 1.3.99 (A')'' = 0. QED
This result will he generalized very far. Corollary 1.3.31. Eoery nil, tnultiplicatively closed subset of un ndimensional algrbrii over a.field F is nilpotent of index < n.
Proof.
Inject into M J F ) via the regular representation.
QED
$1.4. Identities and Central Polynomials for Matrix Algebras, and Their Applications t o Arbitrary PI-Algebras
As we shall see, identities are a basic tool to transfer information from matrix algebras to other algebras. In this section we shall obtain a good variety of identities by analyzing the standard and Capelli polynomials on matrix algebras and their subalgebras and by constructing and examining central polynomials for matrix algebras. At the end of the section it is
81.4.1
Identities and Central Polynomials
21
shown how to obtain structural information of an algebra from a few select polynomials, thereby focusing the PI-theory on algebras satisfying the identities of matrix algebras. The Amitsur-Levitzki Theorems
There are many known identities for matrix algebras, as we shall see below (and in the exercises). To start with, since M,(C) is an n2-dimensional C-algebra, we see immediately that any (n2 1)-normal polynomial is an identity of M,(C). In particular, S n 2 + 1 is an identity of M,(C), a fact recognized by Kolchin and Levi [49]; one of the early problems in PI-theory was to find the minimal t such that S , is an identity of M,(C). As noted by Razmyslov [74a], this question is answered via the Cayley-Hamilton theorem by a proof which is conceptually easy but notationally cumbersome.
+
Theorem 1.4.1 (Amitsur-Levitzki). S , , is an identity of M,(C) for every commutative ring C. Proof (Razmyslov). By Proposition 1.3.7, we need only show that S,, is an identity of the generic matrix ring Zn{ Y } , viewed as a subring of M n ( Q ( < ) ) . So consider the generic matrix Y,. By Newton’s ( - l ) k a kY;l-k = 0, where ak has the form formulas, Y;l + ( - l ) k ~ u q u t r ( Y , U ’ ) . . . t r ( Y , U ’for ) suitable q u E Q and suitable j-tuples 11 = ( u l , ..., u j ) suchthat 1 < u1 < u2 ,< ... ,< ujand u l + . . . + u j = k . Cuqutr(X;l)...tr(X;’)X;-k, with u as before, Now let $ ( X I )= X; and write A$(X1,.. ., X , ) for A 1 2 A 1 3 ~ - ~ A l(cf. n $ Definition 1.1.21). Since the trace is additive, we have 0 = A$(Yl, ..., x )
xi=l
=
c
Kl...ynll
neS,nr(ii) I1
+CC C k= 1 u
q u t r ( ~ , , . .~ .u l ) . . . ~ ( n - k + l ) Y n ( n - k + 2 ) . . .KO;
ntSym(n)
intuitively, we have “multilinearized” $. Now, replacing each yi by [YZip1, Y,,] and applying Proposition 1.2.18 we get 0 = 2”S2,(Y1,..., Y,,,) +p( Y,, . .., Y,,), where p is a sum of terms of the form
’..)
~ u ~ ~ ~ ~ 2 u 1,1 Y ,~m ( I~, , 2 ~ Yzrr(ul)-lr ~ 1 , -
...S 2 ( , 1 - k ) ( Y 2 r r ( k + I ) -
Yzn(u,)))
I , . . . , Y2n(n))
for appropriate c in Sym(n). (With a long enough sheet of paper, the reader should be able to write p out explicitly). But by Remark 1.3.4 each of these
22
[Ch. I
THE STRUCTURE OF PI-RINGS
terms is 0, so p ( Y,,. . ., Y,,,) S2,(Y,, . . ., Y2,) = 0. QED
= 0.
Consequently, 2”S,,( Y,, .. . , Y,,)
=
0, so
Razmyslov [74a] actually proved, more generally, that every identity of M , ( C ) is a “consequence” of the Cayley-Hamilton theorem. a fact to which we shall return in Chapter 2. Let us now demonstrate the sharpness of the Amitsur-Levitzki theorem by examining evaluations of polynomials (on matrices) in terms of matric units, a sound method in view of Lemma 1.1.31. =~j~j~...(Sj,.~i,ei,j,. Thekey fact is thetrivialityei,j,...ei,j, [Incidentally, it is somewhat enlightening to view this situation graphically, letting eij represent the directed edge from i to j ; a monomial in the eij has a nonzero evaluation iff the corresponding graph is a “directed path.” This notion has been useful in organizing complicated proofs, such as some early proofs of the Amitsur-Levitzki theorem. Ironically, the two clearest proofs, due to Razmyslov (presented above) and Rosset [76] (given in the exercises) do not even mention matric units, but rely on the interplay of the symmetric and standard polynomials.] The identity S,, is “minimal” for matrix algebras, as we shall see in the next theorem. The main idea is to examine the following important set of matric units. Example 1.4.2 (Thestaircase). Let {eijl1 < i,j d n3 beaset ofmatric units and r l = e l , , r2 = el,, r3 = e,,, r4 = e 2 3 ,..., r 2 n - l = en,,. Then r l r 2 - - . r z n -=I ~ ~ ~ a n d r , , r , , . . . r , , , , _ , , = O f o r a n y#~ 1 inSym(2n-1).
,,
Lemma 1.4.3. I f R contains a staircase (el e I 2 ,. . .,en,,].of matric units then R has no R-proper identity ofdegree < 2n - 1. Proof. Multilinearizing, we need only prove that R has no R-proper multilinear identity f ( X I ,. . . ,X,) of degree < 211- 1 ; rearranging indeterminates, we may assume f = CntSym(d) u x,,. . . X , , with a,, R # 0. But then f ( e l l , e I 2 ,2e2 . . . . ) = u , e , , e l , e 2 , # ~ ~O,so,fis ~ not an identity. QED
Example 1.4.4.
The algebra of upper triangular matrices R I.
=
x:liiij6n4eij has no R-proper identity of degree t. If p # 0 then Now write q = ~ ~ = , , r ~forA suitable , 0 < i d t , yields 0 = rrri. Hence 0 = r r q ; matching coefficients of since q is invertible, we get r$' = 0. This proves the claim. But R has no nonzero nil ideals. so J = 0. QED
Let us record some obvious facts about R[I], in order to exploit Theorem 1.6.12. I f A c R, write A[A] to denote the set of polynomials of R[A] whose coefficients are in A . Remark 1.6.13. (i) A[A] n B[A] = ( A n B)[A] for all subsets A . B of R. (ii) Z ( R [ A ] )= ( Z ( R ) ) [ A ](iii) . R [ I ] is a central extension of R .
We can now extend our main results from semiprimitive rings to rings with no nil ideals. Write Ad R to denote A is an ideal of R . Theorem 1.6.14. !f NiI(R) = 0 and R is a PI-algebra, then eoery nonzero ideul oj'R intersects Z (R ) nontrivially. Proof. Suppose 0 # Ad R . Then A [ I ] a R[A], which is a semiprimitive PI-algebra, so by 'Theorem 1.5.33,
0 # A [ I ] n Z(R[E.])= A[A] n Z ( R ) [ A ] = ( A n Z ( R ) ) [ I ] ,
implying A n Z ( R ) # 0. QED Theorem 1.6.15. Proof.
!f Nil(R) = 0, then R is admissible.
First inject R into R[A] and then apply Theorem 1.6.4. QED
Theorem 1.6.16.
!fNil(R)
of degree d, then R has PI-class
Proof.
=0
and i f R satisfies a polynomial identity
< [di2].
Immediate from Theorem 1.6.15 and Theorem 1.5.31.
Let us apply Theorem 1.6.15 directly and cleanly. Theorem 1.6.17. Suppose NiI(R) = 0 and R satisfies a polj7nomial identity of degret. < d. I f A is a nil, multiplicatioe subset oj' R , then A is nilpotent ofindex 6 [ d / 2 ] . Proof. Let n = [ d / 2 ] . Injecting R into a direct product of matrix algebras over fields M, (Fk,), YE^, with each k;. < n, we see that each component A,. of A is nil, and thus by Proposition 1.3.30 An = 0. Thus A " = O . QED
Corollary 1.6.18.
!f R is a PI-algebra, then Nil(R) = X(ni1 left ideals
of R). Proof.
Let A
= E(ni1 left
ideals of R ) . Clearly A is an ideal containing
$1.6.1
Injections of Algebras
45
Nil(R). Suppose R satisfies a polynomial identity of degree d d . If L is a nil left ideal of R, then (L+Nil(R))/Nil(R) is nilpotent of index d [d/2], so it follows easily from Remark 1.3.26 that A/Nil(R) is nil. Hence A E Nil(R). Q E D Incidentally, without the PI-assumption Corollary 1.6.18 is a n important open question, due to Koethe; its structural significance is seen in our next injection. The Sum of Nil Left Ideals of Bounded Degree ( N ( R ) )
A nil subset A of R has bounded degree d t if a' = 0 for all a in A . Definition 1.6.19.
N(R) = { a e R J R ais nil of bounded degree}.
Proposition 1.6.20. (i) Suppose Ra has bounded degree d t. Then for all r, rl in R, (rar,)'" = 0. (ii) N(R) is an ideal. (iii) N ( R ) = {a E R(aR is nil of bounded degree}. Proof. (i) (rar1Y+' = ra(rlra)'rl = 0. (ii) Immediate from (i). (iii) Immediate from (i), setting r = 1. Q E D
The next injection is due to Amitsur [55a], and used very fruitfully by Amitsur [71a] and Martindale [72a]. Theorem 1.6.21. Suppose R is an algebra and let be an index set on a 1 : 1 correspondence with R unless R isfinite, in which case we take r = Z'. Let R, = R for each y, let R' = R,, and write ( r y for ) the element of R' whose y-component is rv for all y in r. (i) There is an injection $: R -+ R', given by r + (r,) with each r).= r ; hence R and R' are equivalent. (ii) Zdentifjing R with $ ( R ) G R', we have Nil(R') n R = N(R), inducing an injection R/N(R) --t R'/Nil(R').
nyEr
Proof. (i) Obviously $ is a n injection, proving R < R'. But R' < R by Remark 1.1.19, so R and R' are equivalent. (ii) Let r E R n Nil(R'), and let x = (r,), where { r y1 y E r}= R. Since xr is nilpotent, it follows that Rr is nil of bounded degree, so r E N ( R ) , proving R n Nil(R') G N(R). Conversely, supposer E N(R),sothat Rrisnil ofsome boundeddegreet. Eachcomponent of R'r is in Rr, so R'r is also nil of bounded degree, implying r E N ( R ' ) . Thus N(R) G R n N(R') E R n Nil(R') E N(R), proving N(R) = R n Nil(R'). QED Theorem 1.6.22. R/N(R) is admissible,,for any algebra R. Proof.
Follows from Theorem 1.6.15 and Theorem 1.6.21. Q E D
To my knowledge, Theorem 1.6.22 is the most general positive solution to the injection problem.
46
[Ch. 1
THE STRUCTURE OF PI-RINGS
Corollary 1.6.23. I j R satisjes a polynomial identity of degree d and A is a nil, multiplicative subset of R, then AtdiZ1E N ( R ) . Proof.
Apply Theorem 1.6.22 to Theorem 1.6.17. QED
Corollary 1.6.23 will be further improved shortly. Of course, these last results would be enhanced by more knowledge of N ( R ) . To this end, we need a famous result of Levitzki. Lemma 1.6.24. If A is a nonzero nil algebra (without 1) of bounded degree, theri A ha., a nonzero nilpotent ideal. Proof. Suppose A is nil of bounded degree t . Then a' so "multilinearizing", we have
(i)
xltS!.m(f,a.ql
... a n ,
=0
for all u in A,
=0
for all a,, ..., a, in A. (See $1.1 1 for the presentation of this statement in the context ofidentities inalgebras without 1.) Weinduct on theminimal tsuch that (i)holds.Takinga # Oin Aandimaximalsuchthata' # 0,weseethat (a')' = 0. Let .Y = ai.Now let B = s A ++.Y, a right ideal of A, and B = B/Ann, B. Clearly .YB= 0; thus, for all b , , . .., b , - ] in B and for b, = .Y, (i) yields ~ n E S y m ( , - l , b n l ~ - ~ b= , c 0,implying~,.s,,(,-,,6,~,~..b,,,-,, r-l,s = 0.Bisalso nil of bounded degree < t so by induction B has a nonzero nilpotent ideal I/Ann,B. Then for some m I" L Ann,B, so I"B = 0, implying (AIB)"= 0. Since A I B Q A Be are done unless A I B = 0; in this case 0 # I B a A and (IB)' = 0. QED For any R , either N ( R ) contains a nonzero Proposition 1.6.25. nilpotent ideal or ,Y(R)= 0. Proof. Suppose N ( R ) # 0. Then take a nil left ideal L # 0 of bounded degree. Now A =: LIAnnl, L is also nil of bounded degree and thus has a nilpotent ideal liAnn; L # 0; i.e., LI # 0 and I" E Ann', L for some nl, so LI" = 0. Thus (LIR)" E L(IRL)"-'IR E Ll"-'IR = 0, and 0 # L I R d R . QED
(This notion of passing to LIAnnLLis a very powerful inductive tool in the general theory of rings without 1 and will be used extensively in the theory of generalized identities.) Semiprime PI-Rings
R is called serniprime if A' # 0 whenever 0 # Ad R . Note that R is semiprime iff R has no nonzero nilpotent ideals. Corollary 1.6.26.
