Polynomial identities in ring theory

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

This Page Intentionally Left Blank

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.

This Page Intentionally Left Blank

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-(<), in which we consider formal symbols ti in place of the number i, with the additional stipulation that (titj)= (tjti) for all i, j. Clearly every element of .d(() can be written uniquely in the form (ti,. . . tik),where i , < i2 6 . . . 6 i, ; ,&(<) is called the free commutatiue monoid. The Free Algebra @ { X }

We shall now find a method of building an algebra from a given monoid. Definition 1.1 ' 5 . For an arbitrary set H , define the &module freely generated by H , denoted 4 H , as ahhlahE 4 and ah = 0 for all but a finite number of h } , endowed with operations

(xhr,,

Z a h h + l P h h = X(ah+Bh)h,

a ( l a h h )= E(ctah)h

for a E 4 .

Remark 1.1.6. + H is a 4-module, with 0 element XhE,,Oh and additive inverse - (Ca,,h) = I(- ah)h. Proposition 1.3.7. respect to multiplication Proof.

I f H is a monoid, then + H is an algebra, with ( C f e H a J f ) ( z g E H b g g ) = x h s H ( Z f g = hafPg)h.

Routine verification of the axioms. QED

When H is a monoid, we shall call 4 H the monoid algebra. The elements of 4. U(X) are somewhat unwieldy, as formal infinite sums, so we shall make

{x

obviousidentifications;forexampleXidenotes a,,hla, = Ounless h = Xi,in which case ah = l}. Now we show that & / / ( X ) is the free algebra. Proposition 1.1.8. (i) free commutatiue algebra.

4 //(X)

is the free algebra. (ii)

4. &(()

is the

4

THE STRUCTURE OF PI-RINGS

[Ch. 1

Proof. (i) Let r r : X -,R be a set-theoretic map. We can extend cr uniquely to a monoid homomorphism 0’:-U(X) + R, setting of(I ) = 1 and cr’(Xi,... Xiu)= olXi,)..-cr(Xi,);define 6 by 6(xuhh)= Euho’(h).Obviously 6 extends cr. On the other hand, if rrl is a homomorphism extending cr then and then crl(xahh)= Xahcrl(h) = z a h O ’ ( h ) = 6 ( x a k h ) ,proving u1 = 6. An analogous proof holds for (ii). QED Definition 1 .I .9. Write 4{X} for the free algebra $.#(X), and 4 { X l , ,. . , X,} for the subalgebra generated by X I , . . . ,X,; write 4[;] for the free commutative algebra &@((). Z { X } (resp. Z[<]) is called the free ring (resp.free contmutatioe ring).

If 4 is a domain (i.e., has no zero divisors), then 4 [ ( ] is a commutative domain and thus has a field of fractions, which we denote +(t),the field of “rational functions in countably many indeterminates.”

Identities Definition 1 .I .lo. I f f g 4 { X ) , let f ( R ) = (images o f f under algebra homomorphisms from #(X} -+ R } . Let f ( R ) + be the additive subgroup of R generated by f(R ) .

Intuitively, each element of f ( R ) is the “evaluation” obtained by substituting an element ri of R for Xi. Remark 1.I .11.

If gEf(4{X)), then g(R) E f ( R ) .

Now we can get to the subject of this book. Definition 1.I .12. Iff(R) = 0, we callfan identity of R, and say, “ R satisfies (the identity)f”; when 4 = -7, j is called a classical identity of’ R.

Thus j ’ is an identity of R i f f f E n {ker $1$:4{X} -, R is a homomorphism} ; more intuitively, f is an identity iff each evaluation of f on R is 0. A related definition, also of much importance, is Definition 1 .I .13.

f’is R-central if 0 # f(R)c Z ( R ) .

Remark 1 .I .14. Suppose gEf(c${X))+.I f f i s an identity of R, then g is an identity of R. Iff is R-central, then either g is an identity of R or g is R-cent ral.

In order to talk precisely about identities, we need a few definitions; our object is to be able to describe some nice properties of X , X 2 - X 2 X , , the

$1.1.1

5

Basic Concepts and Examples

identity defining commutativity, so that we can require these properties in certain other identities. Definition 1 .I .I5. A n element f of 4 ( X } is called a polynomial. Writingfas Ia,,h, we call cth the coeficient of h. If ah # 0, then we call a,h a monomial of f , and ah is a coejJcient of f ; f is R-proper if, for some coefficient a of,f, aR # 0 (or, equivalently, a ’ 1 # 0). Definition 1 .I .16. Define deg‘(f) = max{deg,(h)lall monomials h of . f ]; degi(f) = min{deg,(h)lall (nonzero) monomials h of f } , and deg(f) = max{deg(h))allmonomials h off}. A polynomial f is homogeneous in X i if deg’(f) = deg,(f); f is linear in X i if degi(f) = degi(f) = 1 ;f is t-linear if .f is linear in each X i , 1 d i d f ; f is niultilinear if, for each i, either deg’(f) = 0 or f is linear in X i . Definition 1.I .I 7. A classical identity of R is called a polynomial identity if it is multilinear and if 1 is a coefficient; a ring with a polynomial identity is called a PI-ring.

(Of course, every polynomial identity of R is R-proper for every nonzero homomorphic image R of R. We shall elaborate on this observation in Theorem 1.6.46.) Definition 1.1.18. We say X i occurs in a polynomial f if degi(f) > 0; write f ( X , ,..., X,) to denote that X , ,..., X , are the only indeterminates occurring in f: For any algebra homomorphism b{Xf -+ R sending X i to r i , 1 < i < d, we writef(r,, . . . ,rd) for the image off.

Obviously f ( R ) = {J(rl,. . . ,r,)lall ri in R ) in view of Proposition 1.1.8. Thus to check that a polynomial j’ is an identity of R we must show f ( r l , .. .,rd) = 0 for all ri in R. For example, writing [a, b] to denote the “Lie product” ab-ba, we see . . .,X,) is R-central iff [X,., J], but not S, is an identity of R . that f(X,, Given a class of algebras {Ry!]vErj,, we define to be the settheoretic Cartesian product, with operations defined componentwise. Clearly nYErR, is also an algebra and is called the direct product of the R , . There is a canonical projection n Y :n R y+ R, given by taking the ycomponent of an element. A direct product of copies of R is called a direct power of R . We shall now see how identities pass from algebras to related algebras. Write R , G 4 R , if every identity of R , is an identity of R , (as $-algebras).

,

nlGrR,

Remark 1.1.19. (i) If R , E R,, then R 1 GsR,. (ii) If R , is a homornorphic image of R,, then R , < & R , . (5)If R , d , R for all y in r, then S + R . (Proof is an easy verification.)

mYEI.R,)

6

[Ch. 1

THE STRUCTURE OF PI-RINGS

These properties are fundamental to PI-theory and shall be used time and time again (without reference) to transfer information from one algebra to another algebra. Multilinearization

To understand better the definition of PI-ring let us examine multilinear polynomials. These are tied in with the symmetric group on m symbols, denoted Sym(m), which is the group of permutations of (1,. . .,m). (Given K,, n2 in Sym(m) we write n 1 n 2 to denote, “first n1 and then K,.”) If n is a product of elements of order 2, ca!led “transpoz ~ S y m ( r n )then , sitions;” if this product has length k, we write sgn for ( - l)k,well known to be independent of the particular product. Write (ij)for the transposition interchanging i and j. Also define the alternating group Alt(m) = {nESym(m)Isgn = l ) , a normal subgroup of Sym(m) of index 2. Remark 1.1.20. form

Every multilinear polynomial of degree t has the

GX,,;

* *

where each a, E qb .

X,,,

nES)rn(r)

We shall now examine the important process of multilinearization, which ..., X , ) in order to obtain a is applied to a given polynomial f(X,, multilinear polynomial with related properties. The description involves the evaluations off on the ring 4{ X ) ; we use the notation ,f(h,, h,, .. .,hd) for the polynomial obtained by substituting X i - h i , 1 ,< i < d , where hi are polynomials. In this way, f ( X , , . . .,X,) can be viewed as a function from + { X } ‘ , ) to 4 { X j , and for the purpose of facilitating later proofs we work in this more general framework. Definition 1.1.21. Given any algebra R and $ ( X , , . . . ,x d ) : R‘,’ -+ R, we define AijJI:R‘”‘) + R by

A i j $ ( X , . ., x d + 1 ) 9

=

any

function

+ ( X I *..., Xi- 1 , X i + X j , X i + 1, ..., x d ) - $ ( X I ..., xi - x j , xi+ 1 . . . , x d ) - $(XI * * . x i - 1, xi,x i + 1 7 . .. x d ) . 9

9

1 9

9

t

9

For example. i f f = X i X , , then A I 3 f = ( X , + X 3 ) 2 X 2 - X X : X 2 - X 3 X 2 =

x,x~x,+x~x,x,.

This procedure will be described in detail in $2.3; we collect now the most important properties for a polynomial f ( X , , . . .,X,,,). Remark 1.1.22. Suppose that degi(f) > 0, and let g = Ai,,,+, j : (i) degi(g) = deg‘(.r‘)- 1. (ii) If 1 < j d m and j # i, then degj(g) = degj(f).

$1.1.1

7

Basic Concepts and Examples

(iii) g E , ~ ( + { X } ) +(iv) . All coefficients of g are coefficients of ,f: (v) If deg'(J') = deg,(.f')= u, then g ( X , , . . ., X,,, Xi) = (2"-2),f(X,,. . ., X,,,). Remark 1.1.23. Suppose f has "constant term 0" (i.e., each monomial of f has degree >O). If deg'(f(X,, ..., X,)) = u, then Ai,,,,+ I S . . A ~ , , + ,f ~ - is a polynomial which is linear in the indeterminates X i , X,, l,. ..,X,+,Applying this procedure repeatedly yields a multilinear polynomial.

,.

Remark 1.1.24.

Suppose .f is an identity of R , having constant term is also an identity of R . In particular, if R satisfies an identity with all coefficients k 1, then R is a PI-ring. a. Then a ' 1 = f(0, .. . ,0) = 0, implying j - c c

By Fermat's little theorem, the finite field E / 3 Z Example 1 .I .25. satisfies the identity f ( X , ) = X:-X, (as well as the obvious identity [Xl,X2]). Let us linearize f step by step. First, take f,(X,, X,) = A , , f = (X:X2+X,X2X, +X,X:+X:X, +X,X,X,+X,X:). Then take .12(X1, x,, X3) = A23(11) = (X,X3X, + X , X , X , +X,X,X3+X3X1X,+ XI x2x3 +XI x3 X2). Definition 1 .I .26. The symmetric CnE~ym(t)Xn~ ...Xnr.

polynomial

on

t

letters is

Remark 1 .I .27. CnESym(l) X,, ... X,, is an identity of every finite field of t elements. (Proof: If F is a field of t elements, then F - 1 0 ) is a multiplicative group, so, by Lagrange's theorem, X I - ' = 1 for all x in F - ( 0 ) . Hence, X'-X is an identity of F and, as in Example 1.1.25, one sees easily that the multilinearization of XI- X is the symmetric polynomial.)

Normal Polynomials Actually there is a class of polynomials that is far more important Definition 1 .I .28. A polynomial f is t-alternating if the following condition holds for each i, j , with 1 < i < j < t : For any homomorphism $:4{X} - 4 { X } such that $(Xi) = $(Xj) we have $(f)= 0. (In other words, writing Xi in place of X j yields f ( . . . , Xi, . . .,Xi, . . .) = 0.) Definition 1 .I .29. A polynomial .f is t-normal if f is t-linear and t-alternating. Iff(X,, . . . ,X,) is t-normal, we callfnormal.

[ X I , X,] = X , X 2 - X , X , is a normal identity of every commutative ring. The alternating and linear properties are analogous to the alternating and linear properties of the determinant of a matrix. Consequently, one

8

[Ch. 1

THE STRUCTURE OF PI-RINGS

expects to accumulate considerable information concerning normal identities by mimicking parts of the theory of determinants. This idea will become a pillar of our structure theory. t the easiest The standurd polynomial S,(X,, . . . ,X,) = ~ . n G S ~ mI (. -t .)XXTnis exampleofa t-normal polynomial ;cf. Example 1.2.13and Remark 1.2.15below. Let us proceed with easy observations about verifying t-linear or t-normal identities. Given subsets Tl. ..., T of R and a polynomial .f'(xl, . . ., xd),we write ,/'(TIx ... x T) for { . f ( x l , .. ., Sd)I.yi E T } . We write T")for T x ... x 7; the Cartesian product taken i times. So . f ( R t d ' )is the . given sets A, B and a binary operation 0 : A x B -+ G for same as , / ' ( R ) Also, a i o hi for some additive set G, we write A o B for all elements of the form suitable ai in A , b, in B. For example, for a ring R , we write [ R , R ] for Ci[ril,ri,]. Similarly, define A' = A and, inductively, A' = A'A. In the , {finite same spirit, if A. 5 R , for each y in an index set r, define x , , E r A ,= sums of elements taken from the various A }. Note that any element of CycrA, lies in some finite sum of the A,. The one notable exception to this notational rule is that if A c R , we write R - A to denote { r E R l r $ A}.

xi

'

Remark 1.1.30. Iff(X,, . . . , X , ) is t-linear, x , , . . ., x,, r,+ l , . . ., r, are elements of R , and r i j € Z ( R ) ,then

Lemma 1.1.31.

Zff(X,, ..., Xd) is t-linear and T,, ..., T, are subsets

of R, then

Proof.

Immediate from Remark 1.1.30. QED

Proposition 1.1.32. Suppose R is spanned as a Z(R)-module b j a set B. To check that a t-linear polynomial f is an identity OfR (resp. is R-central), it sufices to showj(B'" x R(d-") = 0 (resp. 0 # f ( B @ x) R(,-')) c Z ( R ) ) .

Proof.

Immediate from Lemma 1.1.31. QED

Let us note an important special case. Call R a central extension of a subalgebra R , if R = Z ( R ) R l . Then Proposition 1.1.32 says that every central extension of R , satisfies all multilinear identities of R , . Note that if

$1.1.1

9

Basic Concepts and Examples

R is a central extension of R , , then Z ( R , ) E Z ( R ) . Thus, if R is a central extension of R , , which is a central extension of R , , then R = Z ( R ) R , = Z(R)Z(R,)R, = Z ( R ) R , , so R is a central extension of R,. Proposition 1.1.33. I f f ' ( X , , ..., X , ) is t-normal and T c R, such that Z ( R ) T is a Z(R)-algebra having dimension < t , then ,/'(T(''x R(,-')) = 0. Proof. Clearly we may replace T by Z(R)T. Let B = { b , , . . . ,b,- ,) span T. Some element of B must appear twice in the first t places of any evaluation of f ( B ( ' )x R ( d - ' ) ) ; since f' is t-alternating, obviously f'(B"' x R(d-")= 0. Hence, by Lemma I .I .31, , f ( T ( 'x) R'd-'') = 0. QE D

Examples of PI-Rings

We are ready for the main examples of PI-rings, to illustrate the subject matter. Example 1.1.34. Suppose p R

=

0 for a prime number p . Then for

4 = L,p X , is an R-improper identity of R . (Of course we could also take c$ = Z / p Z , in which case p X ,

= 0.)

Example 1.1.35. If R is an algebra of dimension < t over an arbitrary commutative ring, then every t-normal polynomial is an identity of R. Indeed, R obviously has dimension < t as Z(R)-algebra, so we apply Proposition 1.1.33 with T = R . A special case of this example, "matrix algebras," is extremely important and is the topic of $51.3, 1.4. Example 1.1.36. An element r of R is integral of' degree t (over 4) if r' = x t Z h a i r i for suitable elements ao,. . ., a t - l in 4. R is integral if each element of R is integral; R is integral of bounded degree t if each element of

R is integral of degree bt. (We take t as small as possible.) Long ago, Jacobson proved that every integral algebra of bounded degree is PI, as we show now. Proposition 1.1.37. l f ' R is integral of' degree G t , then ,for airy t-normal polynomial , f ( X , , . . .,xd)the polynomial

,!-([xi, X , ] , [x:-', X , ] , ' . . , [ X , , X , ] , XI+

17

. ' . X,) 7

is an identity OfR.

Proof. ai

in

For any x,, x2 we know that .Y: = ~ ~ : ~ c t ifor x \ suitable [.Y:, x2] = 1;:a i [ x i , x,] (since [ao, x,] = 0). Thus, letting

4, so

10

T

[Ch. 1

THE STRUCTURE O F PI-RINGS

=

{[xi,x , ] I 1 6 i

< t } , we have j'(T'''x R'"")

= 0.

QED

Thus S,([X',, X J , . . . , [ X , , X , ] ) is a polynomial identity of every integral algebra of degree < t , proving Jacobson's result. Example 1 .I .38. Let ,fl(Xl,X , ) = [ X , , X , ] and, inductively, setting = 2i-1 , let j , ( X ,...., X,,,,) = [./i-l(Xl,...., X,,,), . ~ ~ - , ( X ~ + I . . . . , XR , ~ ) ] . is Lie solvable of degree i iff ,f, is an identity of R.

,

Example 1.1.39. Let

.fb= x , ,

. f l ( x , , X , ) = [ ~ l ~ ~ , and 1~

./;~=[.f;-l.X;+ll

(inductively). R is Lie nilpotent of degree i iff ,fi is an identity of R, iff ,fiis R-central.

,

Example 1 .I .40. If R is the algebra of upper triangular 11 x n matrices over 4 (including the diagonal), then [ X I , X , ] [ X , , X , ] . . . [ X 2 n - 1 ,X,,,] is an identity of R. (Indeed, [ R , R ] is nilpotent of index n, so the assertion is immediate.) This identity has considerable importance in the Russian school of PI-theory. Example 1 .I .41. Let bE{x)= b{X)/(ideal generated by all X i X j + X , X , and X f ) . We call &(.Y) the exterior or Grassmarl algebra, and write x i for the image of X i in 6,I.x:. Write B = { x j l l < i < co) and B' - [xjl... xiu10,< u < co and il < ... < i"). For u = 0 we write "1." Each element of & [ x } can be written uniquely in the form x : ( c r , b l x , ~ & , h ~ B ' ) , where all but a finite number of the uh are 0.

Let us develop some easy properties of exterior algebras. If b = xil. . . x i " , we write deg(b) = u. Let 4E(.x}0 (resp. & { x ) ~ ) be the &submodule of bE{.x) generated by all monomials of even (resp. odd) degree. Remark 1 .I .42. If ~ E & { X } ~then , a x j = xiu for all x i . If ~ E $ J , ( . Y ) I , then axi = -xia for all x i . (Proof: We may assume U E B 'and then induct on the length of a.I Remark 1 .I .43. Z(&{x}) = 4,{xJ0. [Immediate from 1.1.42, in view ofthe fact that aEZ(&:{x))iffax, = xiu for all i.] Remark 1.I .44.

Remark

As Z(4,{x))-module, 4E{.~J is spanned by B u { 1 ).

Proposition 1.1.45.

[ X I , X , ] is &(.x).-central.

Proof. By Proposition 1.1.32, we need only [.xi, l ] ~ Z ( & ( x j ) and [.X~,X,]EZ($~{X)). But [xi, xj] E & { . Y ) ~ = Z(&{ x)). QED

Thus (bE{x; is a PI-ring and has a central polynomial.

show that each [xi. I] = 0 and

$1.2.1

Facts about Normal Polynomials

11

$1.2. Facts about Normal Polynomials

Having seen that t-normal polynomials figure prominently in finite dimensional algebras and in integral algebras of bounded degree, we should like to examine t-normal polynomials more explicitly, building on Remark . . .,X , ) is t-linear. Our 1.1.20. Assume throughout this section that f(X,, main object is to find a way to verify when f is t-normal. Definition 1.2.1. If n E Sym(k), k < t , define f ; k , , ) as the sum of those monomials off in which X I , .. . ,X k appear in the order X , , ... X , , ; write h k ) forf;k,l).

For n in Sym(t), write n of for

Definition 1.2.2.

f ( X , , , .’ .

x,,,x,+

..,X , ) .

1,.

f i s t-normal i f ( i j ) o f = -$for

Proposition 1.2.3. Proof.

5

all i < j d t .

Iffis t-normal, then (working in the i and j positions)

(ij)of+f=f( ..., x j)...)xi)...) + f (... xi, ..., xj, ...I = f ( ..., x i + x j , . . , x i + x,... j )-f( ..., xi,..., x i,...) -f( ...)x j)...)xj,...) = 0. )

Conversely, suppose (ij)of = -f for all i < j < t. Then, writing f = we have ( i j ) = - f ; r , ( i j p c ) for all n E S ~ m ( t )Thus, . letting G = {n~Sym(t)ln(i) < n(j)>,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 {<$)I1 < i,j < n, 1 < k < a}. Definition 1.3.5.

4,,{Y}

is the subalgebra of

M,(4[<])

generated by

Yk = (ti:)),1 < k < oc). Each Y, is called a generic matrix. #,( Y,, . . ., yk) is

the subalgebra of 4,,{ Y} generated by Y l , . . . , Yk.

16

THE STRUCTURE OF PI-RINGS

[Ch. 1

C induces a homoinorLemma 1.3.6. Ecery homomorphism $: 4[t] phism b,,{Y ) + M , ( C ) , given by componentwise specialization. -+

Proof. $ extends naturally to a map $: M , ( $ [ t ] ) + M,(C) (according to the entries), which we restrict to cj,{ Y). QED Proposition 1.3.7. b,,{Y} is ,free .for all 1natri.u algebras (over a commutative &algebra). Zn{Y } is freefor all matrix rings. Proof. Suppose R = M , ( C ) and r l , r2 , . . . e R . Write rk = (cf.:’) for each k . If o(&) = r, for some map o, then we define the algebra homomorphism C J ’ : ~ [ -(+ ] C by = c$’, and we obtain the first assertion by using Lemma 1.3.6.The second assertion is immediate (with 4 = Z). QED

d(
Of course, Z,{ l‘} is not itself a matrix ring, and we shall see in Chapter 3 that all nonzero elements are in fact regular. Nevertheless, we call +,,{ Y } the dgebra qf’generic ( 1 1 x 1 7 ) matrices. Remark 1.3.8.

4,,{ Y } <,,,Mn(6[(]). For any commutative algebra C , M , , ( C ) <,,,4,,(Y } .

4,,{Y )

Proposition 1.3.9. Proof.

S M , ( $ [ r ] ) , so

Immediate from Proposition 1.3.7. QED

I f $ : $ { X } -+ +,,{Y) is the homomorphism sending = {identities of M , ( 4 [ 5 ] ) }= {identities of $,{ Y ) ) .I n particular, 4,,{Y j z 4 { X } / { i d e n t i t i e so f ’ M , , ( 4 ~ [ < ] ) ) . Proposition 1.3.10.

Xi++ .for all i, then ker$

Proof. Let W = {identities of M , ( 4 [ 5 ] ) . By Remark 1.3.8 and Proposition 1.3.9, W = {identities of 4,,{Y}}, so W c ker$. But ker$ c W by Proposition 1.3.7. QED Theorem 1.3.11.

4,: Y }

is free

in

the class of all

algebras

S.,>Mfl(+Ct1). Proof.

View $,,(Y} as @{X}/{identitiesof M , ( 4 [ ( ] ) . Given a ring R

< M,(I$[~]) and elements r l , r 2 , . . .in R, define a homomorphism $: $ { X ) -+R such that .Yi-ri, I < i < oc. Then k e r 2 ~ (identities of R } 2 {identities of M,,i@[t])]., so $ induces a homomorphism $:@,{ 1.1 R such that $(

x ) = r i for all i.

-+

QED

Theorem 1.3.1 1 is very significant because, as we shall see, the class of rings < M,,(Z[<]) is very large. Modifying the Algebra of Generic Matrices

Suppose F is a field, and A is a commuting indeterminate over F. Given M , ( F ) , we view M J F ) E M,(F[A]) in the natural way and consider

OE

$1 3.1

Matrix Algebras

17

( l - a ) ~ M , ( F [ l ] ) ; de t (l -a ) is an element p ( l ) of F[,l], called the characteristic polynomial of a. There is some finite extension of F in which p ( l ) splits into linear factors; since p ( l ) has degree n, p ( A ) has n (not necessarily distinct) roots, the characteristic values of a. (The uniqueness of the characteristic values follows easily from Galois theory, but is irrelevant to the subsequent proofs.) Note that we identqy F [ l ] with Z ( M , ( F [ l ] ) ) . Let us now consider "diagonalization" of a matrix. Write F(")for the free F-vector space of dimension n. M , ( F ) is identified with the F-algebra of linear transformations of F'"'; in this way F(") becomes a right M,(F)module. Changing the basis of F(") by means of an invertible linear transformation b changes each matrix u to b - ' a b , as is well known. If a has distinct characteristic values a,, . . . , a,, in F , then it is very easy to show (cf. Herstein [64B, 46.21) that we can choose b such that b - ' a b is diagonal; in this case, a is called diagonalizable. The following fact simplifies many considerations about the algebras of generic matrices. Recall that elements c , , . . . ,ct of a commutative algebra are algebraically independeizt iff(c,, . . ., c,) # 0 for every nonzero f ((,,. . ., 5,) in 4[[]. Lemma 1.3.12 (Procesi). Let 4 be a commutative domain. The characteristic values ofthe geireric rnalrix Y, = L$)ejj ( ~ 4 ,Y, () )are distinct.

Proof. A polynomial p ( A ) has distinct roots iff p is relatively prime to its derivative. If the characteristic polynomial det(l - Y , ) of Y, had multiple roots, then for every matrix s E M,(4[5]), de t(l- s) would have multiple roots, which is absurd (since we can take s = c ( { f ) e i i ) . QED

In fact, the characteristic values of Y, are algebraically independent (cf. Exercise 1). Let 4 be a commutative domain. Then for some Proposition 1.3.13. suitablefinite extension field F qf 4 ( ( ) ,the generic matrix Y, is diagonalizable in M,(F). Proof.

Immediate from Lemma 1.3.12. QED

Definition 1.3*14m

4,,{Y ' } is the homomorphic image of 4 , { Y ) obtained by specializing ( 1 f ) ~ Ofor all i # j. Write Y; for the image of Y,. Proposition 1.3.15. in the class of all algebras

I f 4 is a commutative domain, then 4,,{Y ' } is,free <,.,,M,(4[(]).

Proof. In view of Theorem 1.3.11 we need only show that there is a homomorphism $:4,{ Y ' ) -, Y ) with $ ( Y ; ) = Y, and (I/(&) = & for all k > 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


Let us isolate a key feature of the above argument. Remark 1.3.16. If a, h c M , ( F ) and b is .f({b-'ab) x M , , ( F ) ( d - l ' )= b - ' j ( { a } x M,(F)(d-'')b.

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 <2n-

Theorem 1.4.5 (Amitsur-Levitzki). 1f.f is a multilinear identity qf M,(d,) of degree < 2n then,for some u in 4, f = as2,. Proof. By Lemma 1.4.3, we may assume deg(,f) = 2 n ; write f = ~ , E S y m , 2 n ) u n. .X. Xn nl ( z n lBy . Corollary 1.2.4, it suffices to prove for all i < 2 n + 1 thatcqilCl,,,= -a,.Useamodifiedstaircase:Ifiisodd,writei = 2k- 1 and get

O = f ( e , l . , , , e , l , n 2 , . . . , e n ( &l).ak,enk.nkrenk.nk,enk.n(k+ = (a, +a,ii + 1 )n)eln;

l)~”’~‘nn.~n)

$1.4.1

Identities and Central Polynomials

23

if i is even, write i = 2k - 2 and get = .f‘(en I , n 2 7 ’ . ’ - (an + a(ii+ 1 )n )ex

- 1).nk, enk.nk* enk.nk. enk.n(k

9

1 .n 1

+1

)?

’ ’ ’9

enn.nn? enn,n I

1 ,n I

. QED

Now we can measure how near a matrix ring is to commutativity in terms of degrees of identities; if R satisfies an identity of degree k and R = M , ( C ) , then n d [ k / 2 ] . For example, a matrix ring (over a commutative ring) satisfying an identity of degree 3 must be commutative. Many later results, including Kaplansky’s theorem, may be viewed as attempts to generalize this theorem, and the “staircase” will be a key feature in their proofs. Example 1.4.6. Suppose G is a group. Viewing G as a monoid, we form the monoid algebra $G, which is called a group algebra and is writtefi 4 [ C ] . Suppose G has an abelian subgroup A of index n, i.e., G is a union of n right cosets of A . Then $[GI is n-dimensional over the commutative subring $ [ A ] , so by Proposition 1.3.23 and Amitsur-Levitzki S , , is an identity of #C. In characteristic 0 this result is sharp, in the sense that if $[GI satisfies a polynomial identity of degree 11, then G has an abelian subgroup of finite index; the characteristic p case is also known, and a thorough treatment is given in Passman [77B]. We do not prove this characterization, but only remark that the ultraproduct construction in $7.5 enables one to reduce the assertion to the case that G is finitely generated.

The Role of the Capelli Polynomial

Example 1.4.4 raises the question, “Is there a polynomial that is nof an identity of M , ( F ) but is an identity of every proper subalgebra?” The answer is, “Yes” and is quite easy, using the Capelli polynomial. Proposition 1.4.7.

For e w r y conirnutative ring H , en,€ C , , 2 ( M , ( H ) ) .

Proof. Let t = n2. We order the t matric units e i j by the word ordering on the words ( i j ) . Thus e l l < e l , <*... < el, < e21 < ... < en,,?and we write ek for the kth matric unit. Consider C,,(e,,. . . , e , , e , ,. . . , e t ) = ~ n e S y m , t ) ( s g 7 c ) e... n l e,,e,. el Now elen2e2= ellen2eI2, which is 0 unless en, = e l l . Likewise, e2en3e3= 0 unless en3 = e21.Continuing in this manner, we , e 0, , seethat thereispreciselyonechoiceofr,, ,.... e,,suchthat e , e , , e , ~ ~ ~ e , # and this choice determines 7c (and forces enl = e n , ) . Thus C2,(el,... , e , , e l , .. . ,e,) = ken,,. QED

Now we can derive a very nice property about identities of subalgebras of matrices.

24

THE STRUCTURE OF PI-RINGS

[Ch. 1

Theorem 1.4.8. CZn2 is not an identity of M , ( F ) but is an F-idenrity of every subalgebra,jor a l l j e l d s F .

Proof. Every subalgebra has dimension < n z , so we combine Proposition 1.4.7 with Example 1.1.35. QED Let us backtrack to improve Theorem 1.4.8. Proposition 1.4.9. ring H .

C,,,(M,(H))+ = M,(H) for every commutative

Proof. Let t == 11'. Using multilinearity of C,,, we only need prove that e i j ~ C , , ( M , ( H ) )for + all i, j . By Proposition 1.4.7 and symmetry we have all e i i E C z t ( M , ( H ) )Now . for j # i, (1 - e i i ) ( l + e i j )= 1 ; hence by Remark 1.3.16 , eij = (eii+eij)-eii~C,,(M,(H))+ e i i + e i j= (1 - e i j ) e i , ( l+ e i j ) ~ C , , ( M , ( H ) )so QED This result is \'cry useful, because it gives the following neat way of making central polynomials n2-normal. Proposition 1.4.10. I f g ( X , , . . .,Xd) is l-linear and M,(H)-central, then g(C2,z(X,,.. ., x,,'), xZnZ+ . ., X2,'+d-l) is n2-normal and M , ( H ) central.

Proof.

Straightforward from Proposition 1.4.9. QED

Central Polynomials, Featuring gn Here is the only central polynomial for M,(Q), n # 1 , that was known before 1972! Example 1.4.11. For every commutative ring C , [X,, X,]' is M 2 ( C ) central. [Indeed, for all x1, X,EM,(C), tr[x,,x,] = 0, so [ x 1 , x 2 l 2 ~ C by the Cayley-Hamilton theorem.]

We are ready for the development of central polynomials. The central polynomial we give now (following Amitsur [77]) relies only on fundamental properties of t-normal polynomials, in particular the Capelli polynomial, and in the course of its development we gain valuable insights into how t-normal polynomials act on matrix rings. The original central polynomials discovered by Formanek, as well as some other related central polynomials, are given in Appendix A and its exercises. Theorem 1.4.12. Suppose C is a commutative algebra and f ( X l ,..., X,) is r-normal. Gioen elements x1,..., x, qf M,(C), let V = C f = Cxi, l and lat T : V -+ V be a C-linear transformation. Viewing T

4 1.4.1

25

Identities and Central Polynomials

canonically as an image following,formulas;

of' a

t x t marris (cf. Proposition 1.3.1), we hace the

(i) (det T ) f ' ( x , , . . . , x d ) = , f ' ( T . x l .., , . T-Y,,.X,+~, . . ., x d ) ; (ii) ( d e t ( I - T ) ) f ( x I , . . . , .Y,,) = . f ( ( A - T ) x , ,..., ( A - T ) X ~ , X ~...,. . +x~d ), (where I is a commutative indeterminate iliewed as a scalar t x t matrir); (iii) If'C:=o( - l ) i o $ - i is the charucterisric pnlynornial of'T, with clo = I , then,for 1 < k < n a,f(x,,

. . . ,X d ) = ;, .+ ...1+; - .J'(T''x,,. . . , Titx,, XI.+

. ,x d ) ;

,-k

(iv)

tr(T)f(x,, . . . , x d ) =

l , f ( . ~ l , .. . ,x i -

(i) Suppose T x j =

Proof.