I f ' R is semiprime. then N ( R ) = 0.
$1 6.1 Proof.
47
Injections of Algebras Immediate from Proposition 1.6.25. Q E D
Theorem 1.6.27. I f R is semiprime and satisfies a polynomial identity of degree d , then Nil(R) = 0, R has PI-class < [ d / 2 ] , every ideal qf R
intersects Z(R) nontrivially, and R is admissible. Thus R
< Mldi2,(Z[ k , so, for each xESym(d)-{I}, A,, ’ “ A n d E RBSR. But J ( A l , . . . , A d ) = ( ] , SO RBbR 2 A1”’Ad=(BL-lR)d&. If 11 6 t - 1 , thus (RBb-’R)d-‘ is nilpotent, implying RBb-’R is nilpotent, contrary t o the minimality o f t .Thus n > t - 1, i.e., t < n, as desired. E L,(R), so NiI(R) G L2(R). Since L,(R) is nil we (ii) By (i), (Nil(R))‘””21 get NiI(R) = L,(R). Q E D
Jacobson [64B, p. 2331 gives a n example of Amitsur, of a PI-ring R for which NiI(R)”-’ $ L,(R). Theorem 1.6.36(i) is improved in Exercise 6.
<MJQ)
Arnitsur’s Method of Obtaining Canonical Identities Lemma 1.6.37. With the notation as in Theorem 1.6.21, suppose f is an identity cfR’/Nil(R‘). Then some power f k o f f is ( i n identit!!(J’R. Proof. Write f = f ( X , , ..., X,). Note that there is a 1 : 1 correspondence from R(d’into r, so we label the corresponding elements of r as ( I , , . . ., r d ) for all ri in R. Let ii be the element of R’ whose ( r l ...., i d ) component is r , . By assumption f ( ? l , . . . , ? d ) ~ N i l ( R ’ ) , so for some k f (iI,. . .,id)k = 0. Matching the ( I , , . . . ,r,)-components yields f ( r , , . .. ,rd)k = 0 for all r,, . . . ,rd in R, so f ( X , , . . .,X d ) kis an identity of R. Q E D
$1.6.1
Injections of Algebras
49
Proposition 1.6.38 (Amitsur's method). Suppose % is a class of ulgebras such that for euch R i n % euery direct power of R is in (6 arid R/Nil(R) satisjies an identity .f Then we huve the.followirig conclusions: (i) Eoery algebra in % satisfies a power of ,A i t . , a suitable , f k (where k muy depend or1 the algebra). (ii) I f ; moreover, all direct products of'algebras in % lie in %, theri s o m e f k is a11identity of all algebras in (6.
(i) is a special case of Lemma 1.6.37. Suppose on the contrary that for each k in Z-' we have R , such that (ii) f k is not an identity of R,. Then let R = R,. By (i) some f k is an identity of R, thus of R,, contrary to assumption. QED Proof.
nk..n
Amitsur's method is extremely useful in verifying that a class of algebras is PI, for it reduces the problem to algebras with nilradical 0. We shall make use of this procedure many times, and illustrate it now in connection with a most important decomposition. Definition 1.6.39. An ideal B of R is prime (resp. semiprime) if RIB is a prime (resp. semiprime) algebra. Remark 1.6.40. The following conditions are equivalent for B a R : (i) B is prime (resp. semiprime); (ii) if B c Aid R, i = 1,2, then A , A , $ B (resp. Af $ B ) ; (iii) if a,, a , E R- B then a , Ra, $2 B (resp. a , R a , $ B ) .
Levitzki proved that every semiprime algebra is a subdirect product of prime algebras. We use a slightly easier result which is more to the point, in view of Theorem 1.6.27. Remark 1.6.41. Suppose S is a multiplicatively closed subset of R not containing 0. Then by Zorn's lemma there is an ideal P of R maximal with respect to empty intersection with S, and any such P is prime. (Proof: If P c A , and P c A , then there exist elements si in A in S , i = 1,2, SO s l s zE A , A , n S , implying A , A , $ P.) Proposition 1.6.42. Suppose Nil(R) = 0. Then R is a subdirect product of prime algebras, each kai?itig riilradical 0. Proof. For each nonnilpotent element r in R , let P , be an ideal of R, by Remark 1.6.41 P, is prime. maximal with respect to P, n {r'li > 1 = 0; We claim that Nil(R/P,) = 0. Indeed, otherwise we have A / P , = Nil(R/P,) # 0 for some A d R. But P, c A , so some riE A ; since A / P , is nil, some rik E P,, contrary to construction of P,. Thus the claim is proved. Now ( I { P , l r is not nilpotent) is an ideal of R missing all nonnilpotent elements of R, and thus is a nil ideal of R and therefore 0. Thus, R is a subdirect product of the RIP,. QED
50
[Ch. 1
THE STRUCTURE OF PI-RINGS
Lemma 1.6.43. Suppose R is prime and B,\y E r) is ajaniily of ideals ?f’R with intersection 0. Then.fhr m y nonzero subset A sf‘ R. [B,IA $ B Y ) = 0.
0
Proof. Let A , = n { B y l A G B Y ) and I = ( 7 { B , , I A s i B y } . Then A1 G A , I E ,4, n I = 0, implying I = 0. QED
Proposition 1.6.44. I f R is prime and semiprimitive and 0 # A G R , then n{primitivr.ideals ofR not containing A } = 0. Proof.
Special case of Lemma 1.6.43. QED
These useful general structure results can be put together to obtain a nice theorem of Amitsur [71a]; actually we present a slightly weaker form here, leaving the “best” version for the exercises. Proposition 1.6.45. I f R is prime and NiI(R) = 0, and proper identit-v of degree d d, then R has PI-class 6 [ d / 2 ] .
iff
is an R-
Proof. We may assume that f is multilinear (by multilinearizing). i.e., aR # 0 for some coefficient a off: Thus aR[I] # 0. But R [ I ] is prime and
semiprimitive, so by Proposition 1.6.44 nrprimitive ideals of R[I] not containing a R [ A ] } = 0. Thus R[A] is a subdirect product of primitive images on whichfis a proper identity, which must therefore have PI-class 6 [ d / 2 ] by Kaplansky’s theorem. Thus R[A] has PI-class 6 [ d / 2 ] , so R also has PI-class 6 [ d / 2 ] . QED (Amitsur). Suppose ,/’is an identity qf’ R which is Theorem 1.6.46 R-proper .for every nonzero homomorphic image R of’ R, and deg(,j) = d . Then R satisfies 2 1 ,for some k E L’. I f R is semiprime, then s Z [ d l 2 ] is an identity OfR. Proof. Case I. NiI(R) = 0. Then by Proposition 1.6.42 we may assume also that R is prime, so we are done by Proposition 1.6.45 and the Amitsur-Levitzki theorem. Case 11. R is arbitrary. Coefficientsofj’generatean ideal Icontaining I (for otherwise ,f’ is not R/I-proper, contrary to assumption). Thus we may apply Amitsur’s method to Case I, to conclude some S ; l d i 2 l is an identity of R . Finally, note that if R is semiprime then we have just shown R is a PIring, so NiI(R) = 0 and R satisfies s , [ d , , I by Case I. QED
(Note that applying Proposition 1.6.38(ii)we can show k depends nor on R , but only o n Also, we could have used other identities in place of S2,d,2,.) Amitsur’s theorem shows that our definition of polynomial identity is completely general.
$1.7.1
Central Localization of PI-Algebras
51
$1.7. Central Localization of PI-Algebras
In view of Theorem 1.4.26 we are most interested in a ring R for which C 2 n L + 1is an identity and g n is central, with 1 e g n ( R ) R (or even better, 1 E g , , ( R ) + ) . Now the first two conditions say that R satisfies the identities C2,,2+land [X4nZ.k2,gn],which in particular is true if R is semiprime of PIclass n, so the only hindrance to a very useful structure theory is the third condition 1 c g n ( R ) R .In this section we shall see how we can transform R into a ring R , with 1 eg,,(R,), instantly proving many useful theorems. The procedure is called central localization and is defined as follows: Write Z = Z ( R ) and let S be a submonoid of Z (i.e., S is a multiplicative set with 1). The Cartesian product R x S = { ( r , s)lr E R, s E S} has a relation -, given by ( r l , s l ) ( r 2 , s 2 )iff ( r l s 2 - r 2 s 1 ) s= 0 for some s in S ; is easily seen to be an equivalence. Let rs-' denote the equivalence class of ( r , s ) and let R , be the set of these equivalence classes. R, can be given the following operations for all a in 4, ri in R, si in S : rls;' +r2s;' = (r1s2+~zsl)(s1s2)-'; (rls;')(r2s;') = ( r l r 2 ) ( s l s 2 ) -;' a ( r l s ; ' ) = (c(rl)s; The reader is invited to check that these operations are well defined and make R , into a n algebra; the verifications are tedious but not much more difficult than the usual construction of Q from Z.Note that the multiand the "zero" element is 0.1 plicative unit of R , is 1.1
-
-
'.
-'.
-',
Remark 1.7.1. There is a canonical homomorphism vs: R -+ R , given . sins, b y r - + r l - ' , a n d k e r v , = { r ~ R ~ r s = O f o r s o m e s i n S }Forevery s l - ' has the inverse 1s-l. Proposition 1.7.2. Gizien an algebra R' and a homomorphism + R' such that v(s) is invertible i n R' for all s in S, we have a urzique hornonzorphism $ " : R s -+ R' such that $ ) ( r l - ' ) = v(r) for all r i n R. Then ker$, = { r s - ' l r E k e r v } . v: R
Any such homomorphism must satisfy $ v ( r s - l ) $ v ( s l- ' ) = $,,(rl-') = v(r), implying i+hY(rs-') = v(r)v(s)-', and this map is indeed a homomorphism. Moreover, ker$, = {rs-'Iv(r)v(s)-' = 0} = {rs-'lv(r) = O } . QED Proof.
(This proposition is of the utmost importance in utilizing localization in later sections.) Corollary 1.7.3. I f S E S', then there is a canonical homomorphism $: R , + R,. given by r s rs- I . II; moreover, vs(s) is invertible for all s in S', then $ is an isomorphism. -+
Proof.
Take v = v s f , and $
=
$vs,
in Proposition 1.7.2. Under the
52
[Ch. 1
THE STRUCTCKE O F PI-KINGS
additional hypothesis we construct 1.7.2. QED
I,-
by taking
1'
= v, in Proposition
We shall say an element z of Z is regular if Ann,z = 0; S is regular if every element of S is regular. Call R torsion.frer over Z if Z - {O) is regular (implying, in particular. that Z is a domain). Recall from Remark 1.6.29 that every prime ring is torsion free over its center. If S is regular then ker(v,) = 0, and so we shall view R c R,s. Let us now look at Z(R,s).T o do this, we adopt the convention that for any subset A c R , Asdenotes ( a s - l l a ~A , s € S ) . Proposition 1.7.4.