=

2

cnl ,I

'' 'c

. ., xd).

Txi,x i +

x3:=c i j x i .Then

n ~ ~ t f ( XI ?r

.' '

9

-Ynf?

lr

...

9

xd)

3

...

n ESym(r)

= =

(sg 7c)cn 1.1 ' . c r i , i . f ( x '

n ESym(f)

(det T)f(xl, . . . , x d )

13

. . x ~xi, + 1 . ?

9

xd)

by definition of determinant.

(ii) Follows from (i), using the C[I]-linear transformation place of T. (iii) Follows from (ii), matching coefficients of (iv) Take k = 1 in (iii). QED

(A - T ) in

Theorem 1.4.12 has far-reaching applications in the theory of prime PIalgebras, to be discussed later in this section. For the time being, we use it to derive a central polynomial of Razmyslov [73a]-Bergman [75P]-Amitsur [77]. Corollary 1 .4.13.

!fa, h, sI, . . . ,s d E M,,(C)and if'j'is n2-normaI, tkeri It 2

tr(a)tr(b)f(x,, . . . ,x d ) =

C , f ( x , , . . . ,a.xib,...,x d ) . i= 1

Proof.

As usual it suffices to assume C is the free commutative ring Z[(] and that each x i is the generic matrix yi, 1 < i < n 2 . Then it suffices to assume C = Z ( 0 , a field, over which Yl, . . . , & is clearly a basis. Consider the F-linear transformation T:M , ( F ) --* M , ( F ) , given by T ( x ) = axb. Writing a = (aij) and b = (bij), we have T ( e i j )= auibj,,eu,, whose coefficient of e j j is aiibjj. Hence tr(T) = ajibij= a,;) (Cy= b,) = tr(a) tr(b), and Theorem 1.4.12(iv) yields the desired assertion. QED

x:,,.=

(xy=

zitjx

26

[Ch. 1

THE STRUCTURE OF PI-RINGS

Theorem 1.4.14.

a

Given

multilinear

n2-normal

polynomial

f(x,, . .., xd), we have a multilinear polynomial ,?(x,, . . ., x d + i ) such thatfor every commutative algebra C and all x l , . . .,xd, a in M,(C), f ( x l , .. . ,xd?a ) = tr(a)tr(f(x,, . . . .xd)).

Proof. Write

We may assume xl,. . . , x d , a are generic matrices. Let t

i= 1

n2.

i

f

1

=

. . ., xi- L

f(xI7

1

xd+

1 xixd+ 2 , xi+

1,

.,'

7

xd)

=

1

,hl x d +

1xixd+

2.h.2*

i= 1

for suitable polynomials.fil(Xl,. . .,X i . . , X d )and.fi,(X,, . .., X i - I, ., X d ) .Then by Corollary 1.4.13, writing Xi+

.. ., x i - xi+ 1, . . ., xd) = 111 ri2 = f l A x l , . . .,x i - 1, x i + 1,. ..,xd) ril

1 9

9

we have for all b i n M , ( C ) tr(a) tr(b)f(x,, . . .,xd) =

i

I

i= 1

i= 1

1f ( x l , . . .,ax,b, ...,x d ) = 1 ri,axibri2.

Taking traces of both sides, we have tr(tr(U) tr( f'(X,, ... ,X,j))b) = tr(a)tr(f(x,, . . . , x d ) )tr(b) i

= t r ( t r ( a ) t r l b ) f ( x , ., . .,x,)) =

i

tr(rilaxibriz)= i= 1

tr(rizr,1axib) i= 1

so tr((tr(a)tr(f(x,,. . ., rizrilaxi)b)= 0. Thus by Remark 1.3.2 tr(a)tr(,f(.ul,...,x,)) = rizrilax,; letting .f = , . / i 2 . / , 1 ~ d t ,x,, we have proved the theorem. QED

x:=,

Corollary 1.4.15. Suppose f is n2-normal and multilinear and f(M,(H))+ = M,(H). Then in the notation of Theorem 1.4.14 ,i' i s M,(H)central.

Now by Proposition 1.4.9 we could t a k e f = CZnL, to conclude that (;2n2 is M,(H)-central for every commutative ring H. For the remainder o f t h e book, we write 9, for the n2-normal polynomial obtained from C2nLusing Proposition 1.4.10 Thus gn also is Mn(H)-central. Remark 1.4.16. Iff is 1-linear, then f ( R ) + is a Z(R)-module. [Proof: If r,, . . . ,r, 6 R and z E Z(R), then z f ( r l , . . .,r,) = f ( z r , , . . . ,rm)E ~ ( R ) . ]

4 1.4.1

Identities and Central Polynomials

Theorem 1.4.17.

27

For every commutative ring H , g , ( M , ( H ) ) +

=

H.

Proof. We have shown 0 # g , ( M , ( H ) ) c H for every commutative ring H . Let I = g , ( M , , ( H ) ) + , an ideal of H , by Remark 1.4.16. If I # H , then g , ( M , ( H / I ) ) + = 0, contrary to the first sentence of this proof. Thus I = H . QED (Actually, we could use C 2 , , - instead of 9,; cf. Exercise 9.) Let us throw in a useful, easy fact about identities and central poiynomials of matrix rings. [f g ( M , ( H ) ) c H , then g is an identity o f M , ( H )

Proposition 1.4.18. .for all t < n.

x:,j=l

Proof. The t x t matrices xk = aijkeijcan be viewed as n x n matrices in the obvious way, by adding extra zeros for row n and for column n. So 0 = e,,,g(xlr... ,xd)enn,implying g ( x l , .. ., xd)= 0 [since g ( M , ( H ) ) c HI. Hence g is an identity of M , ( H ) . QED

Corollary 1.4.19.

M , ( H ) G hM , ( H ) f o r all t d n.

Properties of n2-Normal, Central Polynomials of Arbitrary Rings Now that we have g,, let us see what we can do with it. (Here we follow Amitsur [75] and Rowen [75P].) Writing g, as gn(Xl,. . .,x d ) for suitable d , define n2+

gb(x17 ...

9

x d + 1)

=

1

1 (-

)ign(xl

9

. ., > x i -

1 7 xi+

1 3 . ' ' 3

xd+

1

.

i= 1

+

Obviously g; is (n2 1)-normal (cf. Corollary 1.2.7) and is thus an identity of M , ( H ) . But we can translate this into a statement about matrices. Let t = n2. Suppose xl,. . ., x, are given in M,(F) for F a field. In view of Theorem 1.4.12(i),x l , .. . , x t are a basis of M , ( F ) iff there exist x , + ~..., , xd such that g,,(xl,.. .,xd) # 0. Thus g, distinguishes bases from nonbases. But moreover, setting z = g n ( x l ,..,xd),we have for any element x in M,(F), 0 = gb(.x,

. . ,xd) = -zx

+

f

( - l)i-lgn(x,

. . )xi- 1 , x j + 1 , . . . ,.xd)xi,

i= 1

so x = (( - l)i-lg,,(x,x l , . . . , x i - 1 , x i +l , . . . ,xd)z-')xi. In this manner we have explicitly computed the coefficients cli in F such that x = mxi. This is quite remarkable, especially in view of the fact that once we had the polynomial gn the rest was formal. Let us now use the same argument in general, for an arbitrary algebra R . Lemma 1.4.20.

Suppose C2,+, is an identity of R , with t = n2. For each

28

[Ch. 1

THE STRUCTURE OF PI-RINGS

t-normal polynomial g ( X , . . . . ,X,,), and for all r l , . . . , r d . k in R , I

g ( r l , .. . , rd)rd

I

=

C (-

,,

I)i-+lg(rd+r l r . .. , r i -

i= 1

ri

l,.

. . ,r d ) r i .

Proof. Let $ ( X I , . . . ,X d + I ) = ( - l ) i g ( x l ?... ) xi- 1 , . .. X d ) x i . Clearly g is (t + 1)-normal and thus is an identity of R by Proposition 1.2.12. Hence i(r,,+ r , , r 2 , .. . ,r,,) = 0, and the assertion follows immediately. QED 9

This easy fact is at the heart of much PI-theory because if we have an n2-normal, R-central polynomial g, we now see exactly how an arbitrary element r d + is Z(R)-dependent on a given set r l , . . . ,rd. This idea is used in the next result. Theorem 1.4.21. Let t = nz. Suppose Cz,+ is an identit). of R , and R has a t-normal, R-i.entra1 polynomial g ( x . . ., xd), For ellery element x of g(R)R, R x is conrained in a finite-dimensional Z(R)-submodule of R. I n particular, if 1E g ( R ) R , then R is a .finite-dimensional Z(R)-module. Moreorer, ij' .Y E g ( { r , } x .. x (Iz) x R',,"')R and s r # 0 jbr all nonrero r iu R , then r l , . . . , I , are a basis of a,free Z(R)-submodule of R containing R r ; if.. = 1, then R is itselfthis free Z(R)-module.

x:=

Proof. Write x= g(rjl, . . . ,r j d ) x j for suitable elements r j l , .. . ,rid, x j of R . We shall show that R x G I:=, Z(R)rji. proving the first assertion. Well, for any r in R, by Lemma 1.4.20 k

rx=

k

x!=l

l

1 r x j g ( r j,,..., r j d ) = C 1 ( - I y + ' g ( r x j , r j 1,...,r j , i - l , r j , i t,,..., rj,,)rji, j = 1i= 1

j= 1

as desired. In particular, if 1 Eg(R)R, then taking x = 1 we see that the rji span R as a ZfRkrnodule. Now suppose, furthermore. for 1 d i < t that rji = ri for allj. Then as just shown, r l , . . . ,rl span a Z(R)-submodule of R containing R x . On the other. hand, suppose ziri = 0 for suitable zi in Z ( R ) . Then for each i d t we have

xi=

-Cg(r, -

...., r , - l , z i r i , r j +,.... l r l , r j , , +,..., l r.)~-=z.\ i d - j 1 . 7

J

so zi = 0; thus r , , . . , r, are Z(R)-independent. QED

4 1.4.1

29

Identities and Central Polynomials

This approach has another crucial result. Lemma 1.4.22. Suppose R has an n2-normal central polynomial g ( X , , . . . , X,,), and CZnz+ is an identity of R. Then.for every c in g ( R ) and for every ideal d of R we have cd G ( A n Z ( R ) ) R . Proof. Write c = g ( r l , .. .,r,,) for suitable ri in R. For any a in A we (' g ( a ,r , , . . . ,ri- ri+ . . . , r,,)ri, have by Lemma 1.4.20 ca = implying caE (d n g ( R ) ) RE ( d n Z ( R ) ) R . QED

xi=,

,, ,,

Theorem 1.4.23. Suppose C?,,I+I is an identity of R and g is an ti2-normal,R-central polynomial such that I E g ( R ) R .Then,for every ideal d of R we have A = ( A n Z ( R ) ) R . Proof.

Immediate from Lemma 1.4.32. QED

Thus under hypotheses of Theorem 1.4.23 not only is R a finite Z ( R ) module but also we have a 1 : 1 correspondence from {ideals of R } to {ideals of Z ( R ) } ,given by A -+ Z ( R ) . We shall now see that this correspondence is onto, and shall record other, easily related results about algebras with I-linear central polynomials. Theorem 1.4.24. Suppose R is an algebra with a 1-linear central polynomial g ( X , , . . . ,X,) and write G = g(R). Let A , B be additive subgroups OfZ = Z ( R ) .

(i) ( A R n Z ) G = AG. (ii) I f A, B are ideals ( ( A + B ) n Z ) GG A+B.

xi

(i) Suppose z = airiE Z for suitable a, in A. For any element in G we have

Proof.

c

of' R , with d n G c A and B n G G B, then

= g(x,, . . . ,x,)

zc

= zg(x,, =

..., x,)

= g ( z x , , .. ., x,)

1aig(rix,,. . .,x,)

E

= g ( ( E a i r i ) x l.. , .,x,)

AG .

i

Hence ( A R n Z ) G G AG, and the reverse inclusion is immediate. and & E B ,such that c i + h ~ Z For . any c = g ( x l , ..., x,) (ii) Let c i ~ A G , we have (ci+ 6)c = g( (ci+&)x,,, . . ,x,) =g(cix, ,..., x,)+g(6xl ,...,. x , ) E A n G + B n G s

A+B.

in

QED

Corollary 1.4.25. Zf g is 1-linear and R-central with 1 E g ( R ) + , then for every ideal A o f Z ( R ) we have A R n Z ( R ) = A.

Proof.

Immediate from Theorem 1.4.24(i). QED

30

[Ch. 1

THE STRUCTURE OF PI-RINGS

Let us collect some of the results obtained above. Theorem 1.4.26. Suppose R has an n2-normal central polynomial g , with 1 Eg(R)+,and suppose that CZn2+,is an identity of R . Then there is a 1 : 1 correspondencefrom {ideals of Z ( R ) } to {ideals of R } given by A + AR, and the inverse of this correspondence is the map A’ + A n Z ( R ) .Moreover, R is afinite-dimensional Z(R)-module that is free if 1 E g(R).

Having obtained valuable information from identities of matrix algebras, we should thus like to focus on the formal connections between rings through the identities they satisfy. Write R , <mult9R2if R , satisfies all multilinear identities of R, (as &algebras). Definition 1.4.27. R , and R , are mult-equivalent over 4 if R , <mull(bR2 and R , Gmmultd,R,. R, and R , are equivalent over4 if R , d e R , and R , d g R , .

It will turn out that the interesting instances of mult-equivalence and equivalence rarely depend on the particular choice of 4, so, for convenience, we shall often omit “over 4” when talking of mult-equivalence and equivalence. Example 1.4.28. Proposition 1.1.32.

R is mult-equivalent to every central extension by

Example 1.4.29. For every commutative ring H,M,,(H) M,(B), because multilinear identities of M , ( H ) are checked by evaluations on matric units.

R has PI-class n if R < m u l t Z M,(Z) and g,(R) # 0. Definition 1.4.30. We immediately have the following basic information about PI-class. Remark 1.4.31.

For every commutative ring H,M J H ) has PI-class

n. All central extensions of an algebra have the same PI-class. If R has PI-

class t

< n then by Corollary 1.4.19and Example 1.4.29R

<mu,tk

M,,(Z).

Every algebra of PI-class n has the n2-normal, multilinear, central polynomial g,,. In view o f Theorem 1.4.21, 1.4.23,and 1.4.24,a major goal in our analysis will be to demonstrate that a given algebra has some PI-class n. As one example, let us restate a special case of these theorems. . Theorem 1.4.32. Suppose R has PI-class n, and l ~ g , ( R ) + Then there is a 1 : 1 correspondencefrom {ideals of Z ( R ) } to {ideals of R } , given by A -+ AR, and the inverse of this correspondence is the map A’ -+ A n Z ( R ) ; also, R is a finite-dimensional Z(R)-module. Moreover, if g,(R) contains an invertible element, then R is a free Z(R)-moduleof dimension nz.

91.5.1

Primitive Rings, Kaplansky’s Theorem

31

Linear Dependence through the Capelli Polynomial Here is a very simple property about vector spaces, possibly infinitedimensional, that yields a very important property of the Capelli polynomial. The idea is to reduce the verification of linear dependence of vectors to t components.

nIYsl.

Lemma 1.4.33. Suppose V = F,, where each F, zz F ; i.e., V is a direct product of (possibly infinite) copies of F. Suppose we are given ui = (ayi),1 d i d t . Then u , , . . . ,u, are F-independent i f f o r some y I , . . . ,y, the vectors (ayli,..., aYti),1 < i < t, are F-independent.