Z,
c Z ( R s ) ,with equaliry holding i f S is regulur
Proof. Suppose zls;' EZ, for z l in Z , s1 in S. For all rs-' in R,. we have ( z l s ; ' ) ( r s - ' ) = ( z I r ) ( s l s ) - '= ( r z 1 ) ( s s l ) - ' = ( r s - ' ) ( z , s ; ' ) . proving 2,sEZ(R,). Now suppose S is regular and suppose rls;'€Z(Rs). For all r in R O = [ r 1 - ' , r l s ; ' ] = [ r , r , ] s ; ' , so [ r , r l ] = 0, proving rl E Z ; thus r l s; E Z,. OED
Proposition 1.7.5. I f R is prime (resp. serniprime with S regular), then R, is prime (resp. semiprime). Proof. If r , s ; ' R , r 2 s i 1 = 0 then r l R r 2 = 0, so r l = 0 or rz = 0. (The proof for semiprime is analogous, taking r l = r2 and s1 = s2.) QED l s i and If rls; . . . , rks; are elements of R , then, letting s = ".sk, I < j d k , we have r j s i = x j s - ', 1 d j d k . Thus -yj = r j s l " . s j we may always assume that a given finite set of elements of Rs hurr the same denominator. This observation will be used without further ado.
nf=
',
+
The Algebra of Central Quotients
Definition 1.7.6. For S = {all regular elements of Z ) , R , is called the algebra qfcrntral quotients o f R , and is written Qz(R).
Suppose S is given. Write R1- ' for ( r l -'Ir E R } = \qs(R),a homomorphic image of R. Lemma 1.7.7. Proof.
l [ f i s a multilinrar polynomial, then ,f(R,) = ,f (R),.
Immediate from Remark 1.1.30. QED
Proposition 1.7.8. to R s . Proof.
R,s
are algebraicaily C-independent ,for some u < k . Then some subset of (.Y~,. . .,.sk} which coiitairis (.sl,. . .,xu: is a transcendeiice basis of H (oL1er C). IIx
Proof. Expand (s,, . . . ,s,) to a maximal algebraically C-independent subset S of i.s,, . . . ,x k ) ,and let C‘ be the C-subalgebra of H generated by S. Any element x of H is algebraic over C[x,, . . . ,xk], which in turn by Remark 1.10.12 is algebraic over C’; thus Y is algebraic over C‘. Therefore S is a transcendence basis of H. QED
Theorem 1.10.14.If trdeg(H/C) = t < cx;, then derice basis of H has exactly t elements.
every
tvunscen-
Proof. Suppose {x1,.. ., x,] and [.s’,, ..., .Y;] are transcendence bases of H (over C ) ; using Lemma 1.10.13 we can conclude with a transfer argument analogous to Herstein [64B, lemma 4.61 to find a transcendence basis of < t elements which contains x i , . . . ,xi,#, proving m < t ; thus m = t . QED
Remark 1 .I0.15. If C is a commutative algebra over a field F and A is a proper ideal of C, then every regular element c E A is riot algebraic over F. (Indeed, if c were algebraic then c would be invertible, contrary to 1 $ A . ) Remark 1.10.1 6. Suppose R is an algebra over a valuation ring C and r e R is algebraic of degree I I over C. Then r satisfies an equation C;=, ciri = 0 with some c j = 1. (Indeed, write C;= f i r i = 0; some c j divides all the ci,so we divide through by c j . ) Consequently in any homomorphic image R of R, I. is algebraic over C.
Note that by Theorem 1.7.9, for any prime PI-ring R with center Z , Q , ( R ) is algebraic over Q z ( Z ) ,implying R is algebraic over Z. We shall use this fact to apply valuation rings to transcendence degree. Lemma 1.10.17 (Procesi). Suppose R is a prime PI-algebra over a ,field F, with t = trdeg(Z(R)/F) < w .Giver1 arbitrary P ESpec(R), write ,for the canonical image in RIP. The11trdeg(Z(fT)/F) < t .
84
[Ch. 1
THE STRUCTURE OF PI-RINGS
Proof. Write Z = Z ( R ) , S = Z-{O), and F, = Z,. We work in R,, which is algebraic over F,.By Theorem 1.10.10 there is a valuation ring V of F , such that R‘ = R V has a prime ideal P’ lying over P . Let = R’,’P’; then we can view R 2 ( R + P ’ ) / P ’ G R‘. By Remark 1.10.15 Z has a transcendence basis (over F) containing an element of Z n P ; this is also a transcendence basis of V over F, implying trdeg(V/F) < t . Clearly R‘ is algebraic over I., so is algebraic over P by Remark 1.10.16;thus t > trdeg(Z(R)P/F) > trdeg(Z(R)/F),identifying F with F. QED
R.
R.
Theorem 1.10.18 (Procesi). Suppose R is a p r i m PI-algebru o w a field F and trdeg(Z(R)/F)= t < co. Then rank(R) d t .
Suppose 0 c f c . .. c p k is a chain in Spec(R), and let W = R/P,. By Lemma 1.10.17 trdeg(Z(R)/F)6 t - 1, so by induction rank(R) Q t - 1 . But O C : ~ ~ ~ C * *in . CSpec(R), P, so k-1 G t - 1 , implying k Proof.
6 t . QED
Ranks of Affine PI-Rings
If R, E R and x , , . . ., X,ER write R,{x,, . . ., ?Ik) to denote the subring of R generated by R , and x,, . . . ,x k . Definition 1.10.19. R is a finitely generated extension of R, if R ..., x,} for suitable xi in C , ( R , ) , 1 6 i 6 k, for some k ; in case R , is a field F we say R is a , n e (over F). = R,{x,,
We aim towards a major theorem of Procesi (some of which was also discovered by Markov [73]), characterizing the rank of a prime affine algebra in terms of the transcendence degree. (Finitely generated extensions are discussed in greater detail later in Chapters 4 and 5.)
.
Proposition 1.I 0.20 (Procesi). Suppose R = R, { x,, . . . .xk} [with xi in C , ( R , ) ] is prime of’ PI-class n, and let F,,F be the respective fields of fractions of’Z(R, I,Z(R). Then F is generated as ajield over F, by 6 kn2 +n6 elements. Proof. Localizing at Z(R,)-{0}, assume R , is simple with center F,. Let T = Q,(R), 4 = C,(Rl). T is simple and [ T : F ] = n z , so A is an Falgebra of some dimension t d I?; moreover, by Corollary 1.8.26 A is aijaj,1 < i < k , and simple. Let a,, . . . , a t be an F-basis of A ; write xi = a.a. , = xt,= aijuau,1 Q i, j G t , for suitable aij, aijuin F. Let K be the subfield of F generated (over F,) by the ( k t + r 3 ) elements {aijI 1 < i 6 k, I 6 j < t } u {aij,,I 1 G i , j , u < t } . We shall conclude by K a j , a K-algebra of dimension t proving K = F. Well, let B =
x:=,
,
&,
$1.10.1
Valuation Rings
85
(because the aj are K-independent). BF = A , so B is prime (and thus simple) of PI-class Hence Z ( B )= K . Now each s,= cliiaJ€B , so R = BR, 2 B O F R , , , implying Z ( R ) c Z(BR1)= Z ( B ) = K . Therefore F = K . QED
&.
Now we do the "rank 0" case, i.e. when R is simple. Zf R is simple and finitely generated over C c Z(R), trdeg(Z(R)/C) = 0 (i.e.,Z ( R )is algebraic over C ) .
Lemma 1.10.21. rlieii
Proof. Let Z = Z(R), a field, and let F be the field of fractions of C. Clearly F c Z, so R is affine over F , and we may assume C = F. By Proposition 1.10.20 Z is generated over F by a finite set {z,,. . . ,z k } , which we can shrink to a transcendence basis { z , , . . . ,z,} of Z over F for suitable t , where the zi have been reordered if necessary. Let H = F [ z , , . . .,ZJ E Z. For all i, 1 d i d k , each zi is algebraic over H , and thus integral over H [ p ; ' ] for some pi # 0 in H . Let p = p1 " ' p k # 0, and put S = { p i l i 3 0 ) . Each zi is integral over H,, implying Z is integral over H,; thus rank(H,) = rank(Z) = 0, so H , is a field.
Since zl,. . . ,z, are algebraically independent, there is an algebraic extension F , of F , with elements p 1 ,. . . , p l in F , and a homomorphism $: H -+ F , given by $ ( z i ) = pi, 1 < i < t , with $ ( p ) # 0. Then tc/ extends to a homomorphism t,hs: H , -+ F , ; since H , is a field, ker($,) = 0, implying H , is algebraic over F . Thus t = 0, so Z is algebraic over F. QED Remark 1 .I 0.22. If x,, . . . , x, are algebraically C-independent elements of H then { p ( x , , . . .,x,)IO # p€CIA1,.. . ,A,]} is a submonoid of H not containing 0. Theorem 1.10.23. Let R be a prime, afine PI-algebra over afield F , atid let t = tr deg(Z(R)/F). Theti t < co, rank(R) = t , and, in fact, there is a cliain 0 c P , c . * * c PI in SpecJR).
Proof. Let Z = Z(R). By Proposition 1.10.20 Q z ( Z ) is a finitely generated field over F , implying t < co. Hence by Theorem 1.10.18 rank(R)
< t.
We shall finish by proving inductively on rank(R) that there is a chain 0 c P, c ... c P, in Spec,(R). There is nothing to prove if t = 0, so assume t 3 1. Note that 0 # g , ( R ) ' d Z , so by Remark 1.10.15 we have a . S transcendence basis c , , ..., c, of Z over F , with c , ~ g , ( R ) + Let = { p ( c , ,. . . ,c,-,)10 # p e F [ A , , . . . ,2,- ,I}. Then R , is finitely generated over F, c R,, but q 1 - l is not algebraic over F,, so by Lemma 1.10.21 R, is and not simple. Taking a nonzero prime ideal P of R with P n S = 0. letting denote the image in RIP, we see ... ,c,-1 are algebraically F~
, 2. Clearly rank(R) = rank(R/Nil(R)), so we may assume NiI(R) = 0. Then R has some PI-class n d [d/2], and one sees easily that R 6 hf,(F[ i >, i,. (The underlying motivation here is that 0 is divisible by everything, so the "smaller" something is, the more elements divide it ; this idea permeates all the proofs about completions.) Defining operations componentwise, we get a ring Cauchy(C) of all the Cauchy sequences of C. Define Cauchy,(C) all j 3 i) .
=
[(ci)ECauchy(C)/foreach c there is some i such that c I c j for
Cauchy,(C) is called the set of null sequences of C,and is obviously an ideal of C. Define the completion of C to be Cauchy(C)/Cauchy,(C). Remark 1.10.46. In a valuation ring C , if c,c21c3c4,then either c1 Ic3 or c21c4. (Otherwise el = c 3 p , and c2 = c4p2 for suitable p L .p 2 in Jac(C), implying c 1 c 2 = ~ ' 3 ~ 4 ~ which 1 ~ 2 ,is inconsistent with clc21c3cs.)
5 1.10.1
Valuation Rings
91
Lemma 1 .I 0.47. Suppose (ci) is LI nonnull Cauchy sequence of a iw'uation ring C. Then, ,for some c E C , there esists i, such tbat c [ c j,fi)r ull j > i,. Proof. By definition, for some C E C we have an infinite set o f j in H + for which cXcj. But, for some i,, we have cI ( c j - c i ) for all j > i 2 i,; choosing i, large enough, we may assume c[ci,,. Hence c/cj for all j > i,. Q ED
Theorem 1.10.48. Let C' he the completion of C. There is a canonical injection C -+ C', given by c + (c,c, c; . . .). C' is a valuation ring lying above C. C' is its own completion. Proof. Straightforward (using Remark 1.10.46 and Lemma 1.10.47 repeatedly). Q E D
Remark 1.10.49. If C is a valuation ring of an algebraically closed field, then its completion is also a valuation ring of an algebraically closed field. (Just take an algebraic equation and view it componentwise.)
Let us see how this ties in with rank 1 valuation rings. Lemma 1.10.50. Suppose C is a valuation ring, and P = Jac(C). Rank(C) = 1 g j o r all x E P, y E C - (O), we have ylx",for suitable 11. Proof. (3)We are done (with n = 1 ) unless P. In this case P/Cy is the only prime ideal of C/Cr and is thus nil, proving X"E C y for some n. (+ Suppose 0 # B ~ s p e c ( C ) For . any X E P and 0 # J'E B , we have some ~ " E Bso, ~ E Bthus ; P c B , proving P = B. QED
Proposition 1.10.51. C has rank 1 i f t h e completion o f C has rank 1 . Proof.
Straightforward, using the condition of Lemma 1.10.50. Q E D Valuation Rings and the Integral Closure
Here are some pretty results of Nagata on the integral closure. Proposition 1.10.52. If'C is u doniein contained in a valuation ring V of'ajeld F, then the ititegrcrl closure of'C in F is contained in V . (111particular, every ~aluatioriring V is norniul.)
Suppose X E F is integral over C. We claim X E V. Otherwise and x is integral over V, so by Remark 1.9.42 X E V, as desired. Q E D Proof.
x-'
EV
Theorem 1 .I 0.53. Suppose C is u quasi-local ring contained in a field F . Then n(ualuation rings o f F lying above C } is the integral closure o f C in F.
92
[Ch. 1
THE STRUCTLIRE OF PI-RINGS
Proof. In view of Proposition 1.10.52,we need only show that for every x in F not integral over C there is some valuation ring V of F nor containing s. i.e., . Y - ' EJac(V). Well, let P = Jac(C) and C' = C[z-'] c_ F ; P C ' + x - ' C ' # C'. for otherwise 1 = r i s - i for ~EZ!'. ~ , , E Pr.i E C , so 1 - p o = ~ ~ = l r , x - =i , ~ ~( 1~-~~)-'r~x'-',contrarytoxnc)tintegral ~ ' over C. Now by Theorem 1.10.10 there is a valuation ring C, 2 C' with .u-'C'+PC' c Jac(C,); get V by passing to the integral closure o f C, in F and applying Theorem 1.10.10again. QED
po+z:=, z:=,
Theorem 1.10.54. Suppose C is (1 caluation ring of a ,fitdrl F, F' is un algebraic estensioti of' F , and C' is the integral closure of'C i n F . For any tialuation ring I/ 01 F' lyirig ubove C, we have V = C;,, where P = Jac( V ) r,,C'. Proof.
By Proposition 1.10.52 C' E V. Let S = C'-P. Then Ck E V,
= V, so we need to show that every element X E V is in C$. Write ~ ~ = , , c i = 0 for some t with c, # 0, c , E C . By Remark 1.10.16 we may assume some c j = 1 ; taking j as large as possible, we may assume c .i €.Jac(C) for all i > j. Now let a = xjIA c i s i + - j and b = Ltxj+, c i s ' - J . Then b E Jac(C)V
c Jac( V ) ,and ux + 1 + b = 0. For any valuation ring V ' of F' lying above C,we havexEV'orx-'EV'.If.uEV'then b E V ' a n d u = - ( l + b ) x s V ' ; if x - ' E I/' then a6 V' and b = - ( a x - ' + 1)E V'. Hence by Theorem 1 A0.53 u, bEC'and b E C ' n J a c ( V ) = P , s o .Y = - a ( l + b ) - ' E C . ~ QED Algebras over Valuation Rings
We shall now study algebras over valuation rings, as the last preliminary to the big Bergmiin-Small theorem. These results are from Bergman-Small [751. Remark 1.10.55. If R is an algebra over a valuation ring C and A d C, then every element of A R has the form ar for suitable a in A , r in R . (For a typical element airi, note that for some j we have a: in C such that a I. = a .Ja ! i d i d f ; t h e n ~ : : = , a i r i = i i j x ~ = l a I r i . )
xi=
19
Definition 1.10.56.
Suppose C is a commutative domain. S = IS almost n-dimensional (over C) if [ R , : C,] = n ; i.e., n is the maximal number of C-independent elements of R . In this case, write dim(R .C) = n.
C - (O}, and R is a C-algebra. R
Suppose C is a valuation ring, R is a torsion-free Lemma 1.10.57. C-algebra, P€Spec(C). and P = C n P R . Then, ident$ving C / P with (C P R ) / P R c R 'PR, we hatie dim(R/PR ; C / P ) < dim(R ;C).
+
Proof.
Let - denote the canonical image in R
=
R / P R . We show for
xi
$1.10.1
Valuation Rings
93
any C-dependent elements r , , . . ., r, of R that f l , . .., f , are C-dependent. ciri = 0. Some cj divides all ci, 1 < i < n ; dividing out by Well, write cj, we may assume c j = 1, so C:= FiYi = 0 with Cj = 1. Hence dim(R ;C) < dim(R :C). QED
x:=,
,
Lemma 1.10.58. I f R is u torsionTfree algebra over a valuation ring C and P E: Spec(C) such that C n PR = P , then R/PR is torsionTfree over C/P. Proof. Let - be the canonical image in R/PR. Suppose crE PR for some c in C - P , r in R. By Remark 1.10.55 cr = p x for some p in P , x in R. Moreover, since c $ P , we have p = cc, for some c1 in C (and thus in P ) , so r = c , x e P R . In other words, if Z= 0 and C # 0, then f = 0. QED
If R is a C-algebra and PeSpec(C), write P-Spec(R) for (BeSpec(R)IB lies over P } . We shall have repeated occasion to use Proposition 1.10.59 in the subsequent proofs. (Note that over a finite index set, direct product and direct sum are the same.) Proposition 1.10.59. Suppose C is a valuation ring, and R is a torsion-free C-algebra with dim(R ;C ) = n. Suppose PeSpec(C) is arbitrary. (i) INC holds,fbr the evtension R of’C. (ii) !fB’~Spec(R) such that P E B‘, then there exists B ~ P - s p e c ( Rwith ) B E B‘. (This is called “going down.”) (iii) xBsEP-Spec(R,dim(R/B ; C / P ) < n.
Proof. (i) Weshow incomparability in P-Spec(R),and aredone unless PSpec(R)is nonempty, whereby C n P R = P. Write- for the image in R/PR, and let S = - {O}. Then B -+ B, gives a natural set injection from P-Spec(R) into Spec(B,s).But csis a field,over which R,s is d n-dimensional by Lemma 1.10.57, so R,/Nil(R,)issemi-simpleby Corollary 1.7.35.Henceall primeidealsofR,are maximal, and thus incomparable, showing P-Spec(R) is incomparable. (ii) Continuing the logic of (i), we see that P-Spec(R) has only a finite number of primes B , , ..., B , ; thus by Corollary 1.3.31 0 = Nil(R,v)” 3 ((B,),... (Bt),Y, so ( B , ... B,)” c B’. Hence some Bi c B’. (iii)
c
n 2 dim(R ; 2
e)= [ R , : c,] >, [R,/NiI(R,)
1
[ ( R s / & ) : C s ]=
dim(R/B;C) BE P - S p e c ( R )
BEP-Spec(R)
2
: C,]
dim(R/B ;C / P ) . QED
Even if [R : C ] is finite and C C E R ; cf. Exercise 4. Theorem 1.10.60.
=
Z(R), we d o not necessarily have G U for
Suppose R is torsion ,free over
u
uuluation ring C
94
[Ch. 1
THE STRUCTURE OF PI-RINGS
and dim(R ;C) = n. Thetijbr unj' gii-en set .& of'incompurable prime ideals of' R, we haue .dim(RIB :C / ( Bn C)) < n.
xBe
Proof. If an infinite sum > 11 then some finite subsum > 17, so it suffices to consider the case .H is finite. Let .P= ( B n CIB E .&j, and let m be the number of members of ..P (in Spec(C)). If m = 1 then we are done by Proposition 1.10.59; in general the proof is by induction on i n . Let Po be a maximal member of 9.let .Yl = . P - ( P o ) , and let PI be a maximal member of .PI.By Corollary 1.10.3 PI c Po. Thus using Proposition 1.10.59(ii) and putting .d= -8n (Po-Spec(R)u P ,-Spec(R)), we can define a map p : d -, P I-Spec(R)such that p ( B ) E B for all B E . P / . Write d(B) for dim(R/B ;C/(C n B ) ) . and p ( . d ) for ( , p ( B ) I BE .d]. By passing to RIB' and applying Proposition 1.10.59(iii), we have I,,(,,,= ,,.d ( B ) < &B') for all B' E P l-Spec(R). Thus
c d(B) c d ( B )+ =
nE
8
BE .d
BE+- d
d(B)
r each p ~ S p e c ( W )that every chain of P-Spec(R) has length < ( k - 1)n2+1. Then we may assume R is prime and I' = 0, so W is prime. Localizing at Z ( W ) - lo}, we may assume Z ( M ' ) is a field. Then W is simple, so R 2 W O FF ( r , , .. . ,r k : . Pass to the affine case. using Exercise 1.8.1.) 2. If C is a principal ideal domain and V is a valuation ring of the field of fractions of C. then V = C, for somc P ~ s p e c ( C )hence ; V has rank < 1.
3. Construct a valuation ring with valuation group Z,x h, (cf. Bourbaki [72B, 46.3.41). 4. (Bergman -Small [ 751) A counterexample to GU. Let C be as in Exercise 3. and view C in its field of fractions F. Take u, h in C, such that every proper principal ideal of C has the form a"C or o"h"(' for suitable m E Z + , EL. Let R = C e , , + C [ a - ' ] u , , + b C [ a ~ ' ] e , , + C [ U - ' ] Ec~ M ~ , ( F ) . Then R is finitely generated over C = Z(R), but GU fails from C to R. On the other hand, taking % = {prime PI-rings whose center is a valuation ringj in Exercise 1.9.3. shows (v) holds but (iii) fails.
Ch.
11
Exercises
107
5. Ifg,(R)+ = Z ( R ) and R has a set ofri x 1 1 niatric units. then R :M,(Z(R)I. [Him: Clearl! . R and any .Y in T, show R :M , , ( T ) for some ring T. But for matric units e , , e , , _ . of [.Y. R]g,(e,, e 2 , .. .) = 0.1 6 . (Rowen [74a]) If Z(R) is a field and g,, is R-central, then R is simple. [Hint: R/NiI(R) is simple; split and lift idempotents, and then apply Exercise 5.1 Now sharpen Theorem 1.8.48. We lead to Bergman’s example of a finite ring which is not admissible (cf. $1.6).
7. If R is quasi-local and e is idempotent. then 4- # 0 for all z # 0 in Z(R). 8. If R can be injected into M , ( C ) then R can be injected into M,(C,). Thus, for each idempotent e of R and for all c in C such that ce = 0 we have ReR n cR = 0. [Hint: Take P 2 Ann,.(ReR n c R ) . ] 9. Let G be the additive group LipZ @ L i p 2 L . and let R be the ring of (group) homomorphisms from G to G . Let e l , e2 be the respective projections of G t o the first and second components, let s, be the natural group injection Hips + H / p 2 Z , and let Y, be the canonical homomorphism E / p Z L + E/pE. given by I ++1. Then c , . e 2 . Y,. s 2 have respective annihilators p, p2. p. p, and R has p 5 elements. Now e l is idempotent and pel = 0, but 0 # pe, = x2el.xi. Hence, by Exercise 8. R is riot admissible.
nPFSpec(C,
$1.ll Exercises 1-5 show that the PI-theory without I is parallel to the usual PI-theory. Let R be the reduced ring with 1 of R,.
If R, is semiprime then Z ( R ) is the reduced ring with 1 of Z(R,). I f R, is prime (resp. primitive) then R is prime (resp. primitive). 3. If R, is primitive with PI (i.e., simple PI) then R, = R. 4. Define central localization for algebras without 1. If R, is prime PI then Qz(R,) I.
2.
Qz(R)5. If R, is semiprime PI then R is mull-equivalent to R,, and every nonzero ideal of R, intersects Z(R,) nontrivially. =
Here is a sketch of the theory of quotient rings of semiprime PI-rings, as developed by Fisher [73]. Page [73a], Martindale [73], Armendariz-Steinberg [74], and Rowen [74c], with a little twist. Assume through Exercise 17 that R is a semiprime PI-algebra (not necessarily with 1 ) having center Z. Recall Exercise 1.7.4. Also. use Lambek [66B] as a general reference.