Proof. (e) is trivial. Conversely, assume u l , . . . , u, are F-independent. By induction on t , we can find y l , . . . , Y , - ~ such that the vectors (ayli,..., a,*_Ii), 1 d i d t - 1, are F-independent. For convenience write f l i j for 1 d i < t, 1 d j d t - 1 , and let b denote the ( t - 1 ) x (t-1) matrix (Bij), 1 d i , j 5 t - 1. Then det(b) # 0. We need to find some y, such that the (flil,. . myti), 1 d i d t, are F-independent. So assume the contrary; i.e., for all y we can find elements ~ ( y ) ~1 ,d i d t - 1, such that fllj = p(y)ipij, 1 d j d t - 1, and a,, = 1;:p(y)ia,i. Treating the ~ ( y )as~ variables in the equations f l r j = z i i ; p ( ~ ) ~ fwe l ~ have ~ , (by Cramer’s rule) a unique solution for the ~ so for (independent of y), because det(b) # 0. Thus we can write ~ ( y =) pi, p i a y i ,proving u, = 1;:piui. QED all y, a,, = Theorem 1.4.34. Given r l , ..., r, in R = M,(F), the following statements are equiualenr: (i) r I ,. . . ,r, are F-independent ; (ii) C,, - , ( { r l } X . * * x { r , } x R ( ‘ - ’ ) )# O ; (iii) 1 ~ R C , , ~ , ( { r , } x ~ ~ ~ x { r , } ~ R ~ ~ ~ ” ) R . Proof. (iii)* (ii) is trivial, and (ii) (i) is clear because Cz,-l is t-normal. To prove (i) * (iii), assume r , , ...,r, are F-independent. Write rk = (a$)). By Lemma 1.4.33 for some i , , jl,. . .,i,, j , the vectors (a(ifil,.. . ,a!!)*), 1 < k d t , are F-independent. Writing b for the t x t matrix whose entry in the uk position is we see easily that, for any i, ~ j , l C ~ f - ~ ( r ~ , . . . , r , , ~ j , i ~j ~, e- lji~~ i) ~e=j, *det(b)ejj . .j . , e # 0. Thus ejjE RC,, - ( { r l } x . . . x {r,} x R“ - ‘ ) ) Rfor all j , proving (iii). QED

z:Z

This theorem, giving a polynomial condition for elements to be linearly independent, is very useful; a generalization is given in $7.6. $1.5. Primitive Rings, Kaplansky‘s Theorem, and Semiprimitive Rings

In this section we review the structure of primitive algebras to obtain Kaplansky’s theorem that primitive PI-algebras are “central simple.” In the

32

[Ch. 1

THE STRUCTURE OF PI-RINGS

course of the proof we develop techniques that immediately yield important information about maximal commutative subrings of central simple algebras. Then, developing properties of the Jacobson radical. we are able to analyze PI-algebras having Jacobson radical 0. Density

Suppose M is an R-module. Then End, M is an algebra [with respect t o p(ab) = a ( y P ) for all a in #, /3 in End,M and .v in M]. Note that End,(M"') z (End,M)"' for all t in Z". Also, for each y in M , Rq' is a

submodule of M. Call M irreducible if M has no submodules (other than 0, M); thus M is irreducible iff Ry = M for each v # 0 in M. Of course, for T = End, M, we see M is a right T-module under the action y p = p ( y ) for all Y E M and /)€EndRM. Thus we are led to study the endomorphisms of M as right module over a ring T, written End M,, made into a ring by multiplication (/I, o/),)(y) = p1(p2(y))for all v in M (cf. Remark 1.54 below). Lemma 1.5.1 (Schur's lemma). (i) I f M,, M, ure irreducible R-modules, then ecery nonzero $eHom,(M,, M,) is an isomorphism. (ii) If M is irreducible, then End, M is a division ring. Proof. (i) $(.V,) is a nonzero submodule of M,; since M, is irreducible, $(M,) = M,. Likewise ker$ is a proper submodule of M I and is thus 0, proving I) is an isomorphism. (ii) If 0 # BE End, M, then p is an isomorphism by (i), and it is easy to check that p-' E End, M. QED

We shall refer to modules over division rings as uector spaces, since t hey share many important properties, such as bases and dimension. Theorem 1.5.2 (Jacobson's density theorem). Suppose M is an irreducible module. and let D = End, M . I f y , , . . . q',, are D-independent elements of M and if y', ,.. . ,y; in M are arbitrary, then, .for suitable r in R , ryi = yj for all i, 1 < i < u.

.

Proof. Induction on u. The theorem is obvious for u = 1 ; assume the theorem holds for u - 1. Then, viewing ( y , , . . . ,q',- ,) in . M u -,, )an R-module, we have M('-l' = R ( y1 7 ' . ,yu- 1). Claim. There exists ru in R such that rYyV# 0 and rayi = 0 for all i # u. Indeed, otherwise the map $ : R ( y l , . . . , y u - , ) + M given by $ ( r ( y , , .. .,y u - )) = ry, is a well-defined R-module homomorphism, implying t,h~Horn,(A~f'"-'',M) = (End, M)'"-') 4 D"-"; writing $ as D'" - ' I , we have y , = t,h(y,,.. . ,y,- = 1::;yidi, contrary to ( d l , .. ., d , the assumed D-independence of y ,. . . ,y,. '

,

,

61.5.1

Primitive Rings, Kaplansky’s Theorem

33

The claim is established and, by symmetry for each j there exists rj with rl’yj # 0 and rjyi = 0 for all i # j. Since M is irreducible we have rJ such that r j ( r j y j )= y;. Let r = rJrj.Then, for each i, ry, = rjriyi = y:. Q E D

xy=,

Let us record now an easy restatement of the density theorem, for use in Chapter 7. Corollary 1.5.3. Suppose M is an irreducible R-module, v E Z i , and y , , . , .,y , E M are arbitrary. Let D = End,M, and suppose V is a finitedimensional D-subspace of M not containing y l . T h e n there exist elements d , = 1, d,, . . . , d, in D satisfying the,following property:

For each y in M there exists r in R such that rV = 0 and ry, = yd,, 1 6 i

< v.

Proof.

Let V’ be the D-subspace of M spanned by V and y l r . . . r y u . Enlarge a basis of V(over D ) to a basis yi,.,.,y; of V ’ such that y; = y,. Write yi = y(idij for dij in D, and put di = d,,, 1 < i < u. Thus d , = 1. By the density theorem there is r in R such that ry, = y d , and ryj = 0, 2 < j < k . Thus, for each i, ryi = (ry(i)dij= yd,. Q E D

x:=,

x!=

Remark 1.5.4. Let M be an R-module. There is a natural homomorphism $: R + End M,, given by IC/(r):yi+ ry for all y in M ; +(r) = 0 iff rM = 0 , so ker$ = Ann,M. Thus when Ann,M = 0, we can view R E End M,.

We say M is a faithful R-module when Ann, M = 0. Also, write [ M : R ] for the dimension of M over R . (This notation will also be used for right modules, depending on the context.) Corollary 1.5.5. Suppose R has a faithful, irreducible module M , and let D = End, M . T h e n one of the following situations holds:

(i) [M : D ] = t for some t < m, in which case R z M , ( D ) ; (ii) for every natural t , R has a subalgebra with M,(D) as a homomorphic image such that for any a in 4 with aR # 0 we have aM,(D) # 0 under the natural action (induced by the action of I$ on D). Proof. If [ M : D ] is finite then R = End M, z M,(D), yielding (i); otherwise, given t pick D-independent elements y,, . . .,yr of M , and let V = y,D+...+y,D and R, = {rERlrV c V } . There is an obvious homomorphism from R, onto End V, = M,(D) that yields (ii). Q E D

The converse of the density theorem is obvious. If D is an arbitrary division algebra and M is a right vector space over D such that R is a dense subalgebra of End M , , then M is a faithful, irreducible R-module. Now let us see where identities fit in. Let denote the greatest integer of a real number x.

[.XI

34

THE STRUCTURE OF PI-RINGS

[Ch. 1

Proposition 1.5.6. I f R has a faithful, irreducible module M , with D End, M , and if R satisfies an R-proper identity ,f of degree d, then [ M :D] < [d/2],SO R 2 M[,+.D](D).

=

Proof. Suppose [ M : D ] > [d/2].By Corrolary 1.5.5, for some t > [d/2] either R M , ( D ) or M , ( D ) is a homomorphic image of a subalgebra of R. At any rate, ,f is an identity of M,(D), which is impossible by Lemma 1.4.3. Thus [ M : D ] < [d/2],and we are done. QED

A ring is primititJe if it has a faithful irreducible module. A ring is prime if A B # 0 for all nonzero ideals A , B of R. Remark 1.5.7. R is prime iff Ann, L = 0 for every nonzero left ideal L of R, iff s1R.l2 # 0 for all nonzero elements sl, .y2 of R . (Proof is left to the reader; note that for any left ideal L of R, L R is an ideal of R.) Remark 1.5.8. Every primitive ring is prime. [Proof: Suppose M is a faithful irreducible R-module, and A , B are nonzero ideals of R. Then B M is a nonzero submodule of M , implying B M = M ; thus A ( B M ) = A M # 0, so A B # 0.1 Minimal Left Ideals

A left ideal L is minimal if t here is no nonzero left ideal L , c L. Zorn’s lemma does not imply the existence of minimal left ideals, even for primitive rings. The following results show why one wants a ring to have a minimal left ideal. Proposition 1.5.9. Suppose R has a minimal lejt ideal L. ( i ) L is an irreducible R-module. (ii) I f R is prime, then L is faithful as well as irreducible, implying R is primitive. (iii) If R is prime, then every faithful irreducible module M is isomorphic to L. (iv) I f R is prime, then every two minimal left ideals are isomorphic as modules.

Proof. (i) Clearly L is an R-module; any submodule is also a left ideal, so L is irreducible. (ii) Ann, L = 0, so L is faithful. (iii) Pick x,,# 0 in L. Then x o M # 0, so x0y # 0 for some y in M . Hence L y is a nonzero submodule of M , so L y = M . By Schur’s lemma, the R-module homomorphism .Y + x y is an isomorphism of L with M . (iv) Follows from (ii) and (iii) immediately. QED Corollary 1.5.10. If R is a prime ring with minimal left ideal L, then for every faitllful irreducible module M , End, M 2 End, L.

L

Let us apply these results to R = M,(D), where D is a division ring. Then Dei, is a minimal left ideal of R , and D z End, L.

=

xy=

41.5.1

Primitive Rings, Kaplansky’s Theorem

35

Corollary 1.5.11. If R z M , ( D ) .for some division ring D, then D is determined up t o isomorphism and ti is uniquely determined.

Proof. D is determined up to isomorphism by Corollary 1.5.10, and likewise 17 is determined since [ R : D ] = t i 2 . QED The Closure (or Splitting) of a Primitive Ring

We introduce now a very important technique in order to strengthen Proposition 1.5.6. Suppose R has faithful irreducible module M, and D = End, M. Since Z(D) is a field, we can apply Zorn’s lemma to prove that D contains maximal commutative subrings, easily seen to be fields. We focus on one particular maximum subfield F of D. We can view R c End M , c End MF; also we can view F z End M, (under the action taking arbitrary c1 in F to the map ?t:y + JJU for all y in M). Clearly [R, F] = 0, so we can form the subring R F of End M,. Then M is obviously an RF-module whose action extends the given module action of M on R. Any proper RF-submodule of M is an R-submodule, proving that M is irreducible as RF-module; moreover, R F c End M,, proving M is a faithful irreducible RF-module. Proposition 1.5.12. Notation as above. (i) R F is a dense subalgebra qf End M,. (ii) Z ( R F ) = F.

Proof. (i) It remains to show End,, M = F. Write D , = { p E Dl[P, F] Clearly F c End,, M c D,. For any d E D ] , F and d generate a commutativesubringofDcontaining F ; bymaximalityofF, weget d E F.Thus D , = F, proving F = End,, M . = 0).

(ii) Z(RF) G {p€EndMF1[p,R] = 0) c (End,M) 2 D. But F c Z(RF), and F is a maximal commutative subring of D. Hence F = Z(RF). QED With the same notation as above, R F is called a closed (or M-closed) primitive ring and is a closure of R. Although the closure of R is not necessarily unique, it is very useful, being a central extension of R. (Sometimes R F is called the “splitting” of R, but I do not like this terminology because End M , is not in general a matrix algebra.)

Kaplansky’s Theorem: Two Proofs and Applications t o Simple Algebras Lemma 1.5.13. If R, is a closed primitiue algebra satisfying an R,-proper identity of degree d, then R , = M,(F)for somejield F and some n

36

THE STRUCTURE OF PI-RINGS

[Ch. 1

6 [d/2]. More generally, every primitive algebra R sarisjjing a multilinear, R-proper identity (qdegree d is mult-equivalent to M , ( F ) for a suitable field F and some n < [cf/2]. Proof. Form R F as in Proposition 1.5.12, which is primitive and satisfies the same identity. Then by Proposition 1.5.6 RF z M J F ) for some I I 6 [d/2]. QED

R is simple if R has n o proper (nonzero) ideals, Definition 1.5.14. i.e., if RrR = R for each r # 0 in R . Remark 1.5.15. M,(D) is simple for any division ring D. If R is simple, then Z ( R ) is a field. Theorem 1.5.16 (Kaplansky’s theorem). Suppose R is a primitive algebra satisfjing some R-proper identity of degree d. Then R has some PI-class ti < [ i f / 2 ] , and R :M , ( D ) ,/or some ditGsion algebra D with 11’ = t Z [ D :Z ( D ) ] = [ R : Z ( R ) ] . Proof. Multilinearizing the identity, we have a multilinear, R-proper identity, so by Lemma 1.5.13 R is mult-equivalent to some M , ( F ) for suitable n d [ d / 2 ] .Thus R has PI-class n. By Proposition 1S.6, R has the form M , ( D ) . In particular Z ( R ) is a field. so 1 Eg,,(R)+.By Theorem 1.4.32, ti2 = [ R : Z ( R ) ]= t Z [ D : Z ( R ) = ] t Z [ D : Z ( D ) ] .QED

The “usual” proofs of Kaplansky’s theorem do not use Theorem 1.4.32 but rely instead on other methods of passing from RF to R. Here is one such approach. Lemma 1.5.17. Suppose R is simple and r , , rz are Z(R)-independent elements of R. Then there exist T i , , ri2 in R, 1 < i 6 k (for suitable k), such that rilrlri2= 0 and Ef=,rilr2ri2# 0.

c!=,

Proof. Suppose not. Then there is a well-defined map t,b: R r , R -, R , given by $ ( ~ i x i , r , x i z=) ~ i x i l r z x i(for 2 all x i , , x i z in R). Clearly t,b(rar’) = r$(a)r’ for all a, r, r’. But R r , R = R ; for u = $ ( I ) , we have for all r in R , ur = $ ( l ) r = $ ( l r ) = $ ( r l ) = ru, implying u E Z ( R ) . and then r z = $ ( l r , ) = u r l . contrary to hypothesis. QED

(This cute argument will be extended several times, culminating in Theorem 7.6.10.) To apply the lemma, if A c R, say elements x,,. ..,x, o f R are A-independent if aixi # 0 for all a, in A (unless a,, .. .,a, are all 0).

,

Theorem 1.5.18.

Suppose R E R , , R is simple, and there are elements < j 6 t. I f the xi are R-dependent, then they are Z ( R )-dependent.

x1,..., x, of R , such that [ x j ,R ] = 0, I

#l.5.]

Primitive Rings, Kaplansky’s Theorem

37

Proof. Induction on t . For t = 1 the result is clear because if r l x l = 0 then 0 = R r , Rx, = Rx,, so in particular x 1 = 0. Now suppose xi= r j x j = 0. If r, = ar, for some a in Z(R) then rl(xI+ c t x , ) + z i = 3 r j x j = 0 so, by induction, {x, + a x , , x 3 , .. ., x , ) are Z(R)-dependent, implying x,, x 2 , . . . ,x, are Z(R)-dependent, and we are done. So assume r1 and r2 are Z(R)independent. Then by Lemma 1.5.17 we have elements r i l , ri2 in R, 1 d i Q k such that for r; = C!= ril rjri2,we get r; = 0 and r; # 0. But clearly 0 = C;= rixj = z ; = , r f i x j , so by induction (x,, . . .,x,} are Z(R)dependent. QED

,

,

Corollary 1.5.19. If R is a simple subring of R, and Z(R) G Z(R,), then every Z(R)-independent set of elements of R is Z(R ,)-independent. Proof. Suppose, with t minimal, r , , . . .,r, are Z(R)-independent and x : = l r i a i= 0 with ai in Z(R,) and a1 # 0. By minimality of t the cq are Z(R)-independent, so by Theorem 1.5.18 the ai are R-independent, contrary to our supposition. QED

By Proposition 1 S.6 R 2 Alternate proof of Kaplansky’s theorem. M,(D) is simple, so, by Corollary 1.5.19 every Z(R)-independent set of elements of R is (RQindependent. But thus [R :Z(R)] = [RF:F] = n2 for some n 6 [d/2] by Lemma 1.5.13. QED To my taste, both proofs are worth knowing well. Here is an interesting application, which indicates the tremendous impact of Kaplansky’s theorem. Theorem 1.5.20. If R is primitive and has a commutative subring C with [R : C] d n, then R is simple and [R :Z(R)] d n2. Proof. By Proposition 1.3.23 and Amitsur-Levitzki, S,, is an identity of R, so we are done by Kaplansky’s theorem. QED

C

Corollary 1.5.21. If R is a simple algebra of PI-class n, and if 2 Z(R) is a subjeld of R, theri [ R : C] 3 17, so LC : Z(R)] < n.

Proof. By Theorem 1.5.16, [R:Z(R)] [R : C ] 3 11. QED

= 11,.

Thus, by Theorem 1.5.20,

Actually, from the proof of Kaplansky’s theorem comes even more information, if we just pause to collect it. Theorem 1.5.22. If D is a division ring of PI-class n, then for every maximal subfield F sf D, [ F : Z(D)] = n. Proof. = End M

Let M = D as a 1-dimensional right D-module. Let R , z D, of PI-degree n. Now R F z M,(F), where t = [ M : F ]

38

[Ch. 1

THE STRUCTURE OF PI-RINGS

= [D:F ] . But RF has PI-class n, so t = n. Hence [ D :F ] = n, so n2 = [ D : Z ( D ) ]= [ D : F ] [ F : Z ( D ) = ] n [ F : Z ( D ) ] . Thus [ F : Z ( D ) ]= n.

QED Suppose an arbitrary division ring D has a niuximal Theorem 1.5.23. subfield K such that [ D : K ] = n. Then D has PI-class n, and [ K : Z ( D ) ] = n. Proof. By Theorem 1.5.20 D is PI, of some PI-class t < n. By Theorem 1.5.22 [ K : Z ( D ) ] = t. Thus t 2 = [D:Z ( D ) ] = [ D : K ] [ K : Z ( D ) ] = nt, implyi n g ? = n. QED

Primitive Ideals and the Jacobson Radical

For any element y in an R-module M , Ann,?, is a left ideal of R, and the map R --+ Ry given by r -+ ry is an R-module homomorphism with kernel Ann, y . Thus we have the next observation. Remark 1.5.24. If M is an irreducible R-module, then for every j in M , Ann,y is a maximal left ideal and M :R/Ann,j. Conversely, if L is a maximal left ideal of R then R/L is an irreducible R-module.

(If R is simple and L a maximal left ideal, then the irreducible module R/L is faithful since Ann,(R/L) is an ideal # R and must thus be 0; thus R is primitive. Kaplansky’s theorem is the converse, under the additional PIassumption.) Call an ideal P of R primitice if RIP is a primitive algebra, i.e., if P is the annihilator of an irreducible R-module. (In general, this concept is not left-right symmetric, i.e., primitive algebras exist that lack faithful irreducible right modules. However, Kaplansky’s theorem states that every primitive ideal of a PI-ring is maximal, which is left-right symmetric.) Define the Jucobson radical Jac(R) = (-){primitive ideals of R}. By Remark 1.5.24 if L is a maximal left ideal of R with A = Ann,(R:‘L) then A is a primitive ideal of R contained in L (since L 2 A . 1 = A ) . Hence Jac(R) G (){maximal left ideals of R}. On the other hand, if P is a primitive ~ ) ideal then for some irreducible R-module M , P = Ann, M = {Ann, y I y M 2 {maximalleft ideals of R}, so weconclude Jac(R) = {maximalleft ideals of RJ. An element r of R is lefi (resp. right) quasi-invertible if, for some r’ in R, (1 - r’)(1 - r ) = 1 [resp. (1 - r ) (1 - r ’ )= 11 ; r is quasi-inoertible if r is left and right quasi-invertible. (This terminology is borrowed from the nonassociative literature and is used to stress the role of 1 ; a more customary term is “quasi-regular.”) The following results of Jacobson are well known, but are clearer in the presence of 1.

n

n

g1.5.1

Primitive Rings, Kaplansky’s Theorem

39

Remark 1.5.25. In any algebra, if . K ~ . Y= ~ 1 and s3.v1 = 1 then x 3 . [Proof: s2 = ( s ~ . Y = I ) .s Y~ ~ ( . Y ~ . Y= ~ ).y3.]

.y2 =

A subset of R is quasi-invertible if each element is quasi-invertible.

Lemma 1.5.26. If L is a left ideal of leji quasi-invertible elements, then for each x in L there exists x’ in L such that (1 - x ) (1 - x’) = (1 - x’)(1 - x ) = 1 ; thus, in particular, L is quasi-invertible. Proof. Take x’ in R such that ( 1 - . ~ ’ ) ( l -.v) = 1. Then s’= .Y’.Y-.sE L. Now take x” in R such that (1-x”)(l-x’) = 1. By Remark 1.5.25 x” = x . QED

Proposition 1.5.27. Jac(R) is a quasi-invertible ideal of R, containing every quasi-invertible left ideal. Proof. First suppose rEJac(R). Then r is contained in every maximal left ideal, so ( 1 - r ) is not contained in any maximal left ideal. Thus R( 1 - r ) = R, proving r is left quasi-invertible. Therefore Jac(R) is quasi-invertible by Lemma 1.5.26. Conversely, suppose B is a quasi-invertible left ideal, and B $ Jac(R). Then B +L = R for some maximal left ideal L. Hence b + x = 1 for some b in B and x in L. Thus x = 1- b E L is invertible, contrary to L being proper. Thus B c Jac(R) after all. QED

Corollary 1.5.28.

For any ideal A

of R, (Jac(R)+A)/A E Jac(RI.4).

Proposition 1.5.27 shows Jac(R) is the same as what we would get using a right-handed version of all the definitions. Of course, Kaplansky’s theorem says that the Jacobson radical of a PI-algebra is the intersection of the maximal ideals. Semiprimitive Algebras

To fully appreciate the Jacobson radical, we recall that an algebra R is a subdirect product of algebras (R;.)y€T)if we can identify R as a subalgebra R;. such that x,.(R) = R,, for all y in r, where x;.is the projection to of R.;,

nYEr

Remark 1.5.29. Every subdirect product R of {R;.lyEr} is equiva(Indeed, every identity of R is an identity of each R?, lent to nyerR7. implying n , . , , - R , d,,,R; obviously R <,,,m;ErR;,.) Proposition 1.5.30. Suppose each R , y E r, has PI-class I? . [ f ‘ n = max(ti ( y E T} is j n i t e , then every subdirect product R 01 ( R (yE r) has PI-class n.

40

THE STRUCTLIRE OF PI-RINGS

[Ch. 1

Proof. By Remark 1.5.29 we may assume R = n;.€rR;.. By hypothesis some R;. has PI-class n, so we are done by Remark 1.4.31. QED

Subdirect products enter naturally into the structure theory of rings because of the following easy result. Proposition 1.5.31. Suppose R has a set subdirect product of {R/B;I?;Er)!$-(-)n;.El-B;. = 0.

of’ ideals fB;ly E r>.R

is a

Proof. (-=) Suppose n B ; , = 0. Then we define the homomorphism I,$: R + RIB:. by $(r) = (. . . ,r + By,.. .); i.e., the 11-component of +(r) is the coset r+B, in RIB,.. Clearly kert) = (rERlrEeach B.;} = n i E r B ; .= 0, so R is a subdirect product of the RIB,.. ( a ) is obtained by reversing this argument. QED

n;,,,

Now we put the pieces together. Call R semiprimitive if Jac(R) = 0 or, equivalently, if R is the subdirect product of primitive homomorphic images. Theorem 1.5.32, Any semiprimitive algebra R satisfying a polynomial identity of degree ri has PI-class < [d/2]. Proof. R is a subdirect product of primitive algebras R7, each of which has PI-class d [d;2] by Kaplansky’s theorem. Hence R has PI-class d [d/2] by Proposition 1.5.30. QED Theorem 1.5.33. Any notizero ideal of a semiprimitive PI-algebra intersects the center nonrrivially.

Proof. Suppose R is a subdirect product of primitive algebras R,, having PI-class n;,, and let A be an arbitrary nonzero ideal of R. Let n),denote the projection from R onto R;.. Then ny(A)is an ideal of the simple algebra R;,, so n,(A) = 0 or n , ( A )= R,. Let rl = fyET(n,,(A)= R,), and let n = max(P1class of R , Iy E r ;. Now for some y’ in r Ry, has PI-class n. Pick elements xl,. . ., x d in R,. such that gn(xl,.. ., x d ) # 0, and pick a, in A such that rc.,,,(uJ = x i , 1 < i < d. Let a = gn(alr... , a d ) €A. Clearlya # 0. Moreover, foreachyin r1,n,(a) = .qn(n,,(ul) ,..., np(ad))~Z(R,.) in view of Remark 1.4.31; for y&T1, n,(a)En,(A) = 0. Thus 0 # a E A n Z(n,,,R,.) = A n Z(R), proving the theorem. QED

These are the two main structure theorems for semiprimitive algebras, which we shall generalize to semiprime rings in the next section. 51.6. Injections of Algebras, Featuring Various Nil Radicals

In this section we try to enlarge the class of algebras having PI-class by various general injections, also to be used later in other contexts. One basic

41.6.1

Injections of Algebras

41

strategy is to show that certain algebras can be injected into a related algebra, which is the direct product of primitive algebras. Using Kaplansky’s theorem we then can conclude (in the presence of polynomial identities) that the original algebras have PI-class; also, nil subalgebras are locally nilpotent, and ideals of semiprime PI-rings intersect the center nont rivially. The two injections described here are R -+ R[A] and R (rlR;.)/Nil(rlR;,),where the R;. are copies of R , and Nil(R) denotes the largest ideal of R containing only nilpotent elements. Combined, they inject an algebra without nilpotent ideals # O into a semiprimitive algebra, preparing for the application of the results of $1.5. Thus, a program is obtained for proving an arbitrary theorem about PI-algebras by proving the theorem step by step in the following cases: (i) primitive; (ii) semiprimitive and prime; (iii) prime and no nil ideals; (iv) no nil ideals; (v) general case. As an application, Amitsur’s theorem is given, which essentially says that the definition of “polynomial identity” given in $1.1 is equivalent to seemingly weaker definitions. --f

Admissible Algebras and the Injection Problem Definition 1.6.1. An algebra R is admissible if there is an injection from R into a mult-equivalent algebra which is a direct product of closed primitive algebras.

This concept is very interesting for PI-algebras because of the next result. Proposition 1.6.2. If’ a PI-algebra R is a direct product of’ closed primitive algebras, theti R is isoriiorpliic to afinite direct product ] J k M k ( H k ) , where each H , is a direct product qf’fields, arid R has some PI-class. I i i particular, every admissible PI-algebra has some PI-class. Proof. Suppose R satisfies a polynomial identity of degree d. Then R 2 n Y E r M k , ( F ;for . ) a (possibly infinite) set r, where each F;. is a field and k;, d [d/2]. Letting H , = = k l F;. for each k d [d/2], we see that R 2 n k M k ( H k ) , a finite direct product (taking only those k for which H , # O), and clearly then R has PI-class d [d/2]. QED

nlyerIk,

Corollary 1.6.3.

Zf’R is admissible with PI-class ti, theii R 6,. M,,(Z[5]).

Proof. In view of Proposition 1.6.2 it suffices to show that if k < t i then for any field F , M k ( F ) 6,M,,(Z“[]). But M , ( F ) < M , ( F ) by Corollary 1.4.19, and M n ( F )d M,(Z[l]) since M J F ) is a homomorphic image of Mn(L[t]). QED

42

THE STRUCTURE OF PI-RINGS

[Ch. 1

The “injection problem” is whether a given algebra is admissible. Theorem 1.6.4.

Every semiprimitive algebra is admissible.

Proof. If R is the subdirect product of primitive algebras R , YE^, then R is mult-equivalent to R,,.But each R;. is mult-equivalent to its closure, so the theorem follows easily. QED

n;.ET

Here is a cute application of Theorem 1.6.4. Proposition 1.6.5.

I n a PI-algebra R, r 1 r 2= 1 implies r2r1 = I .

Proof. This is immediate for R = M , ( F ) in view of Remark 1.5.25 and the Cayley-Hamilton formula. If Jac(R) = 0 then we are done by applying Theorem 1.6.4 and checking at each component. Finally, in the general case, suppose r 1 r 2= 1 Then in R = R/Jac(R) we have F2F1 = 1. implying 1 - r z r l E Jac(R), so r2r1 = 1 - ( I - r 2 r l ) is left invertible. Suppose r 3 ( r 2 r 1 ) = 1. Then ( r 3 r 2 ) r 1 = 1 = r l r 2 ,so by Remark 1.5.25 r3r2 = r 2 . QED Theorem 1.6.4 has often been reformulated as, “Every semiprimitive PIalgebra can be injected in M J H ) for a suitable commutative ring H.” Indeed, for n > I\ define $: M , ( H , ) --* M , ( H , ) by $ ( X a i j e i j )= Xaijeij (cf. Proposition 1.4.18); then n k M k ( H k ) is naturally injected in some M,,(nkHk).Unfortunately, this argument is wrong for algebras with 1, because $ does not send I to 1. For example, M 2 ( Q )x M 3 ( Q ) cannot be injected, as ring with 1, into any M , ( H ) . Nevertheless, if we drop the multequivalent” condition the result can be salvaged. Remark 1.6.6.

There is an injection of Mk(R)into M,,(R), given by rijei+uk, j + uk‘

X f , j = rijeij CZ::b X!,j= -+

Corollary 1.6.7. Any semiprimitive PI-algebra can be injected into a suitable matrix algebra over a direct product ofjelds.

Proof. Using Remark 1.6.6 and Theorem 1.6.4 we see in fact that if R is semiprimitive of PI-class n then R can be injected into M , , ( H ) for some direct product of fields H . QED For our purposes Corollary 1.6.7 is virtually useless, and we are much more interested in Theorem 1.6.4. Our present objective is to find a general class of admissible PI-algebras. One reasonable question is, “Is every algebra of PI-class n admissible?” By the argument of Corollary 1.6.7 any such algebra would be a subalgebra of a suitable matrix algebra (over a commutative algebra). Bergman [74] showed that this fails even for a certain finite ring having p s elements. In $4.4 we give the original counterexample, due to Small [71a], which has other nice properties. Another interesting counterexample is in Lewin [73].

51.6.1

Injections of Algebras

43

PI-Rings With Nilradical 0

Nevertheless, Theorem 1.6.4 can be extended with the aid of a number of structure theorems. Recall Definition 1.3.25 (of “nil”). Lemma 1.6.8. then B is nil.

If A is a nil ideal of R and BIA is a nil subring of RIA,

Proof. For any a E A and b c B we have b J E A for some j , implying ( a + by’ E A. But thus ( ( a+ b)J)k= 0 for some k. Hence A + B is nil. QED

Proposition 1.6.9.

R has a unique maximal nil ideal.

Proof. By Zorn’s lemma R has a maximal nil ideal N . If B is a nil ideal of R then ( B + N ) / N is nil, so, by Lemma 1.6.8 B + N is nil, implying B c N . QED

The unique maximal nil ideal of R is called Nil(R), the nilradical of R. (Sometimes the nilradical is called the “upper nil radical”, to distinguish it from other nil radicals). Remark 1.6.10.

Nil(R/Nil(R))= 0 by Lemma 1.6.8.

Remark 1.6.11. Nil(R) is quasi-invertible. In fact if r’” = 0 then = xyL-iri. Thus Nil(R) c Jac(R).

(1

In the foregoing, I denotes a commutative indeterminate. The main tool in passing to Nil(R) is the following famous theorem of Amitsur; the neat proof here comes from Goldie [75B]. Theorem 1.6.12.

ZfNil(R)

= 0, then Jac(R[I]) = 0.

Proof. Let J = Jac(R[A]). We claim, that viewed as polynomials in I the leading coefficients of the elements of J are a nil ideal of R. Indeed, suppose p = Cf= rilliE J . Then l p E J , so, by Proposition 1.5.27 there exists an element q in R[A] such that (1 - l p ) q = q ( l -Ap) = 1. Then q - l p q = 1, so q = Apq + 1. This is the case m = 1 of the formula

(i) q

=

Impmq +Cy=-iAipi,

which we verify now by induction on m - 1 ; we write q = lm-lpm-lq

+

+

1 Aipi =

c

m- 1 i= 0

as desired

Indeed, assume (i) holds for

m-2 i=O

= I”pmq

M.

lipi,

Am-’pm-’(Apq+

1)

+

m-2

1 lipi

i=O

44

THE STRUCTURE OF PI-RINGS

[Ch. 1

' r: in R and take m > 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[<]).