6 . (Martindale [73]) If J is a large left ideal of R, then J is a semiprime PI-algebra without I , and Z ( J )c Z(R). 7. A left ideal J of R is large iff J n Z is large in Z . 8. Let Y’ = [large ideals of R \ and define an equivalence on the set Q = [ ( / L E ) J E EY and p : E + R is a 2-module homomorphismj. by ( p l r E 1 ) ( p 2 , E 2 ) iff p i and p 2 have the same restriction to some E in Y . The set of equivalence classes Q/- is an algebra (under and [(PI,EI)][(PZ~E,)I = the actions [(pi,E,)]+[01,,E2)1 = [ ( P , +pZ.E1n E 2 ) I [(pipz, (E2n Z)E,)] anda[(p, E ) ] = [(ap. E)]); wecallthisalgebraQ(R).Q(R)ischaracterized by the following four properties: ( I ) there is a canonical injection R -Q(R), given by right multiplication, sending 2 into Z(Q(R));(2) for any E in Y and p in Hom,(E, R). there is someq in Q(R)suchthat.uq = p(.u)forall\-inE;(3)foreachq # OinQ(R),O # Eq E RforsomeEin !P;(4)y = 0 ilTEq = 0 for some E in 2’.Q(R) satisfies the same homogeneous identities as R. 9. Z(Q(R)) = Q(2).( H i n t : Use Exercise 7.) 10. For any large left ideal J of R, Q ( J ) = Q(R). 11. If A , a R , i = 1. 2, and A , = AnnZ.4, and A, = Ann,A,, then Q ( R ) : Q(R/A,)@ @RIA,). ( H i n t ; A , @ A, is a large left ideal.)
-
108
THE STRUCTURE OF PI-RINGS
[Ch. 11
12. Suppose PI-class(R) = n. Define N , = n { P ESpec,(R)) and, inductively, given N , . , ,.... N , , define N , = n [ P e S p e c k ( R ) I N i pP for all i > k : . Then Q ( R ) = Q ( R / N , ) @... @ Q ( R / N , ) . 13. If 1 e R and K is an A,-ring with central idempotent e, then eR is an 4,-ring with multiplicative unit e. 14. If every nonzero ideal of Z contains a nonzero idempotent of y,(R)' then Q ( R ) is an .4,-ring. [ H i n t : Using Zorn's lemma, find idempotents e , in g.(R)+ such that 0 ,R e ; is large in R , S O Q ( R ) = Q ( o ; R ~ ~=; ~ ) ;Q(RU;)).) IS. A ring is oon Yeumann regular if. for each x there exists y with X ~ = Y x. Q ( Z ) is von Neumann regular, and every ideal of Q ( Z ) contains a n idempotent element. Thus. if g,(R)' is large in Z , Q ( R ) is ;in ,&-ring, 16. Q ( R ) is a finite direct sum of A,-rings, and so is Azurnaya. (Use Exercise 12.) 17. Q ( R ) is the maximal left quotient ring of R . and is also the maximal right quotient ring of R . 18. If L is a large left ideal of a semiprime ring R then L is mult-equivalent to R . 19. If R is a PI-ring without 1. spanned additively by nilpotent elements. then R is nil. [ H i m : If r is nonnilpotent then some prime ideal of R misses all powers of r. Hence we may assume R is prime. Taking a suitable central extension, we may assume R = hf,(F), which is absurd.) 20. (Procesi [73B. p. 1521) If R = { x , ...., x k ) is a C-algebra without 1 satisfying a polynomial identity of degree 2n and if all the monomials in the x, of length < ti2 are nilpotent then R is nilpotent. [lfint: As in Exercise 19, reduce to matrices. which are spanned by the monomials in the x, of length < nz (prove!). Thus R is nil, and hence locally nilpotent.] 21. (Schelter-Small [76]) A PI-ring whose maximal left quotient ring is nor PI. Let F be a field, K be an infinite direct product of copies of F , and R be the F-subalgebra ( F e l l +FuZZ+KP~,)
of M z ( K ) .
Then Fe,, + K e , , is a large left ideal of R contained in all large left ideals. so the maximal left quotient ring of R IS End,(F @ K), which is mi a PI-ring. Nore on how PI-theoryjits into general structurr rheory. The ideal f x E R 1 Ann,x is large) of Exercise 1.7.4 is called the singular ideul: R is nonsingular if its singular ideal is 0. Since every semiprime PI-ring is nonsingular, PI-theory sometimes is viewed as a special case of the theory of nonsingular rings.
CHAPTER 2
THE GENERAL THEORY OF IDENTITIES, AND RELATED THEORIES In Chapter 1 we studied the structure theory of PI-rings by focusing on particular polynomials (e.g., the standard polynomial, the Capelli polynomial, and a multilinear n2-normal central polynomial of M , ( F ) ); the theory of Chapter 1 could well be called the theory of rings satisfying a polynomial identity. In this chapter, the approach is altered in two ways. Most importantly, the emphasis is shifted to the set of all identities of a ring; this enables us to study “relatively free” PI-rings, which are so important in the study of finite-dimensional algebras (cf. Chapter 3). Also, we introduce two extensions of the PI-theory-PI-rings with involution and “generalized identities”-in order to pave the way for interesting applications in Chapters 3, 7, and 8. Possibly the most useful Pl-theoretic result in this chapter is Corollary 2.3.32: that when 4 is an infinite field, any algebra is equivalent to any of its central extensions. It follows easily that all simple algebras of PI-class n are equivalent; this result has a very easy proof, given in Exercise 2.3.4, and implies Corollary 2.4.10, which in turn is sufficient to yield the results of Chapter 3. The reader may well take this route as a shortcut to the very important theorems on central simple algebras. The less direct route taken in the text of 92.3 provides a considerably deeper understanding of 7’-ideals and the multilinearization process, including the situation for arbitrary 4.
52.1. Basic Concepts In this section we introduce the underlying notions of this chapter, namely, “relatively free PI-rings,” “T-ideals,” “rings with involution,” and “generalized identities,” and indicate how they will be useful to us later in the book. Relatively Free Algebras and Their Relation to T-Ideals
We have seen in 91.3 how the generic matrix ring simplifies various proofs about matrices. Our first objective is to show that the generic matrix I09
110 algebra
THE GENERAL THEORY OF IDENTITIES
[Ch. 2
4,,{Y ) is so useful because it acts like a free algebra
Definition 2.1.I. An algebra U is relatiuely,frer if U = 4 { X } / l such that, writing X i for X i + I , we have the following property: For every R satisfying all identities of U , each map { X I , Xz,. . .} -,R can be extended to a unique homomorphism U -+ R.
Write .B(R) = {identities of R } . For I = ,B(U), we see that U is the ”free” algebra satisfying all the identities I . Obviously 4 ( X ) is relatively free. By far the most important other example is c $ ” { Y ] ,which is relatively free, by Theorem 1.3.11. Relatively free PI-algebras are usually called “universal PIalgebras” in the PI-literature. Let us now characterize relatively free algebras. Proposition 2.1.2.
I f U = 4 ( X ) / I is relaticelyjree, thvr~I = .f ( U ) .
Proof. - U ( U )c_ I , by definition of identity. On the other hand, if j ’ ( X , ,. . ., X,) E 1 then . f ’ ( X , ,. . . , X,) = 0, so, setting R = L; in Definition 2.1.1, we see that j ’ ( r , , ..., r m )= 0 for all elements r in U , implying j ‘ ~ . 1 ( U ) . QED
Definition 2.1.3. An algebru endomorphism of R is an (algebra) homomorphism R 3 R. Definition 2.1.4. A is a T-ideal of R (written AQ,.R) if A a R and $ ( A ) E A for every algebra endomorphism $ of R . Remark 2.1.5.
For every algebra R, .P(R)Q ,.4{Xj.
Remark 2.1.6. If Aa,c${X) and fc A, then j‘(4{X]) E A. [Indeed, for any h,, . . .,h, E 4 ( X } we have an algebra endomorphism of 4{XI taking 1 < i < m, implying f ( X , ,... ,Xm)+,f(h1,.... h,), so Xiwhi, .f(h, . . - , h m ) ~ A . ] Theorem 2.1.7. Suppose AQ T4[Xj,. (i) & { X ) / A is relatit.ely,free. (ii) !fBa,c,b{.u)and A c B, then B / A a , c $ { x } / A . Proof. Write - for the canonical image of 4{X} in b [ X } / A . First observe that for any algebra R and any map 0:{X,, X,,. . .) -+ R there is a unique homomorphism $ : 4 ( X ) --* R such that $(Xi) = .(Xi) for all i. [Indeed, noting q{X} is free,just define $ such that $(Xi) = O(Xi).] Let us show that A E .f(@{X}/A).Indeed, suppose thatj’(X,, ..., X,)EA. Given a homomorphism $:4{X} 3 ( b { X ] / A ,we can write $(Xi) = Li,1 d i < m , for_suitable hi in c,b{X}. But by Remark 2 . 1 . 6 f ( h , , . . . , h , ) E A . Thus ____ $ ( . f )= . f ( h ,,... , / I r n ) = 0,sof€f(4{X),’A). (i) Suppose f(R) 2 .f(+[X;/,4),and there is a map 0: :XI. ,qz, . .I -, R .
$2.1 .]
Basic Concepts
111
We can lift (T to a homomorphism $ : 4 { X ) + R , such that $ ( X , ) = o ( X , ) for all I ; since A G Y ( R ) , we have $ ( A ) = 0, so I) induces a homomorphism 4 [ X } / A + R extending cr. Clearly it is unique. (ii) For any algebra endomorphism cr of # ( X } / A , we take the homomorphism t,h:+{X} + 4 { X ) / A such that $ ( X , ) = a(%) for all i ; as above, t,h induces an algebra endomorphism I,& of 4 { X } / A , and I,&(X,) = o ( X , )for each i, implying cr = $ (since @ { X } / A is relatively free). Then __ o(B)= $ ( B ) = I)@) G B.Thus B $ [ X ) / A . QED
a,
We now have the pieces for an important, although easy, result. Theorem 2.1.8. equivalerit :
For urq’ I
a 4{X )
the following statemerits are
(i) 4 ( X f / f is relatively free; (ii) l Q T 4 { X } ; (iii) I = f ( R ) f o r a suitable algebra R . Proof. (i) 3 (iii) by Proposition 2.1.2; (iii) => (ii) by Remark 2.1.5; (ii) * (i) by Theorem 2.1.7(i). QED
In view of Theorem 2.1.8, we are now interested in 4 ( R ) ,and not merely in its multilinear elements (which sufficed for Chapter 1). This observation sets the tone for the remainder of this chapter; in particular, we want to find a method of transferring all identities from an algebra to a central extension (cf. $2.3). Before we continue our study of f(R),we shall introduce three more theories, which will be of use later in the book; our point in introducing the theories now is to be able to avoid duplications in statements and proofs which hold both for these theories and the usual PI-theory.
Generalized Identities
The first theory is initially motivated by the observation that sometimes, in the study of polynomials in matrix rings, we wish to focus on evaluations in which certain indeterminates are always sent to given elements. (Theorem 1.4.34 is a very good example.) Thus it would be useful to have a theory admitting “polynomials” whose coefficients could be taken from all the elements of a ring (instead of from a central subring). To illustrate this idea, note that as n becomes larger, the minimal identity SZn of M , ( F ) becomes increasingly complicated (having ( h i ) ! monomials), whereas e , X , e , , X , e , - e l lX,e,,X,e,,, viewed in the obvious way, is a “generalized” identity of M J F ) for all i t . The basic notions of generalized identities (often called
,
112
[Ch. 2
THE GENERAL THEORY OF IDENTITIES
“generalized polynomial identities” in the literature) are a bit more cumbersome than the classical PI-analogs, but once mastered they lead to better organization of several important PI-t heorems, as well as some new PI-theorems. We shall give the cumbersome definitions now, leaving the GI-theory for Chapter 7, with Chapter 8 being an important application. Definition 2.1.9. Let R be a given monoid. The free monoid ouer R (written .N(R; X ) ) is defined as the set of strings ( w l X i , k 3 0, w j E Q, with multiplication given by
( t o , X ; , x i k w k +I)(w;xj, ...xj,,m;+ *) = ( W I xi,. . . xi,,( O k + , w; ) X j , . . . x,,,w:+ I )
3
where ( ~ ~ + ~ denotes w ’ , ) the product in R. Clearly .N(R;X) is a monoid [generalizing the free monoid . / / ( X ) in 81.1, which one gets by setting R = { l}], so we can form the monoid ring Z.K(R;X), called thefree ring ouer R. For our purposes, 0 will in fact be a ring whose multiplicative structure is the monoid structure of R. Definition 2.1.lo. Given a ring homomorphism $,:R + R, we say a X ) -+R is +,-admissible if o t+ $ o ( w )for all (u in R. homomorphism L.~V(R; Remark 2.1.11.
Given a homomorphism
Go :R
{ rl, r 2 , .. .} c R , we have a unique +,-admissible Z.h’(R; X ) + R such that Xir-*rifor all i.