Proof. N ( R ) = ObyCorollary 1.6.26.Bycorollary 1.6.23,(Nil(R))'Ji21 = 0, implyingNil(R) = 0,TheotherconditionsfollowfromTheorems1.6.14,1.6.15, 1.6.16. Q E D

Theorem 1.6.27 is fundamental to PI-structure theory. The part about ideals intersecting the center is due to Rowen [ 7 3 ] ; the rest is due to Amitsur in the 1950s. Corollary 1.6.28. R is simple. Proof.

I f R is a semiprime PI-ring and Z(R) is afield, then

Immediate. Q E D

Remark 1.6.29. Suppose R is prime. Then zr # 0 for all z # 0 in Z(R) and r # 0 in R. (Indeed, if zr = 0 then 0 = Rzr = zRr, so r = 0 o r z = 0.) In particular, Z(R) is a domain. Corollary 1.6.30.

I f R is a prime algebra over afield F with [ R : F ]

< 'XI, then R is simple. Proof. [Z(R): F ] < co ; since Z(R) is a domain, Z(R) must be a field. Thus R is simple by Corollary 1.6.28. Q E D

Nil Subsets of PI-Rings

Let us improve Corollary 1.6.23 to its best possible version, with the help of another radical due to Levitzki. Definition 1.6.31. subset is nilpotent. a

A subset of a ring is locally nilpotent if every finite

~~ ideal of R and ( A + N)/N is Lemma 1 .6.32. I f N is u l o c ~ a l lnilpotent locully nilpotent subset of RIN, then A is locally nilpotent.

Proof. Let A , = { a , , ..., a,: be an arbitrary finite subset of A. By hypothesis, Ad, G N for some d . Let N o = Ad,. No is a finite subset of N , so N t = 0 for some k ; thus A t k = 0. QED

Proposition 1.6.33.

There is a uriique maximal locally nilpotent ideal

qf R, which we call Lev(R), and Lev(R/Lev(R)) = 0.

Proof.

Analogous t o proof of Proposition 1.6.9. QED

48

THE STRUCTURE OF PI-RINGS

[Ch. 1

Lev(R) is called the Levitzki (locally nilpotent) radical of R. Proposition 1.6.34. Every nil, multiplicatii~elpclosed subset .4 of a PI-ring R is locally nilpotent.

Obviously R/Lev(R) is a semiprime PI-ring, so Nil(R/Lev(R)) implying NiI(R) is locally nilpotent. By Corollary t 6.23. A/NiI(R) is nilpotent; hence by Lemma 1.6.32 A is locally nilpotent. Q E D Proof.

= 0,

Definition 1.6.35. L,(R) = (nilpotent ideals of R ) and, moreover, L,(R) = E ( A a RIA/L,(R) is nilpotent in R/L,(R)}. Theorem 1.6.36 (Levitzki-Amitsur). Suppose R satisjes a polynomial identitj qf’degree d. (i) For any nil, multiplicatively closed subset B of R, we have B[d’21z L,(R). (ii) NiI(R) = L2(R). Proof. (i) Let n = [d/2]. We shall show for all b , , _ .. ,b, in B that {bl, ..., b,}” is contained in a nilpotent ideal of R , which will prove our assertion since h , . . . b , E [ b ,,,.., b,}”. So let B , = ( h ,,.... b “ } . By Proposition 1.6.34 B , is nilpotent. so obviously RBLR is nilpotent for some t ; we choose t minimal in this regard. We need to prove t < n. cl,Xnl . . . X n d . For Well, let .f(.U,, . . . ,X d ) = X 1 X 2 . . Xd+CnrSym(q,-Ill 1 < i < t , put A , i - = Bb-iRBh- and A Z i= B;-’RBb. Clearly A,A, G RBbR whenever 1 > 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 <

R 1-

R,

R. If'S is regular, then R is mult-equivalent

< R. JfS is regular then R c R,,, so R

d R,.

QED

$1.7.1

Central Localization of PI-Algebras

53

(It is a simple matter to extend Lemma 1.7.7 to completely homogeneous polynomials-cf. Exercise 1 t but in fact we have R , ,< R, to be proved in Theorem 2.3.37.) At any rate, Proposition 1.7.8 is all we need to obtain some striking results about the structure of PI-rings. The idea is to study Q , ( R ) through Q z ( Z ) , which is the well-known ring of quotients of a commutative ring. Our first result generalizes the well-known fact that if Z is a commutative do~nai~i (i.e., Z has no divisors of 0 other than 0 itself) then Q z ( Z ) is a field, called the,field of,fructions of Z . Theorem 1 .7.9.

! f R is a prime PI-ring, then Q , ( R ) is simple.

Proof. R, is a semiprime PI-ring by Proposition 1.7.5 and Proposition 1.7.8; moreover, Z(R,) = Z,. But Z is a domain and Z , is isomorphic to the field of fractions of Z , so Z(R,) is a field. Hence R , is simple by Corollary 1.6.28. Q E D

Theorem 1.7.9 with this proof is due to Rowen [73] ; the result was found independently and virtually simultaneously by Markov [73], Formanek, Martindale, Procesi, Schacher, and Small. This is an improvement of an important theorem of Posner [60], and enables us to handle prime PI-rings in much the same way as commutative domains; we shall call it the "Posner-Formanek-Rowen theorem." Localizing at a Set Intersecting g,(R)R Nontrivially

Another important case of central localization occurs when P is a prime ideal of Z ; then S = Z - P is easily seen to be a monoid, and we write R , for R,. This technique is analyzed in depth in $1.9. Meanwhile, let us obtain one easy but surprising result, which will enable us to strengthen Theorem 1.7.9. Proposition 1.7.10. 1 E ~ ( R ) RThen . 1E g ( R ) ' .

Suppose g is multilinear and R-central and

Proof. By Remark 1.4.16, g ( R ) ' Q Z . I f g ( R ) + = Z then we are done. Otherwise, let P be a maximal ideal of Z which contains g ( R ) + ;taking any maximal ideal B of R,, let R = R,/B. Now l€g(R,)R,, and g ( R , ) = g ( R ) , G Z , , implying g is R-central. Thus 0 # g(R)' 4Z ( R ) . But l? is simple, so Z ( R ) is a field and y(R)' = Z ( R ) . Hence ( Z , + B ) / B E Z ( R ) = g ( R ) + = (g(R,)'+B)/B, implying l . l - ' ~ g ( R , ) ' + B . Write 1.1-l = zis-' +b for suitable zi in g ( R ) , s in S , and b in B. Then b = 1.1-lzis-' EZ, n B G P,, implying 1.1 - E P,, which is absurd. Q E D

x:=, x!=

'

We are now ready t o generalize Theorem 1.7.9 nicely.

54

THE STRUCTURE OF PI-RINGS

[Ch. 1

Theorem 1.7.11. ZfCZn2+and [X4n2+2, g,] are identities ofR and if S ng,(R)R # 0,then the following conclusions hold: Z, = Z(R,) = gn(Rs)+; there is a 1 : 1 correspondence from {ideals of Z,} to {ideals of Rs}, given by A H AR,, and the inverse of this correspondence is the map A I+ A nZ , ; [Rs:Zs] isfinite. Proof. By Proposition 1.7.8 C Z n Z t and 1 [X4nz+Z, g,,] are identities of R,; thus g,, is Rs-central. Moreover, by Lemma 1.7.7 gn(Rs)Rs = (g,(R)R), so 1 Eg,(R,)R,, and by Proposition 1.7.10 1 Egn(Rs)+.But g,,(R,)+ a Z(R,); thus Z(R,) = g,(R,)’ = (g,,(R)+), G Z, c Z(R,), so equality tiolds at each step. The rest of the theorem follows from Theorem 1.4.26. QED

Note that as a special case the hypotheses of Theorem 1.7.11 are satisfied if S = {all regular elements of Z} and Z n g,(R)R contains a regular element ; in this case R, = Qz(R). Theorem 1.7.11 has very useful applications t o the prime spectrum of a PI-ring of class n, but we defer these applications until $1.9,where the prime spectrum will be studied in several aspects. Semiprime PI-Rings

Let us now analyze the algebra of central quotients of a semiprime PIring. The following example shows that we must not set our sight too high. Definition 1.7.12. The direct sum of {R,lyEr}, denoted OYErR,,is the set of all (x),)in R;. such that x y = 0 for all but a finite number of y. If r is isfinite, the direct sum does not have 1, and, as an algebra without 1, has no regular elements.

nyer

Example 1.7.13. A commutative, semiprime ring R = Q,(R) with Jac(R) # 0. Let Ci = Q[Ai], 1 < i < 00, where each Ai is a commutative indeterminate; viewing JJl b i < R ) C i as a Q-algebra in the natural way, let C be the Q-subalgebra generated by 1 and 0 4 i < Ci. In other words, we can view C as a ring of polynomials in an infinite number of commuting indeterminates, subject to the condition that A i l j = 0 for all i # j. Now obviously an element x of C is regular iff the “constant” term of .Y is nonzero. Thus, letting S = {regular elements of C} and A = C-S, we see that A C. Let R = C,. Then A, is the unique maximal ideal of R because {regular elements of R) = (invertible elements of R j = R-A,. In particular, Jac(R) = A, # 0, and R = Q,(R).

a

We see from Example 1.7.13 that the process of taking central quotients is not particularly useful for semiprime rings in general. Thus, we can either limit our attention to certain kinds of semiprime rings, or can look at more

41.7.1

Central Localization of PI-Algebras

55

general quotient constructions. We choose the first option now, and later shall describe briefly the other quotient constructions, with the use of several exercises. Definition 1.7.14. A PI-ring is semisimple if R is isomorphic to a finite direct sum of simple PI-rings.

Obviously if R is semisimple then R has only a finite number of maximal ideals and their intersection is 0. We shall shortly derive the converse, a useful fact. Lemma 1.7.15 (Chinese remainder theorem). Suppose R is semiprime, with a Jinite set of ideals A , , . . . , A, such that A , n ... n A , = 0 and.for all i, 1 6 i < t , A,+ ( A , . .. A , - Ai+ . .. A , ) = R. Then the canonical injection R -+ ] ]:= (RIAi)is an isomorphism.

, ,

Proof. By symmetry, it is enough to find the preimage of ( l , O , ..., 0). But by hypothesis 1 = a + x for some U E A , ,X E A,... A,; clearly x is the desired element. QED

Proposition 1.7.16. [ f R is a PI-ring with ajinite number of maximal ideals P , , . . . , P, with intersection 0, then the canonical injection R -, JJ:= (RIPi)is an isomorphism, and thus R is semisimple.

,

Proof. F o r e a c h j # i , P j $ Pi,so P , . . . P i _ l P i + l . . . P , ~ P i ; s i n cP,is e maximal, P , + P , . . . P i - , P i + , '"Pi= R , and we can apply the Chinese remainder theorem. QED

These two results do not enter directly in the next set of results, but they do serve as a model, motivating our characterization of PI-rings whose rings of central quotients are semisimple. Lemma 1.7.17. I f R is a semiprime PI-ring with Z E Zand Ann,z then z is regular (in R). Proof. 0 = Ann, z = Z n Ann, z. But Ann, Theorem 1.6.27. QED

Proposition 1.7.18. Proof.

za R ,

= 0,

so Ann, z = 0 by

If R is semiprime PI, then Z(Q,(R)) = Qz(Z).

Immediate from Lemma 1.7.17and Proposition 1.7.4. QED

Proposition 1.7.19.

Q,(fl,,, R,)

: n Y e r Q , ( R , ) canonically.

Proof. It is easy to see that (z,) is regular iff each z , is regular, so the result is immediate. QED

Theorem 1.7.20.

lf R is a semiprime PI-algebra and Q z ( Z ) =

56

[Ch. I

THE STRUCTURE OF PI-RINGS

n:=

Fi, with each Fi a field, then Q z ( R ) % with center Fi.

n:=Ri .for suitable simple Ri

Proof. Let e, be the multiplicative unit of Fi and let Ri = e,Qz(R). Each R i is a semiprime PI-algebra with center Fi,a field, so Ri is simple, and clearly Q,(R) z = eiQz(R)= R i . QED

n:

n:=

Let us now try to view Theorem 1.7.20 more intrinsically in terms of R.

.

Lemma 1.7.21. ! f a PI-algebra R has aJinite set { P I , ., . P , ) of prime ideals with intersection 0 and if no subset of { P I .. . . , P , ) hus intersection 0, then Q z ( R ) z Qz(R/Pi),and so Q z ( R )is semisimple.

n:=

Proof. For any given j , 1 < j < t , there is an element r . such that r j $ P j but r j E P i for all i # j . (Indeed, just take nonzero r j in O i z j P i . )Now let -denote the canonical homomorphic image in RIPj. Z ( R )n v #0, so take some element z j in RrjR such that z j ~ Z ( a )Then . for all r in R , [zj, r] E P j ; also z j E P i for all i # j , so [zj, r] = 0. Hence z j E Z . Now let Ri = R/Pi, 1 ,< i < t , and consider the natural injection R + Ri given by I’ -+ ( I + P , , . . ., r + PI);z j is identified with an element in n R i whose ith component is 0 for all i # j. If an element z of Z is regular then zzj # 0 for all j , implying z is regular in n R i . Thus by Proposition 1.7.2 we can identify Q z ( R ) as a subalgebra of Q z ( n R i ))v n Q z ( R i ) . Now Then z is regular and (0,..., 0,1,0 ,..., 0 ) = z - l z j ~ Q Z ( R ) . take z = Thus we have r I R i E Q z ( R ) ;every regular element of Z ( n R i ) is invertible in Q z ( n R i ) ,and so is regular (and thus invertible) in the subalgebra Q,(R), implying Q z ( n R i )= Q,(R). But each Q z ( R i ) is simple, so we are done. QED

n:=

z:Jz1z,.

Proposition 1.7.22. If P , n . . . n P, = 0 ,for suitable prime ideals P I ,. . ., P , o f a PI-algebra R , then Q z ( R )is semisimple. Proof. Just take a minimal subset of { P , , . . . , PI; having intersection 0 and apply Lemma 1.7.21. QED

One instance where we can verify the’hypothesis of Proposition 1.7.22 is as follows: Definition 1.7.23. An ideal A of R is an annihilator ideal if A # R and if A = Ann, B for some B c R. R has ACC(annihi1ator ideals) if there is no infinite chain A , t A, c A 3 c ... of annihilator ideals of R. (Here ACC is an abbreviation for “ascending chain condition.”) Remark 1.7.24.

For any subset A of R , A G Ann(Ann’A ) .

Remark 1.7.25.

If A , c A , then Ann A , 2 Ann A , . If A is an annihilator ideal, then A

Remark 1.7.26.

=

Ann(Ann’ A ) .

$1.7.1

Central Localization of PI-Algebras

57

(By Remark 1.7.24 A c Ann(Ann’ A ) . But for some B c R, A = Ann B and B E Ann’(Ann B ) = Ann’ A, implying A = Ann B 2 Ann(Ann’ A). Thus A = Ann(Ann‘ A ) . ) Thus in Definition 1.7.23, taking B = Ann‘ A, we may assume B a R . In order to investigate annihilator ideals we would like to introduce an interesting concept of Procesi, which will recur often in this book. Definition 1.7.27. If A denoted C , ( A ) , as { r E R 1 [ r , A] R = R,CR(R,). Remark 1.7.28.

A

E

R , define the centralizer of A (in R ) , R is an extension of a subalgebra R , if

= 01.

For any A

s R , C,(A) is a subalgebra

of R, and

CR(CR(A)).

Remark 1.7.29. Every central extension is an extension. Also, every algebra is an extension of its center.

Actually, these are the motivating examples of extensions, and often facts to be proved in each case can be merged into facts about extensions. A more detailed study is given in $1.9, but meanwhile we have the following results.

If R is an extension of R , and A Q R , , then R A Remark 1.7.30. C , ( R , ) A = A C , ( R , ) = A R and is an ideal of R.

=

Proposition 1.7.31. I f R’ is an extension of R and R’ satisjes ACC(annihi1ator ideals), then R satisJies ACC(annihilator ideals). Proof. Suppose, on the contrary, that A , c A , c ... is a chain of annihilator ideals of R . Write T = C , . ( R ) and for each i write Bi = Ann; Ai and A : = Ann,.(BiT) 4 R ’ ; then one sees easily that A i = Ann, B j = Ann,(B,T) = R P, A:, so A’, c A; c ..., contrary to R‘ satisfying ACC(annihi1ator ideals). Q E D

Remark 1.7.32. Suppose R is semiprime and A Q R. Then AnnA Ann‘A.[Indeed,letB = Ann’A.Then(BA), = B(AB)A = 0 , s o B A = Oand R E Ann A ;likewise, Ann A E B.] Thus by Remark 1.7.26, Ann A # 0 if A is an annihilator ideal. =

Let us now see how ACC(annihi1ator ideals) relates to Proposition 1.7.22. Proposition 1.7.33. 1f’R is semiprime and satisjies ACC(annihi1ator ideals), then R has a j n i t e set qf prime ideals with intersection 0. Proof. Let .d= {annihilator ideals of R } , and .’P = {maximal members of .d}. If A 2 P , B 3 P are ideals of R, and P E . with ~ A B c P, then A(B(Ann‘ P)) = 0, implying A E .dand so A = P (by maximality of P in .d).