R and also homomorphism
-*
Definition 2.1.12. Let id denote the identity map of R, and let 9, be the set of elements of Z.&”(R;X) lying in the kernel of every idadmissible homomorphism Z. H(52;X ) + R.
.Yo would be a candidate for the set of generalized identities of R, except that it has a number of “trivial” elements, such as w 1 +w, - (wl +a,)for all w i in Q and, more subtly, Lo,X , ] for all o in Z(Q). So we reject this candidate, but can use it in the “correct” definition. Define a T-ideal owr Cl of Z , K ( C l ; X ) to be an ideal .4 such that $ ( A ) G A for all id-admissible ring endomorphisms $ of Z.,H(R; X ) . Definition 2.1.13. Let .Y, be the T-ideal over R of H . / / ( R ; X ) generated by all elements of the form [w,Xi], WEZ(R).and all elements of 9,not involving X at all. Define R{X} = Z.X(R; X)/.Y,. To underline the fact R is a ring. we now write W instead of 52 and consider W ( X ] .
The reader may be familiar with W { X >as the free product of W and Z { X ) over Z , where Z = Z ( W ) , but I like the above construction because it gives W { X ) in terms of ”relatively free rings over Q.” In particular, the reader should ha\e no trouble verifying the next remark.
$2.1.]
Basic Concepts
113
R is a W-ring if there is a canonical homomorDefinition 2.1.14. phism W + R such that Z ( W )-+ Z ( R ) . If R l , R , are W-rings, a Whomomorphism $: R , .+ R , is a homomorphism such that $(wr) = w$(r) and $(rw) = $(r)w for all r in R, w in W. Remark 2.1 .I 5.
W{X} is the free W-ring, in the sense that for any
{ r,, r,, . . .} in a W-ring R there is a unique W-homomorphism $: W{X} -+ R such that $(Xi) = ri for all i. (i) Suppose A 4 R and R is a W-ring. Then RIA is a Remark 2.1 .I 6. W-ring, because the canonical homomorphism w H w . l induces a homomorphism w H w . l + A sending Z ( W) -+ Z ( R / A ) . (ii) If ( R , ( y E r ) are W-rings then n,,,,-R, is a W-ring, in view of the homomorphism w H (w * 1?). (iii) If R is a W-ring and W * 1 c R , E R , then R 1 is also a W-ring. (iv) If R is a W-ring and A is a C-algebra with C 5 Z ( R ) then R OcA is a W-ring. [Just map w + (w. 1) 0 1.1 If W is commutative then obviously every W-ring is a W-algebra; this is the point of our definition. In this spirit, the elements of W{X} are called W-polynomials; the canonical images of elements of A‘(W;X) in W{X} are called W-monomials. Definition 2.1 .I 7.
Suppose R is a W-ring.
9 ( R ;W) = n{ker$l$: W{X} -+ R is a W-homomorphism}
called the set of W-identities of R . When W is unambiguous (usually R itself), 4 ( R; W) is called the set of generalized identities (or GI’s) of R (with coeflcients in W). Although we have taken considerable care to rule out “trivial” GI’s, there is still a problem when Z(W) is not big enough. Example 2.1.18.
Let
Obviously W,{X} has no nonzero GI’s as a W,-ring. But as W-ring, Wl{X} does have a nonzero GI, (Ael,)Xlel, -e12Xl(Ae,,). A related problem is that there is no good way to write generalized polynomials uniquely as sums of W-monomials. For example, (w, +w,)Xlw,+w,Xlw4 = w,X,w,+w,X,(w,+w,); there is no rational reason why one writing should be preferred over the other. We shall return to these difficulties shortly. (The latter one, we shall see, actually is irrelevant.)
114
[Ch. 2
THE GENERAL THEORY OF IDENTITIES
Identities and Generalized Identities of Rings with Involution
The next theory is a PI-theory for “rings with involution.” To motivate this theory, we shall indicate one very important way the notion of ‘‘involution” (to be defined below) enters into ring theory. Consider the Brauer group Br(F), where F is a field. Virtually everything about the Brauer group is important; in particular we are interested in subgroups, one of which is Br,(F) defined as { [ R ] E Br(F)I[R]’ = 1). Lemma 2.1.19. I# R , and M,(R,) ’c M , ( R , ) , then R , R , .
R,
are
simple
PI-algebras and
Proof. R , = M , , ( D , ) and R , 2 M,,,(D,), for suitable division algebras D , and D,, so M,nnl(Dl)2 M,,,(D,). By-Corollary 1.5.11, D , 2 D, and mn, = mn2, so n , = n 2 and R , 2 R,. QED Proposition 2.1.20.
Br,(F)
=
{ [ R ]EBr(F)IR 2 RDP}
Proof. Suppose R is central simple, with n = [ R : F ] . If R = RDP then R@,R”P 2 M,(E’), so [ R ] ’ = 1. On the other hand, if [ R I 2 = 1 then R O F R 2 M,(F). and m = n (seen by checking dimensions). Thus M,(R”P) : hf,,(F)@ R”1’ : ( R @ R ) @ R”” : R @ ( R @ R””) : R @ M , , ( F ) :‘M,,(R)so by Lemma 9.1.19 R :R””. QED
Thus we are very interested in the contingency R 2 RDP.What does this mean? Given algebras R , and R,, say a module homomorphism $: R , -+ R , is an anti-homomorphism if $ ( r 1 r 2 )= $ ( r 2 ) $ ( r 1 )for all r l , r 2 in R , ; if, moreover, $ is a module isomorphism we call $ an antiisomorphism. and if R , = R,, we call $ an antiautomorphism of R,. Remark 2.1.21 .
The map r H r from R to RDPis an antiisomorphism.
Corollary 2.1.22. Suppose that R is a simple PI [ R ] E Br,(F) iff R has ail antiautomorphismfixing F .
F-algebra.
Proof.
R
2
In view of Remark 2.1.21, R has an antiautomorphism iff RoP (seen by composing maps), so apply Proposition 2.1.20. QED
We shall see in Chapter 3 that in case R is simple PI with an antiautomorphism, R has an antiautomorphism 0 such that D~ = 1, i.e., o-(a(r))= r for all r in R . Accordingly, we make the following definition. Definition 2.1.23. 1.
An inuolution is an antiautomorphism
D
such that
(i2=
Example 2.1.24. ( x a .I J. erJ. . ) *= EM..^.. IJ J I .
M , ( F ) has the transpose inoolurion (*), given by
$2.1.]
Basic Concepts
115
In what follows, we write (*) to denote a given involution of a ring, much as *’+” might be used to denote its additive structure; “(*)-algebra” means “algebra with involution.” To study (*)-algebras properly, we treat (*) as an intrinsic part of the algebraic structure. Write r* for the image of r under (*).
Thus writing ( R , *) to denote the algebra R with involution (*), we define a homomorphism I):(R,, *) + ( R 2 ,*) to be a homomorphism $: R , --t R , such that $(r*) = $(r)* for all r in R ; we often say, equivalently, Ic/ is a (*)homomorphism. A is an ideal of ( R , * ) , written A a ( R , * ) , if A U R and A* c A ; we often say, equivalently, A is a (*)-ideal of R . If A a ( R , * ) then obviously (*) induces an involution on RIA by ( r + A ) * = r * + A, and the canonical map R + R / A is a (*)-homomorphism; conversely, the kernel of every (*)-homomorphism is a (*)-ideal. As one learns about rings by studying the PI-theory, one may learn about (*)-rings by studying their PI-theory. [Indeed, this is emphatically true for central simple (*)-rings.] Thus, we want a PI-theory involving (*) intrinsically. Actually, it is just as easy to introduce a GI-theory involving (*), so we shall develop this third theory, because there are several proofs which are easier in the more general setting. Throughout, (W, # ) is a ring with a given involution # . [In the PI-case, W is commutative and ( # ) is the identity map; the reader may prefer to focus on this more special situation.] Definition 2.1.25. ( R , * ) is a ( W , #)-ring if there is a canonical homomorphism (W, # ) + ( R ,*) sending Z(W) + Z ( R ) . [If W is commutative and # is the identity, then we are only saying that ( R , * ) is a W algebra.] If ( R , , * ) and ( R 2 , * ) are (W, #)-rings, we say I):R , + R , is a (W, #)-homomorphism if I) is both a (*)-homomorphism and a Whomomorphism.
We need a free (W, #)-ring. This could be found by a suitable monoid ring construction, but instead we shall take a shortcut, making use of some involutions already at hand. Remark 2.1.26. If ( R , * ) is a ring with involution then z * E Z ( R )for all Z E R . Thus (*) induces an automorphism of Z ( R ) ,of degree 1 or 2. Remark 2.1.27. If R is a W-ring, then ROP also has a W-ring structure because w hw # . 1 gives the desired homomorphism from W to ROP.
W e shall refer to this structure implicitly in what follows below. Definition 2.1.28.
The reversal involution # on W { X } is the
116
THE GENERAL THEORY OF IDENTITIES
[Ch. 2
composition of the W-isomorphism W { X )+ W-(X}"p,given by X i-+ X i for all i, with the canonical antiisomorphism from W { X } " pto W {X } . For example, we see the action of # on a typical W-monomial is the , X i Awk+ w wk + X i k . .. X i ,w 1Hw,f+ I X i , X i wf . composite map w l X i ... Thus the reversal involution replaces each w by w # and reverses the order of each W-monomial, showing that it is indeed an involution. The reversal involution is interesting, but for our purposes there is another involution which is more useful.
,
Definition 2.1.29. The canonical involution (*) on W { X ) is the composition of the reversal involution and the ring endomorphism given by X 2 i - 1- X Z i and X 2 i ~ X 2 i - for 1 all i. (Thus, pairing off the indeterminates, we switch them.) ( W ( X ) , * ) always denotes W l X ) with the canonical involution, where we "rename" the indeterminates, writing X i for X 2 i - and XF for XZi.Elements of ( W { X ) *) , are called ( W ,*)-polynomials (or sometimes "generalized (*)-polynomials"); each ( W , *)-polynomial can be written (not uniquely) as a sum of ( W ,*)-monomials,which are strings of X i , X l , and elements of W . [For example, X , w X : X : X , is a ( W , * ) monomial, where \v E W . ] Proposition 2.1.30. ( W { X ) , * ) is the ,free ( W , #)-ring with incolution, in the sense that ,for any ( W , # )-ring with inoolution (R. * ) and for any r l ,r z , . . . in R , there is a unique (W, # )-homomorphism $: ( W {X I , *) 3 ( R ,*) such that $(Xi) = ri,for all i. Proof. = ri and
Obviously we can define a W-homomorphism such that $(Xi) $ ( X , * ) = r:. But this is also a (*)-homomorphism. QED
Definition 2.1.31. Suppose ( R , * ) is a ( W ,# )-ring with involution. . P ( ( R , * ) ; W= ) i ~ , : k e r $ I I C / : ( W f X ~ , * ) - t ( R , * ) is a ( W , #)-homomorphism).
Call . f ( ( R , * ) ; u')the set of GI's q f ( R , * ) (or, equivalently, (*)-GI's oJ'R) with coejicients in W . The (*)-GI theofy is the most general theory considered in this book, although there are several more general theories worth attention. K harchenko (see bibliography) proved several general PIand GI-structure theorems for rings with a finite group of automorphisms; there is a connection between such a theory and group algebras, although its exact nature is not well known. Probably one could extend Kharchenko's results for rings with a finite group of automorphisms and antiautomorphisms. There are also satisfying results for rings naving a finite grade, by S . Westreich. Little is known about rings with derivation. In general, given a family of n-ary operations, one can construct a theory of identities with respect to any given family of n-ary operations. This
$2.1.]
Basic Concepts
117
viewpoint is developed by Neumann [67B], which is mostly about the identities of groups; in Appendix C we shall see some aspects of nonassociative PI-theory. Special Rings with Involution
The first question to ask is, "Is the (*)-GI theory any richer than the GItheory?" Let us rephrase this question more explicitly. Definition 2.1.32. First define the procedure p on ( W { X > , * to ) be the replacement of Xi (resp. Xi*) by X2i-l (resp. X,,), thereby giving us back our original copy of WCX}. (For example, p(XlXTX2- ( X T ) 2 X 5 ) = X,X,X,-X:X,.) Suppose (R,*) is a ( W ,#)-ring, andfE(W{X},*);fis (R,*)-special if p ( f ) is a G I of R. (R,*) is special if every G I of (R,*) with coefficients in R is (R, *)-special.
Now our question is, "Which rings with involution are not special?" Well, for a field F , we can define (*) to be the identity map, in which case X, - X : is an identity of ( F , *), whereas p(X, -XT) = X, -X, is not an identity of F . Therefore ( F , * ) is not special. This example will be generalized to a large class of nonspecial rings with involution. Nevertheless, many rings with involution are special, and we present now an important prototypical example. Remark 2.1.33. Suppose W has an involution ( # ) . If R is a W-ring then R 0 RoPhas an involution ( 0 ) given by ( r l , r 2 ) 0= (r2,rl), and, in view of the map W H ( w .1 , ~ " .l), (R 0ROP,o)is a ( W , #)-ring with involution. Definition 2.1.34. The involution ( 0 ) on R Remark 2.1.33 is called the e.wchange involution. Proposition 2.1.35.
0RoP described in
(R 0RoP,o)is special.
Proof. Suppose I):W ( X l + R @ RnPis an arbitrary W-homomorphism. Letting ni denote the projection of R 0RoPto the ith component, i = 1,2, we have a homomorphism I)1= I)n, :W ( X ) +R, inducing a homomorphism I);:(W(X], *)+(R@Rol',o),given by I ) ; ( j ' ) =( + l ( p f ) , I)l(pf'*)). Iff'isaGI of (R @ R"'', 0)then I)',(j')= 0, implying I),(pJ) = 0. Likewise, define I); by t,&(,j') = (n,(I)(p,f'*)),n,(I)(pj'))). I f f is a GI of (R 0R"I',o), we likewise concluden,($(pf)) = 0,so I)(p,f')= 0.Thuspj'isan identityofR @ R"",i.e.,fis special. QED Generalized Monomials
We return to the difficulty stated earlier that, unlike the PI-case, there is no obvious way of writing an element of W ( X ) uniquely a5 a sum of W -
118
THE GENERAL THEORY OF IDENTITIES
[Ch. 2
monomials; this difficulty will be bypassed by the introduction of “generalized W-monomials,” which will take the place of W-monomials. Since any W-polynomial can be viewed as a (W, *)-polynomial in which no X r occur, we shall actually deal with (W,*)-polynomials. from which considerations about W-polynomials follow as a special case. Recall that an! (W,*)-monomial h is a string of various X i , X f , and elements of W , in any order. The elements of W that appear are called the coeficients of h ; the ( W ,*)-monomial obtained by erasing all the coefficients is called label(h), and h is pure if h = label(h). For example, if h = w l X I X ~ w , w , X , ,then w , , w Z are the coefficients of h, and label(h) = X l X T X 2 . So we see that a pure (W,*)-monomial really has nothing to do with W, and will thus be called a (*)-monomial. Suppose we write a (W,*)-polynomial f as a sum x f = , h , of (W,*)monomials. We say the coeficient set off is { w E W J wis a coefficient of some hi). Strictly speaking, this concept is not well defined, but we shall only use it when there is no doubt as to the particular choice of the hi. For a given pure (*)-monomial h, we say the generalized ( W ,*)-monomial off with lube1 h is x{hillabel(hi)= h ) ; this notion is well defined fie., not depending on the particular choice of h i ) as we shall see now. Proposition 2.1.36. Every (W’,*)-polynomial can be written exactly one way as a sum ofgeneralized ( W ,*)-monomials. Proof. From the definition of the involution (*) on W { X ) , one sees immediately that it suffices to prove that every W-polynomial f can be written exactly one way as a sum of generalized W-monomials. Let $: Z , & ( W ; X ) W { X } be the canonical homomorphism of Definition 2.1.13 (with kernel Given a pure monomial p, let J(, = {generalized monomials having label p}. Then for all distinct p l , .. ., p f , we have $ - 1 ( V l ) n x t = 2 $ - 1 ( V , ) c .I,,implying V , V , = O.Soiff’= L:=l,fi = .L% i= -I l j y , where ft and jy are generalized W-monomials having label p i , with p i , . . . , p f distinct, then 0 =.f’-f= x:=l(,h-/i‘); thus each (,ji.-.L’) = 0, being the generalized W-monomial of 0 with label pi. QED --t
(Actually, the above proof really should be viewed in the context of “graded rings,” and works because .fI is a “graded ideal.”) Proper (Generalized) Identities and (*)-Identities We still have not pinpointed the GI’s [or (*)-GI’s] that interest us. Recalling from Chapter 1 that a proper identity was a sufficient condition on a (primitive or prime) ring to push through the Kaplansky and Posner-Formanek -Rowen theorems. we want to generalize the notion of “proper.”
$2.2.1
PI-Rings Which Have an Involution
119
Definition 2.1.37. A (W, *)-polynomial .f is (R, *)-proper if one of its generalized ( W ,*)-monomials is riot a GI of ( R ,*); ,f is ( R ,*)-strong if ,f is ( R ,*)-proper for every homomorphic image (R, *) of (R, *). ~
~
Remark 2.1.38. A (*)-polynomial f is (R,*)-proper iff f has a monomial whose coeficient does not annihilate R. Thus Definition 2.1.37 does generalize Definition 1.1.1 5. In Chapter 7, we shall build a structure theory of primitive and prime rings based on proper GI’s and (*)-GI’s, and shall prove that if R [resp. (R,*)] satisfies a strong GI then R is a PI-ring. Presently, we record information through use of improper GI’s. Remark 2.1.39. R is prime (resp. semiprime) iff for all nonzero a, b in R,aX,b(resp.uX,a)is riotaGIofR.Thus,ifweknow {improperGI’sofR),wc also know whether or not R is prime (resp. semiprime). Remark 2.1.40. By Theorem 1.4.34, elements r , , ..., r, of M J F ) are F-dependent iff CZf-l ( r l , .. . ,rt, X , , l , . . .,X 2 r - l )is an (improper) GI of R. This description leads to a fairly trivial proof of a more general result, in 47.6. Degree and Related Concepts
We close this section with some technical definitions to permit us to examine ( W ,*)-polynomials. For a (*)-monomial h, write deg,(h) to denote the number of times X i and Xi* occur in the formation of h. For example, for h = XyX:X$, we have degl(h) = 1, deg2(h)= 3, and degi(h) = 0 for all i >, 3. Define degi(f) = rnax{deg,(label(h))lh is a (W,*)-monomial off}, and degi(f) = min{deg,(label(h))lh is a ( W ,*)-monomial off}. Writef(X,, . . .,X,) to denote that degi(f) = 0 for all i > t . Given a map X iH ri (for ri E R), we denote the corresponding image off(X,, . . .,X,) in R as , f ( r 1 , . . .,rt). In the involutory case, when we wish to emphasize (*) we shall write f(X,,X:,. . . , X , , X : ) and f ( r l , r y , . . . , r , , r : ) in place of .f(xl,. .. ,X , ) andf(rl,. . .,r0. Call f homogeneous in the ith indeterminate if degi(f) = degi(,f); j’ is completely homogeneous if f is homogeneous in each indeterminate. Call f linear in the ith indeterminate if degi(f) = degi(f) = 1 ; f i s multilinear iff is linear in each indeterminate occurring in f : For example, X , X T - X i is completely homogeneous ;X X y is not multilinear.
,
$2.2. PI-Rings Which Have an Involution
In this section we study the basic structure theory of a (*)-ring R , under the assumption that R is a PI-ring. The main theorems of $1.5 and $1.6 are
120
[Ch. 2
THE GENERAL THEORY OF IDENTITIES
carried over intact, with some modification in order to account for (*). First some easy remarks. =
Remark 2.2.1. If AU(R, *) then A* = A. [Indeed, ( A * ) * A** (since * has degree 2), so equality holds at each stage.]
c A*
G .4
a
Remark 2.2.2. If A 4 (R, *) for each y in I-, then (nyEl A , ) (R, *). Thus, if r E (R, *). we can define the (*)-ideal generated by r, denoted ( r ;*), to be n{all ideals of (R, *) containing r } . Remark 2.2.3.
If A a R then A * a R . (Thus AA*
Remark 2.2.4.
If A d R , then ( A + A * ) a ( R , * )and ( A n A*)a(R,*).
Remark 2.2.5.
RrR +Rr*R = ( r ; *).
Remark 2.2.6.
If ~ E Z ( Rand ) z* = + z then Rz = (z;*).
E A nA * . )
Remark 2.2.7. If B c Z(R) and B* = B, then Ann, B d ( R , *). If R is semiprime and A a ( R , *), then Ann, AU(R, *) by Remark 1.7.32. Proposition 2.2.8.
Nil(R)Q (R, *) and Jac(R)a (R, *).
= (rk)*= 0; hence If rfNil(R) then r" = 0 for some k, so Nil(R)* E Nil(R). Likewise, if rEJac(R) then (1 - r ) is left and right invertible, so (1 - r*)- = (( 1 - r ) - )*, implying (Jac(R))* is a quasiinvertible ideal of R ;thus (Jac(R))* E Jac(R). QED
Proof.
'
Thus we can study (R, *) by passing t o (R/Nil(R),*) and (R/Jac(R), *). This technique is very important, enabling us to parallel the methods of Chapter 1. For example, Kaplansky's theorem says that every semiprimitive PI-ring is a subdirect product of simple PI-rings. We can extend this theorem quite nicely to the (*)-case. Definition 2.2.9. (R, *). Lemma 2.2.10.
( R , * ) is simple if 0 and R are the only ideals of
( R , * ) is simple iff R has a maximal ideal A such that
A n A* = 0.
Proof. If (R, *) is simple then for a maximal ideal A of R, A n A * a ( R , *) and thus must be 0.Conversely, suppose R has a maximal ideal A with A n A* = 0, and BQ(R, *). If B $ A then
B+A*
=
R(B+A*) = (B+A)(B+A*) E B+AA*
=B,
implying A* c B; thus A = ( A * ) * E B* E B, implying B = R (since A is a maximal ideal). Otherwise B E A, and B = B* c A*, implying B c ( A r8,4*) = O . QED
$2.2.1
121
PI-Rings Which Have an Involution
Remark 2.2.11. Suppose B a R and B n B* = 0. Obviously RIB 0 RIB* has an involution (*) given by ( r , +B, rz +B*)* = (rt +B, r: +B*), and the canonical injection of R into RIB 0 RIB* [given by rt-+(r+B, r + B * ) ] is actually a (*)-injection. In fact, (*) indices an anti-isomorphism from RIB to RIB*, so RIB* z (R/BYP, yielding a canonical injection ( R , *) + ( R / B 0 (R/BYP,0 ) . Proposition 2.2.12. Suppose (R,*) is simple. Then either R is simple, or R has a simple homomorphic image R , such that ( R ,*) z ( R , @ RYP,0)[in which case (R,*) is special]. Proof. Immediate 1.7.16. QED
from
the
above
results
and
Proposition
Theorem 2.2.13. If Jac(R) = 0 and R has PI-class d, then ( R ,*) is a subdirect product of simple ( R y ,*) such that for each y, either R , is simple of PI-class < d or ( R , , * ) z ( R , , 0 RYP,,O)for a suitable simple image R 1 , of R , o ~ P I - c ~are (*)rings then n ( R ,17 Er} has an involution, given by the componentwise involution; i.e., we define (ry)*= (r;). As in $1.1 we have the following fundamental observations : Remark 2.3.5. If R , c R or if R , is a homomorphic image of R, then R, < R. If R , < R for all y in r, then ( n , , , R , ) < R. Remark 2.3.6. If (R,, *) is a subring or homomorphic image of *) (R, *), then ( R , ,* ) < (R, *). If (R,, *) < (R, *) for all y in r, then G (R,*).
nyer(R,,
$2.3.1
Sets of Identities of Related Rings
125
One can rephrase Remark 2.3.6 by saying the class of (*)-rings < ( R ,*) is closed under taking of subrings [with (*)I, homomorphic images, and direct products (and thus also under subdirect products). The converse is also true (cf. Exercise 2 ) .