58

[Ch. 1

THE STRUCTURE OF PI-RINGS

This proves that all the members of .Y are prime ideals. We shall prove the proposition by showing in fact that ./p is finite, with intersection 0. Well, take distinct P , , P,, . . . in .Y, and note that Pi$L P j for all i # j. For each k, let B, = Ann(():, Pi).Then for all k Ann Pk+ $ B,i(for otherwise . “ P k -= 0 G Pr+ implying Ann Pk+ c f k + , so (Ann fk+,), (Ann Pk+,)P1 = 0; since R is semiprime. we would have Ann Pk+ = 0, contrary to Remark 1.7.32).But clearly Ann P , + E B, + This proves Bk # B k + , for each k. Thus we have a chain B , c B , c ... of annihilator ideals, and by hypothesis this chain cannot be infinite. Therefore B has a finite number of elements. Now suppose that Y = { P , , . . . ,Pk).We Pk = 0. Well, otherwise B, E d,so want to conclude by showing B, E Pifor some i, 1 < i < k ; reordering the elements of .P if necessary we may assume Bk G Pk. But then (B,(n:z: pi)), c B,(n:= p i ) = 0, implying B k = BkPl (since R is semiprime). which we showed was impossible. Thus P , = 0. QED

,

,

,

,

.

n:=

nr=,

The idea of looking at d is due, I believe, to Herstein (cf. Jacobson [64B, appendix B]), and can be used to study rings without any chain conditions whatsoever. Let us now conclude with a comprehensive structure theorem. Theorem 1.7.34. The following conditions are equivalent j ) r a semiprime PI-algebra R (wirh center Z ) : (i) Q z ( Z ) is a,finite direct product of.fields; (ii) Q , ( R ) is semisimple; (iii) R sarisfies .4CC(annihilator ideals) ; (iv) 2 satisfies t\CC(annihilator ideals) ; (v) R has a.finite set of prime ideals with intersection 0 ; (vi) 2 has a,finite set qf prime ideals with intersection 0. Proof.

-

(i) (ii) This is Theorem 1.7.20. (ii) 2 (iii) Obviously Q z ( R ) has ACC(annihi1ator ideals) so by Proposition 1.7.31 R has ACC(annihi1ator ideals). (iii) => (iv) True by Proposition 1.7.3I . (iv) = (vi) True by Proposition 1.7.33. (vi) => (i) True by Proposition 1.7.22. (iii) (v) True by Proposition 1.7.33. (v) (ii) True by Proposition 1.7.22. QED Corollary 1.7.35. Zf‘R is a finire dimensional semiprime ulgebra over a jield F , then R is semisimple. Proof. Every ideal is an F-subspace, so obviously R satisfies ACC (annihilator ideals), implying R has a finite set of prime ideals P , . . . . ,P , whose intersection is 0. But each RIPi is simple by Corollary 1.6.30, so each Pi is maximal and we are done by Proposition 1.7.16. QED

$1.8.1

Tensor Products

59

For the reader interested in quotient rings, we survey now what happens with more general quotient constructions. Suppose R is a semiprime PIring. In general Q z ( R ) is not the classical quotient ring of R (cf. Exercise 2); moreover, R may even fail to have a classical quotient ring (cf. Exercise 3). However, in view of Exercise 4 (first proved by Fisher [73]), R does have a “maximal” ring of left quotients Qmax(R), which Martindale [73] showed is also the maximal ring of right quotients and is a PI-ring. These results were improved by Armendariz-Steinberg [74] and by Rowen [74d], the latter paper characterizing Q,,,(R) in terms of ideals of Z ( R ) , yielding all the earlier results without difficulty. Another quotient ring construction is given in Fisher-Rowen [74]. The key result in obtaining these theorems is proved very easily for rings without 1, so we shall present the theory of quotient rings of semiprime PI-rings in the exercises of $1.11. If R is PI but not semiprime then Q,,,(R) need not even be PI, as shown by Schelter-Small (cf. Exercise 1.1 1.2 1 ).

$1.8. Tensor Products and the Artin-Procesi Theorem

Much structure so far has been built by means of polynomial rings and central localization. In fact, these are both instances of an important general construction, the tensor product, which has many uses in PI-theory. In this section we present the basic aspects of the theory of tensor products. Our two main goals are to tie PI-theory to the theory of Azumaya algebras through the Artin-Procesi theorem and to develop enough theory of the Brauer group (of a field) for a smooth treatment of central simple algebras in Chapter 3.

Tensor Products of Modules Assume A , B are given modules over a commutative ring C. Given an abelian group P, we say a map tj: A x B Definition 1.8.1. P is bilirzear (over C ) if for all ai in A , 6, in 5, and c in C $(al +a,, 6,) = $ ( a ~ , b l ) + $ ( a , , b ~ )$4al,bl , + h )= t j ( a l , b I ) + $ ( a 1 , 6 d ,and d4cu1,bl)

+

= $(al, ~ 6 1 ) .

Letting ( A x B ) + denote Z ( A x B ) , the Z-module Definition 1.8.2. freely generated by A x B (cf. Definition I . l S ) , define A O C B= ( A x B ) + / I , where I is the subgroup of ( A x B ) + generated by all (al + a 2 , b l ) - ( a I , 6 , ) - (a2,bl ), (al, 6 62) - (al,61 ) - ( a 621, and (ca,, 6 ) - (al,c6 for all aiin A , 6, in 5, and c in C. Write a @ 6 for the canonical image of (a,6) in A OCB.

+

60

[Ch. 1

THE STRUCTURE OF PI-RINGS

We shall write A @ B in place of A OCB when there is no ambiguity about C. (Note that, so far, A @ B is only an abelian group.) In what follows, “bilinear” means “C-bilinear.” Proposition 1.8.3. The canonical map A x B + A 0B (given by (a, b ) + a 6 b ) is bilinear. Moreoaer, ,for any bilinear mup i,h : .1 x B P, where P is un abdian group, there is an induced group homomorphism 5:A @ B -+ P. such that $(a 0b ) = (a, b),forall a in A, b in B. -+

Proof. Extend $ to a group homomorphism $ : ( A x B)’ + P by $ ( x ( a i ,b,)) = x$(ai, bi).Then $ ( I ) = 0, where I is as in Definition 1.8.2,so the assertion follows forthwith. QED

In fact, one can characterize A @ B (up to isomorphism) by the property given in Proposition 1.8.3, which is used in proving virtually all of the basic results on tensor products. Here is the most fundamental one. Proposition 1.8.4. Suppose A‘, B‘ are arbitrary C-modules, und $, : A and $ 2 : B + B’ are C-module homomorphisms. Then there is a welldefined group homomorphism, denoted $, @ t+b2: A @ B + A’ @ B‘. such that (G1 @ t,b2)(a@ b ) = $ , ( a ) @ t+b2(b),forall a in A and b in B . + A’

Proof. Define $: A x B 4 A‘ 0B’ by $(a, b ) = Gl(a)0$2(b). Obviously $ is bilinear, so $ induces a group homomorphism A @ B + A‘ @ B’ having the stated properties. Q E D

Corollary 1.8.5.

A @ B is in fuct a C-module, with action given by

cC(ai @ bi) = x ( c a , @ b,). Then, with notation and assumptions as in Proposition 1.8.4. $ @ t+h2 is a module homomorphism.

,

Proof. Given c’ in C, define i,hc: A + A by t,bC(a)= ca. Now define c(&,@ bi) to be C$,O l)(&ziO bi) = Cicai@bi. This action is certainly distributive over addition (because $ c @ 1 is a group homomorphism), so A 8 B is a C-module. Then ($1

showing that

0$z)(c(a 6 b ) ) = $ i ( c ~0 ) $2(b) = c$i(a) 6 $2(b) = ~ ( $ 1@ $ 2 ) ( a 6 b ) ,

t+bl

@ t+b2 is a module homomorphism.

QED

Tensor Products of Algebras

Theorem 1.8.6. I f A, B ure C-algebras, then A @ B is u C-algebra, with multiplicnticin induced by (al @ b , ) ( a , @ b 2 ) = (a1a2@ b,b,). Proof.

Fixing i12,b2, define

$1

and

by i+hl(al)= a l a 2 and $ 2 ( h l )

41.8.1

61

Tensor Products

= b , b , for all a , in A , h , in B. Clearly $, and $, are module homomorphisms, so we have t+bl 0$,: A 0B -, A 0B, defining right multiplication 6 y a, 0b,. Doing this for all a, in A, h, in 8, we now reverse the procedure, fixing x in ,4 0 B and defining $ , : A x B -+ A 0B by induces a map K:A 0B GX(a,b ) = x(a 0b ) for all a in A , b in B. Then + A 0 B, corresponding to left multiplication by x. It is now easy to see that A 0B is a C-algebra. Q E D

We can finally restate Proposition 1.8.4 in the proper context. Remark 1.8.7. If A, A’, B, B’ are C-algebras and $ 1 : A + A‘ and $,: B + B‘ are C-algebra homomorphisms, then $, 0$ 2 is also a C-algebra homomorphism (follows easily from Corollary 1.8.5).

From now on, A and B will be C-algebras. We develop other general facts, whose proofs all have the same flavor as Theorem 1.8.6. Proposition 1.8.8.

There is an isomorphism A 0B

+

B @ A given by

z a i 0bi F+ x b i 0ai. Proof.

Define a bilinear map $:,4 x B - + B @ A by $ ( a , b ) = b @ a ; QED

IF is the desired isomorphism.

Proposition 1.8.9. If A , , A , are C,-algebras and A , , A 3 are C2algebras, then there is an isomorphism

( A , O c , A , ) O c , A 3‘ A ,

OCI(A20C2A3)?

such that (al 0a,) 0a3 - a l

0(a, 0a 3 ) .

Fix a in A,3, and define the C,-bilinear map G a : A 1x A 2 ( A , 0 A 3 ) by $,Ja1, a,) = a, 0(a, 0a). Thus we get & : A l 0A , 4.4, @ ( A 2 0 A 3 ) . Now define the C,-biJnear map $ : ( A , 0A , ) x A 3 4 A , @ (‘4, 0 A 3 ) by t+b(x,all 0u,,,a) = t + b a ( ~ a0 , ,a z i ) ; IJ is the desired Proof.

+A, @

isomorphism. QED For any I , s A and I, G B, write I, 0 1 , for { ~ j a l j O a , j l a , j ~One li}. must be careful not to view this as the tensor product of algebras without 1, because the two notions are not the same. (E.g., for A = 2, B = 2/22, and I = 22, we have A Oz B z B and I 0B = 0, but as algebras without 1, I @ B # 0.) I be a homomorphism. Then, letting Proposition I .8.10. Let $: A -+ ; $’ = t+b 0 l : A 0B + A 0B, we have ker$’ = k e r $ @ B.

Proof.

Let I = (kerII/)0B. Clearly I c ker$’, so $’ induces a map

62

[Ch. 1

THE STRUCTURE OF PI-RINGS

-

tj' : ( A 0B ) / I + d 0B, whose inverse we shall now construct. There is a bilinear map A x B ( A 0 B)/I, given by (a, b) + (a @ b)+f. The induced QED homomorphism A 0B -+ ( A 0B)/I is clearly -+

u-'.

Write A 0 1 for { a 0 1 l a € A }

EA

0 B ; likewise for 1 @ B .

Remark 1.8.11. A 0 1 and 1 0 B are C-subalgebras of A 0 B. Moreover, [ A 0 1.1 0 B ] = 0, so A 0B is an extension of A 0 1. There are canonical, onto C-algebra homomorphisms A + A 01 and B -+ 1 0B, givenbyawa0 I andbHl@b. Theorem 1.8.12. If H is a commutative algebra, then A 0H is a central extension of A 0 1, implying A 0H < mult A . Proof.

Straightforward from the above remark. QED

Examples of Tensor Products To motivate further discussion, we shall now see that every example previously given of a central extension is in fact a special case of the tensor product construction. Proposition 1.8.13.

The polynomial ring R[A] 2 R O r Z[A].

x!=

Proof. Define the bilinear map I): R x Z[A] -+ R[A] by $(r, nili) nirjbifor r in R , n, in Z.Then $ induces an onto homomorphism $: R Z [ l ] + R[A]. But clearly every element of R 0 Z [ l ] has the form =

x!=,

$(xr,

x i r i 0Ai. If 0Ai)= 0 then x r i A i = 0, implying each ri = 0 ; it follows that ker $ = 0, so $ is an isomorphism. QED Proposition 1.8.14. R Oz(R)Z(R)S.

If S is a submonoid

of Z ( R ) - { O } , then R ,

Proof. Define tj: R x Z ( R ) , -+ R , by $(r, zs-') = rzs-l for r in R , z in Z ( R ) , s in S. Then $ induces an onto homomorphism (6: R @ Z ( R ) , -+ R,. But any element of R 0Z ( R ) , has the form r 0 Is-' for suitable r in R and s in S . If r 0 Is-' E ker $ then rs-' = 0, implying s l r = 0 for some sI in S ; thus 0 = rsl 0( s I s ) - I = r @ s-', proving $ is an isomorphism. QED

Proposition 1.8.15. Proof.

M,(R) z R g,,, Mn(4).

Define the +bilinear map $: R 0M , ( $ ) $(r,

olijeij)=

x7,j= aijreij,

for

--f

clij

M J R ) by E 4,

rE R .

Thisinducesa homomorphism$: R O,,, M,(c#I) --* M,(R),whoseinverseisgiven by rijeijt-x;,jz r i j0 eij. QED

xZj=

51.8.1

Tensor Products

63

Proposition 1.8.16. Write Z for Z(R). !f [R :Z ] < oc) and S is a submonoid of Z , then End,,(R,) z (End, R ) 0, 2,.

Proof.

We define a bilinear map (End, R ) x Z ,

-+

End,,(R,),

by

(8, Z ) H zp, where zp is the homomorphism sending r s - l to p ( r ) z s - ' . This gives us a homomorphism $: (End,R) 0Z , -+ End,(R,). On the other

xy'

hand, given in End,(&), suppose R = riZ for suitable ri in R. Then P(ri) = xisw1for suitable xi in R, 1 d i < m, and S E S , so $€Endz R ; clearly the map /3 H s p @ s- * is II/ - so II/ is an isomorphism. QED

',

Remark 1.8.17. The tensor product has the following connection with extensions: If R is any ring such that R = A B with [ A , B ] = 0, then there is a homomorphism $ : A 0B -,R , given by $(Ciai0b , ) = x i a i b i ; hence R d A 0 B.

Tensor Products over a Field

One would very much like the maps of Remark 1.8.11 to be isomorphisms. This happens when C is a field, as we shall now see. Theorem 1.8.18. Suppose C is a field and R = AB, with [ A , B ] Then the following statements are equivalent:

= 0.

(i) The map x u i 0 b, -+x u i bi is an isomorphism A 0B zz R. a,,,bi= 0, (ii) For any C-base {qI} ofA andfor all b, in B such that we have all b, = 0, 1 d i < k . sib,, = 0, (iii) For any C-base { b y )qf B andfor all ai in A such that we have all a, = 0, 1 < i d k . (iv) For any C-base (a,,} qf A and { b y }o f B , { q b , , }is a C-base o f R .

xf=

x,$

c,,a,,, where c,, E C Proof. (i) => (ii). Write a typical element of A as and all but a finite number of c,, are 0. Fix p j , and define a bilinear map t+hj: A x B --t B by $j(x,,c,la,,, b ) = c,,jb.Then $ j induces a homomorphism $j: A @ B -+ B. If x:=la,,, = 0 then by hypothesis x ! = lu ,,,0bi = 0, implying, for eachj, 0 = %(If= la,,,@ b i ) = b,. (i) * (iii). Analogous to (i) * (ii). (ii)* (iv). If x i , j c i j u , l , b= , , 0 for cij in C , then 0 = ~ i ( a , , , ~ j c i j b , , ) implying, for each i, x j c i j b , ,= 0 by (ii); thus each cij = 0, proving (iv). (iii) 3 (iv). Analogous. (iv) * (i). We havealreadyproved (i) * (iv),which,in thecaseR = A 0B, says {a,, @ b, 1is a base of A 0B. I f C f = aibi = 0, then writing ai and bi in terms of the given bases and checking coefficients, one readily gets a, 0bi = O . QED

xt=

64

THE STRUCTURE OF PI-RINGS

[Ch. 1

Corollary 1.8.19. I/ C is a jield, then f o r any B the homomorphism .4 --+ A @ 1 ( 5 .4 cX:, B ) is an isomorphism. Proof. Take a basis of B including the element 1. and apply Theorem 1.8.18(iii). QED

We shall also need the following results when studying tensor products over fields. Remark 1.8.20.