Stability of Identities
For the rest of this section, we shall turn to the correspondence of identities between R and its central extensions. Of course, from the point of view of $1.6, the most important central extension is R[A]; passage of a multilinear central polynomial from R to R[n] was a major step in determining the structure of semiprime PI-rings. As we shall see, it is important to determine in general which identities pass from R to R[A] (and, more generally, to all central extensions). First a negative example. Example 2.3.7. If F is a finite field having t elements, then F - ( 0 ) is a multiplicative group of (t - 1) elements, implying X‘, - X , is an identity of F. However, X i - X I is not an identity of the polynomial ring F[A]. Definition 2.3.8. A n identity ,/’of R is R-stable if f is an identity of R[AJ A polynomialfis stable if,fis R-stable for every ring R of whichfis an identity.
Before discussing stability, we should like briefly to treat central extensions of (*)-rings. If ( R l ,* ) is a subring of ( R , *), say R is a central (*)extension of R, if R = Z(R,*)R,. We can produce many central (*)extensions, via the tensor product. Proposition 2.3.9. Suppose ( A , *) is a C-algebra with involution, and B is a commutative C-dgebrci. Then (*) @ 1 is an inoolution of A Bc €3. Proof.
Immediate from Corollary 1.8.5. QED
[The ( W , #)-ring structure passes over, by Remark 2.1.16.1 In particular, for any (*)-ring R, the (*)-structure extends naturally to R[A] .t. R @E Z[A] and to all central localizations. We return again to stability. Definition 2.3.10. An identity .f of ( R , *) is (R, *)-stable if ,f is an identity of @[A], *); a (*)-polynomia1,fis stable iff is ( R ,*)-stable for every ( R ,*) of whichfis an identity [in which case we also sayfis (*)-stable]. Remark 2.3.11. If .fl,fz are R-stable, then J , +f2 is R-stable. [Likewise for “(R,*)-stable.”] Proposition 2.3.12.
I f R is a central extension q f ’ R , , then R anif R ,
126
THE GENERAL THEORY OF IDENTITIES
[Ch. 2
are mult-equivtclent. r f R i s a central (*)-extension of R,, then (R,*) und ( R l , *) are mult-equivalent. Proof.
As in Remark 1.130. QED
In particular, all multilinear polynomials [resp. (*)-polynomials] are stable [resp. (*)-stable]. Proposition 2.3.12 is sufficient for most applications, and leads to the question of how “close” a given polynomial or (*)polynomial is to being multilinear ; intuitively, maybe some polynomials “near enough” to multilinear are stable. (See Exercises 3, 4 for a shortcut when 4 is an infinite field.) In fact, there is an operator, the Aiu of Definition 1.1.21, given, we recall, by . ., X i + X u , .. .,Xm)-,f(X1,.. .. X i , . . ., X,) - . f ( X , , ‘ . ’ xu, ’.,X , )
Ai..f’=.t ( X i , .
1
(where we have specialized Xi respectively to Xi+X,, Xi, and X u ) . For example, iff= X,X: then A12,f= X , X t + X , X ~ ,which is multilinear. Our goal is to use the Aiu repeatedly to “multilinearize” various identities and central polynomials. I n order to optimize the results, we study this procedure carefully and systematically. What we would like is a procedure which takes a polynomial f’to some multilinear polynomial Af, which is an identity of R (resp. R-proper, R-central) whenever ,f is. [Likewise for (*)polynomials.] However, there are three immediate obstacles to a straightforward application of the Ai,. Example 2.3.13. Let R be a commutative ring satisfying the identity X: - X l . Then ( 1 + I)’ = ( 1 + l),so 2 is an identity of R . Let us now cite the three difficulties.
( 1 ) f’= X : - X , + 2 is an identity of R. A l z ( f ) = X , X , + X , X , - 2 , and A I 3 ( A l J ) = 2. Thus the constant term prevents f from being multilinearizable. (2) f = 2X:+.Yt-X, is an identity of R with coefficient I (and is thus R-strong). However, A I 2 A l 3 f = 2 ~ . n t S y m , 3 ~ X n 1 Xisn not 2 X neven 3 R-proper. (3) X: is R-central, but Alz(X:) = X I X z + X , X , is an identity of R since A i 2 ( X : )= Al2(X:-X,)). I n this way we have obtained “too many” identities, killing off central polynomials.
The first two difficulties can be disposed of easily, whereas the third takes more elTort, and can only be overcome completely for rings with nice enough centers. Remark 2.3.14.
Every (pure) monomial is stable and (*)-stable.
$2.3.1
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(Indeed, if a monomial 11 is an identity of R and a is the coefficient of h, then a1 = 0, so h is an identity of R[A].) Thus, in our discussion of stability, we may (and shall) assume that no nonzero monomial of a given identity f is an identity of R [resp. of (R, *)I, for we can subtract these monomials out. Now the constant term of an identity .f is a monomial which -is also an identity, so we may assume it is 0. This disposes of the first difficulty. To smooth the multilinearization process further, we introduce some more notions. Definition 2.3.15. f is blended if deg(f) # 0 and for all i either deg'( /) = 0 or deg,(,f) > 0. If . / is blended, define height(,f) as deg(f) minus the number of indeterminates occurring inf Remark 2.3.1 6.
f i s multilinear ifffis blended with height(f) = 0.
Definition 2.3.17. For a polynomial [resp. (*)-polynomial] f, write f ( R ) + (resp. f ( R , *)+) for the additive subgroup of R generated by f ( R ) [resp.f(R, *I].
Viewing W { X }G (W(X>,*) as rings, we shall treat polynomials as a special case of (*)-polynomials, when feasible. Definition 2.3.18. Say,f, 2 f z if.fl(W{X},*) G ~ , ( W { X } , * ) ;2, ~+~f , if.fi(W{X),*) 5fZ(W{X},*)'.
,
For example, by Proposition 1.2.17 t ! S,+ 2 S , for all t 2 1. +
Remark 2.3.19. Suppose we have polynomials f , 2 + f 2 .Then for every ring R, fl is an identity of R if.f, is an identity of R. Likewise for (*).
2
Remark 2.3.20. Iffl 2 f z >,f3, then,fl 2 f 3 ; if.fl 2 ',f, 2 + f 3 , thenf, f3. Iffl 2 f andf, 2 -',L then.f, +.f2 3 + f +
+
Now we are ready for a more elaborate (but easy) version of Remark 1.1.22. Remark 2.3.21. Suppose j ' ( X , , . ..,X,) is blended, and i , j , u , u are distinct, with deg'( f ) 2 1.
(i) If deg'(f) = 1 then Aiuf= 0. (ii) Aiufis blended. [Just apply ( i ) to each monomial of$] (iii) AiuAjvf= Aj,,Aiu,f (iv) degj(Aiu,f)= deg'(,f). (v) deg'(A,,f) = deg'(f) - 1. (vi) A i u f 2 + J and the coefficient set of Aiuf is contained in the coefficient set off: (vii) If deg"(f) = 0 then height (Aiu.f) = height(f)- 1 .
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The following technical lemma is included because it shows exactly how multilinearization works, but is not used per se in the proofs of this section. Lemma 2.3.22. Suppose ,f(X , , ..., X,,,) is blended and let h, be a monomial o f f with height(h,) = height(f). Let di = deg,(h,), 1 Q i < m ; let c , = m a n d c i = m + ~ ~ ~ ~ ( d j - 1 ) , 2 ~ i < m . Il ,fl ed t iA=i betheidentity operator; otherwise, dejine Ai = Ai,c,+d, - l A i , c , + d , - 2 . . . A i , c , + ,Defrne . A = A , . . . A,. Then Af is multilinear. Moreover, for every monomial h of j Ah = 0 unless depi(h) = di, 1 < i d m, in which case Ah is a sum of 1 IF' di! monomials, which each specialize to h under the map X j c + X i whenever (ci+ 1 ) < j < (ci+di- 1). (In particular, h b each monomial OfAh.)
,
Proof. If h, is multilinear then ,f is multilinear and we are done. Otherwise, take the smallest k such that d, > 1 and let ,I, = A,,,,,+,f: Then there are dk monomials of Ak,,,,+,h, having degree d , - 1 in the kth indeterminate, and height( f l ) = height(f)- 1 ; the proof concludes by an easy induction argument on height, applied toj;. QED
We could now dispose completely with the difficulty described in Example 2.3.13(ii) [cf. Exercise 61, but we circumvent the issue by characterizing R-stable polynomials directly. Note that by collecting monomials having the same degree in each indeterminate, we can write f = where each ,h is completely homogeneous; call the f i the completely homogeneous components off: For convenience, we state the next result only in the noninvolutory case, but the (*)-theorem is analogous.
ui,
Theorem 2.3.23. The jollowing statements are equicalent for mery blended polpomid f ( X , , . . . ,X,) that is an identity of'a gitien ring R. (1) f is R-sfable.
(2) .f is an ideritity of every central extension o f R . (3) f i s in the ,set .Y c .Y(R), defined as follows (inductively on height): (i) Y contains all multilinear identities of R ; (ii) a blended identity of R, which is not completely homogeneous, is in .Y if all of its completely homogeneous components are irr .Y; (iii) a completely homogeneous, nonmultilinear identity h is in .Y' i f A i , , h ~ - Yfor all i,u such that deg,(h) = 0 and deg'(h1 > 1. Proof. By Proposition 2.3.12 the theorem is true for height(f) = 0, i.e., for f multilinear. Also (2) * (1) is trivial, so we need show (1) * (3) and (3) (2) in case f is not multilinear. We shall appeal to induction on height(f), assuming the theorem is true for all g such that height(g) < height( f ). Case 1. f is completely homogeneous. (3). For all i, u such that deg,( f ) = 0 and deg'(j') > 1, we have Ai, f (1)
42.3.1
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2 ' , A so Aiuf'is an identity of R [ I ] and is thus R-stable. But height(Aiuf) < height( f ), so by the induction hypothesis each Ai, f E 9. Thus f ' .Y' ~ by condition (iii). ( 3 )* (2). We want to showfis an identity of every central extension R' of R . Now for all i, u such that deg,(f) = 0 and degi(f) > 1, we have Aiuf E Y so by induction on height Aiu.fisan identity of R'. Looking at the definition of Aiu, we see for each i and all r , , . . . ,r,,, and ri in R', that
, f ( r l , ..., ri+ri ,..., r,) = , f ' ( r l..., , ri, ..., r m ) + , f ( r l , ..,(, . ..., r , ) ;
hence, for all rij in R', 1 6 i ,< m,we have . f ( X j r l j , ..
. ,Cjrmj) = Ij,...., j n J ' ( r l j ,.. , . ,rmj,,).
NOWlet di = degJ For all aij in Z ( R ' )and all rij in R, we get
Zjl
. f ( C j ~ ~ j r l j ~ . . . , ~ j ~ r n= j r , j ) ,....j , , , 4 1 1
" ' a ~ , , , f ( r l j , , . . . , r ~=j ,0., , )
Hencefis an identity of R'. Case 2. f is not completely homogeneous. Let f = Ef,, where thefq are the completely homogeneous components off: ( 1 ) =-(3). We havef(R[A]) = 0. By condition (ii), we need to show that each f , E . Y ; by Case 1 [(I)* (3)] we need merely show each f, is an identity of R[A]. rijAJ,1 ,< i < m, and define inductively So suppose we are given xi = numbers n , = 1 and n i + = 1 + ( k i + n,)deg'(,f). Checking coefficients of suitable powers of A inf(I"'x,, . . . , I n ~ ~ z x=m0,) we see that eachfJx,, ...,x,) = 0. Thus eachf, is an identity of R[I]. (3) * (2). Each ~ , E Y and so by Case 1 is an identity of every central extension R' of R . Thusf' = xji is an identity of R'. QED
Corollary 2.3.24. Let - P ( R ) ,= ( , f ~ . Y ( R ) J d e g j fd< ,for all j } . l / ' t h e completely homogeneous comporierits qf'eachelement O f . f ( R ) are , in .f( R ) ,theri every ,f' in .f ( R ) , is R-stable. Surprisingly, one can push this result one step further if ,f is completely homogeneous (cf. Exercise 10). The hypothesis of Corollary 2.3.24 can be analyzed decisively by a famous argument of Vandermonde.
The Vandermonde Argument and Its Applications
Definition 2.3.25. Given elements cl,. . . ,c, of a commutative ring C , let (c,, ...,c,( denote the determinant of C:,j= l c { - l e i j ~ M , ( C )[By . convention, if ci = 0, then co = 1.1
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For example,
Proposition 2.3.26
(Vandermonde determinant).
I ~ . l . . . . - c r= I
nl