Viewed in A 0B, Z ( A ) 0Z ( B ) C Z ( A 0 B). (Easy.)

Remark 1.8.21. If C = Z ( A ) is a field then Z ( A @ B ) = 1 @ Z ( B ) . (Indeed, Z ( A @ B ) E Z ( 1 @ B ) = 1 0Z ( B ) since B and 1 0B are canonically isomorphic; the reverse inclusion comes from Remark 1.8.20.) ProPosition 1-8-22.

If’ C is a field and A , B are simple with C

= Z ( A ) , then A 0 B is simple.

Proof. Suppose O # I a R = A @ B , and take O # r = ~ ~ = , a j @ h j ~ I with { a l , ...,a,} C-independent, r chosen such that u is minimal. If u 2 2 then by Lemma 1.5.17 there are elements r i l , ri2 in A , 1 < i 6 k, for suitable k such that, letting a; = r i , a j r i 2 , we have a; = 0 and a; # 0. But C ~ = 2 a J 0 b j = ~ ~ = , 2 3 7 = 1 r i , a j r i 2 0 b j = ~ lf)=r (i r( ir2i 0ll @) ~ l , contrary to the minimality of u. Thus u = 1, and r = a , @ b,. But then 1 = 1 0 1 E A u , A 0B b , B c ( A @ B ) r ( A 0B ) G I, so I = R , proving R is simple. QED

xf=,

Corollary 1.8.23.

Proof.

M , ( F ) O FM , ( F ) z M,,(F),for eueryjield F .

Define an F-bilinear map

$1

M,(F) x M J F ) + M,,(F).

by

Il/(ZTj=1 aijeij, Ci,,,=iBuaeua)= Xm. Zi,u=~ ( ~ i j B u ” ~ e i + , u - ~ ~ r n , j + ( i , - l ) r n$; induces a map $: M J F ) O FM , ( F ) + Mrn,,(F),which is clearly an algebra ) M J F ) is simple, so ker 3 = 0 : checking homomorphism. But M r n ( F 0 dimensions over F, we conclude S; is onto, and thus an 1,,=1

isomorphism. QED Example 1.8.24. Let A = Q($). Then A A is not a field, because the kernel of the canonical homomorphism A 0A + A (given by a, 0 (12 4 a, a 2 )is nonzero, containing 1 0 01 .

3-Jz

Tensor products enable us to view part of Section 1.5 in a clearer light, as exemplified in the following important result of Wedderburn.

R‘

Theorem 1.8.25. I f ‘ R‘ is an e.utension qf a simple ring R, then R @Z,R)CR.(R).

$1 2.1

Proof.

Tensor Products

65

Combine Theorem 1.5.18 and Theorem 1.8.18. QED

Corollary 1.8.26. Suppose R' i s an extension of a simple ring R . R' is simple iff C,.(R) is simple. Proof.

Apply Proposition 1.8.22 to Theorem 1.8.25. (*) If C , , ( R ) is not simple then neither is R', by Proposition 1.8.10 and Theorem 1.8.25. QED (=)

The Artin- Procesi Theorern- "Difficu It Direction"

There is a certain tensor product construction which plays an important role in much of this book. Definition 1.8.27. The opposite ring RoPof R is defined to have the same additive structure as R , but with multiplication in the reverse order (i.e., the product of r and r' in R"" is r'r).

be the injection Remark 1.8.28. Suppose R is a C-algebra. Let from R into End,R given by the regular representation (cf. Remark 1.3.21); analogously, define the left regular representation sending an element r of R to the map taking an arbitrary element r' to rr', and note that we thereby get an injection $ 2 : RoP+ End, ( R ). Then we have a homomorphism RoP0R + End,(R), given by ( r l 0r 2 ) :I' 4 r , r r 2 ; we call this the canonical homomorphism. It would be very useful for the canonical homomorphism to be an isomorphism. We start with an important special case, and then generalize it via PI-theory. Proposition 1.8.29. I f R is simple and [ R : Z ( R ) ] = t , then the canonical homomorphism I): ROP O L f R R ,+ M , ( Z ( R ) )is an isomorphism. Proof. RoPalso is simple, so RoP 0R is simple and ker $ dimensions over Z ( R ) ,we see $ is also onto. QED

= 0.

Checking

Theorem 1.8.30. I f C2,,>.+, is an identity of R and 8, is R-central with 1Eg,,(R)R, then the canonical homomorphism Rap@ Z ( R ) R+ End,(@ is an isomorphism.

x:=,

Write 1 = g , l ( r j l ,... ,rjd).yjfor suitable r j l , .. ., rjd, s j in R, let $: R"'' R + End,. R be the canonical homomorphism, where C = Z ( R ) . Also, since q,, is linear in X I , we can find suitable polynomials fUl(X2,..., X , ) and , f W z ( X 2..., , Xd), 1 ,< u < m, for some m, such that gn = ~ ~ = , f , , X , , f Let U 2 t. = n 2 . Proof.

1

< ,i < k , and

66

THE STRUCTURE OF PI-RINGS

[Ch. 1

First we show IL is onto. Take any P in End,(R). For all r in R .

by Lemma 1.4.20.Thus, putting iji=

( - l y - l g n ( x j r , r j l ,..., rj,ipl,rj,i+l,..., r j a ) 6 Z ( R ) ,

we have P ( r ) = /Nxj,izjirji) = xj,izjii?(rji).Now let aUii= ( - 1 )

i-1

j u l ( r j l ,..., r j , i - l , r j , i + l ..... rjd)", '

and b,ji

= , ~ u 2 ( r j ~ , . . . , r j , i - 1 , r j . i +..1.,rjd)P(rji). ,

Then b(r) = ~ , i . i , u a u i i r bfor u i i all r so, identifying aujiwith its corresponding auji@ buji) = P, proving $ is onto. element of R"". we have Next, suppose ~ q , ~ @ q .yq2 l E ker I) for suitable xql in R"", y q 2 in R . Then for all r in R, X 4 v q l r x q 2= 0. Now, for all r l , . . . , r d in R we have

t,b(ri,j,u

I

Writing

we have

But, for each u, i, ~ 4 q ~ q l b u = i . 0, ~ qby 2 hypothesis. Thus g(rl, ..., rd). for each r l ,..., rd in R , implying Z,sql 0.sq2 E Ann,(g(R)R) = 0, proving ker $ = 0. QED

x 4 . ~@x,, q1 =0

$1.8.1

Tensor Products

67

Definition 1.8.31. (Cf. Auslander-Goldman [60, Theorem 2.1 (c)].) R is an AIumaya algebra (over Z ( R ) ) ?/’ rank t if [R:Z(R)] < 03, R”” @ z ( H I R:EndZo,R by the map of Remark 1.8.28, and for every prime ideal P of Z(R), R, is a free Z,-module of dimension t . Definition 1.8.32. R is Azumaj‘a if R is a finite direct sum of Azumaya algebras of various ranks.

We start our discussion of Azumaya algebras with a major theorem, whose proof now is quite easy. Theorem 1.8.33. if C2,’+, is un identity of R and with 1 E g,(R)R, then R is Azumaya Of rank 11’. Proof.

if g is R-central,

a

Suppose P Z(R) is prime. By Proposition 1.7.10, g,(R) $ P , so 1 €g,(R,) and, by Theorem 1.7.11 Z(R,) = Z ( R ) , . Thus by Theorem 1.4.26, R, is a free Z(R),-module of dimension i f 2 . In view of Theorem 1.8.30, R is Azumaya of rank n’. QED Theorem 1.8.33 is the ”difficult direction” of the famous theorem of Artin [69]-Procesi [72a]. The proof presented here has its roots in simultaneous, independent proofs of Amitsur [75] and Rowen [75P], and the current (possibly final) form is due to Schelter. A historical sidelight, satisfying to PI-theorists, is that the early proofs (especially Artin’s) were based on the theory of Azumaya algebras, and made possible the study of PI-rings by means of Azumaya theory. Now the situation has been reversed. We have a purely PI-theoretic proof, and shall indicate by and by how one can use PItheory to study Azumaya algebras. To do this, we must first prove the “easy” direction of the Artin-Procesi theorem. This direction traditionally was proved by appealing to “known” results at the end of Demeyer-Ingraham [71B], which in turn quotes the theorems of Azumaya and Nagata on Hensel rings and Henselization. Fortunately there is now a very easy proof, discovered mostly by Amitsur. To present it, we shall develop some facts about Azumaya algebras, drawing on the original paper of Azumaya [51]. The following important fact is mostly due to Jacobson (also cf. Azumaya [ 5 11) and is often called “Nakayama’s lemma”; the version here, which appears in Schelter [75], is very handy. Proposition 1.8.34. I f M is a Jinite-dimensional R-module with submodule N # M, then there is LI primitive ideal P of R with N + P M # M .

,

Proof. Write M = N + X Y = Ry,, with each y i € M, m minimal; take a Ryi, maximal with respect to y, q! M I . Then submodule M I 2 N MIM, is an irreducible module, so Ann,(M/M,) is a primitive ideal P of R. But P M c M I , so N + P M c M , # M. QED

+xY=2

68

[Ch. 1

THE STRlJCTURE OF PI-RINGS

Proper Maximally Central Algebras

We present now Azumaya’s original definition, which is more compatible with PI-analysis then the more modern notions (which involve projective modules). Definition 1.8.35. R is proper maximally centrui of rank t (over a subring C of Z ( R ) ) if R O P 0, R t End,- R and R is a free, t-dimensional Cmodule. Remark 1.8.36. For any Azumaya algebra R of rank t and any prime ideal P of Z ( R ) R, is proper maximally central of rank t . [Indeed, let Z = Z(R), and H = Z,- Then R;”OI,RP :(R”l’@/H)@,,(H@),R) 2 R”” @ / H R 2 (R”” 0 R ) 0 H :(End, R ) 0 H :End,(R 0 H ) : End,(R,) CC Proposition 1.8.16.1 Theorem 1.8.37. Suppose R is proper maximally central over C . Then (i) C = Z(R), ( i i ) if A Q R theti A = ( A n C)R, (iii) if B C then B=BRnC.

a

Proof. Let . ( r i , . . , r,,,} be a C-basis of R . Then there are f l i in End, R such that Ji(ri) = 1 and [ l i ( r j ) = 0 for all j # i. Now pi can be identified with an element rik, 0 rikZ in RoPOCR, so that Bi(r)= riklr r l k 2 . (i) Suppose =EZ(R).Then zrl = ciri for suitable ci in C. so z = z f l l ( r l ) = Z x k r l k l r l r l k 2 = x.krlklZr1rlk2 = BI(zrl) = c1 EC. (ii) Suppose .4 4 R. If a = ciriE A , then ci = J i ( a )= riklarikZ EA for each i, implying a E ( A n C)R. Thus A = ( A n C ) R . (iii) As in (i), ifc = E b , . y i E C then c = Jl(crl) = x b i b l ( . y i r l )E B. Q E D

zk

xyZl

xy=

xk

xk

Corollary 1.8.38. Suppose R is proper maximally central, with center Z. Then (i) Jac(R) = Jac(Z)R, and Jac(Z) = Jac(R) n Z, (ii) if r l . . ..,rm are

xy= z, where xy= riZ.

elements of R such that R in R/Jac(R), then R =

=

~

denotes the canonical image

Proof. (i) K isa PI-ring, so the Jacobson radical is the intersection ofthe maximal ideals. By Theorem 1.8.37, the maximal ideals of R correspond to the maximal ideals of Z, via intersection with Z, so Jac(Z) = Z n Jac(R).Again by Theorem 1.8.37, Jac(R) = (Jac(R)n Z)R = Jac(Z)R.

(ii) Let N = R = N . QED

xr= r , Z . Then R = N +Jac(Z)R, so by Proposition 1.8.34

Corollary 1.8.39. If’R is Azumaya of rank t , then for every prime P a Z(R), R , is propt’r maximally central of runk t and has unique ma.uima1 ideal pa,.

Tensor Products

31.8.1 Proof.

69

Apply Remark 1.8.36 to Theorem 1.8.37. QED

Thus we are led to study the following kind of ring: Definition 1.8.40. R is quasi-local if Jac(R) is maximal (and is thus the unique maximal ideal of R). Remark 1.8.41. Suppose C is commutative and J = {noninvertible elements of C}. J is multiplicatively closed and contains every proper ideal C. (Indeed, if C is quasi-local and of C. C is quasi-local iff J = Jac(C) iff c E J then CC C , so c E Jac(C), implying J = Jac(C). The other implications are clear.)

a

Ja

Theorem 1.8.42. I f R is quasi-local and proper maximally central of rank t , then t = n2for some n, and R/Jac(R) is simple of PI-class n. Proof. Let 2 = Z ( R ) ,and let r l , .. . , r, be a 2-basis of R. R/Jac(R) is a simple PI-ring of some PI-class n, so n2 d t. Reordering the ri, we may assume that Y1,..., ?,* are a basis of R over Z (where - denotes the canonical image in R/Jac(R)). Then by Corollary 1.8.38(ii) R = 27: r i Z , Le., t = n2 .

Corollary 1.8.43. I f R i s Azumaya of rank t, then t moreover, /or each proper ideal I of R, M,,(Z) R/I.

=

n2 for some n ;

Proof. By Corollary 1.8.39 R, is proper maximally central of rank t and is quasi-local, so t = n2 by Theorem 1.8.42. Now if R, take a prime ideal P of Z(R) containing I nZ(R). Then R,/P,R, is a simple homomorphic image of (R/l), (by the obvious map ( r + I ) s - ' --* rs-'+P,,R,), so M , , ( Z ) R,,/P,R, < ( R / U P <,,,ul,R/I. QED

la

In particular, if R is Azumaya of rank n2 then g , and S2n-2 are not identities of any nonzero homomorphic image of R. A Strengthened Artin-Procesi Theorem and Its Applications

To complete the Artin-Procesi theorem, we need to show R

< M,(Z[<]).

Proposition 1.8.44. Let R be any algebra, and let 9 = (maximal +.+ R , by sending r to ideals of Z(R)). Dejine the homomorphism I):R -+ the element whose R,-component is r l (E R,). Then I) is an injection.

npG

Proof.

Suppose re ker I), and let I = AnnZo,(r} a Z(R). For each P in = 0, implying I 9 P. Thus I = Z ( R ) ,so r = 0. QED

.Ywe have r l - '

To prepare for Amitsur's trick, we quote a well-known result from the theory of central simple algebras, to be proved easily in $3.1.

70

THE STRUCTURE OF PI-RINGS

[Ch. 1

Fact 1.8.45 (Proposition 3.1.6). 1 f R is simple @PI-class n. then rhere esists a basis qf R (oiler Z ( R ) )o f t h e form [yj- ' s i l 1 < i , j < 1 1 ) ,forsuitable sir y in R .

We are ready now for the beautiful trick of Amitsur. Proposition 1.8.46. If R is Azumaya of runk n 2 , then j b r some commuratiw Z ( R )-subalgebra C we have R d M,,(C). Proof. In view of Proposition 1.8.44, we may assume R is quasi-local and proper maximally central. Let Z = Z ( R ) and let - denote the canonical homomorphic image in R = R/Jac(R). Then [ R :Z ] = n2 ; using Fact 1.8.45, take si,y in R such that [yj-'.vil 1 6 i,j 6 1 1 ) are a basis for R over Z . By Corollary 1.8.38 ( y j - ' s i l 1 ,< i, j d n } span R over Z , implying x l , . ..,Y,, span R as a module over the commutative ring Z [ y ] . Moreover, by Proposition 1.3.23, R 6 M , ( Z [ y ] ) . QED

Definition 1.8.47. R is an A n - r i q if R < M , ( Z [ l ] ) and no homomorphic image of R satisfies the identity S2,,-2.

We are ready for the Artin-Procesi theorem, strengthened slightly. Theorem 1.8.48.

Thefollowing conditions are equinalent:

(i) R is an An-ring; (4 R d M , ( Z [ t ] ) , and 1Eg,(R)R; (iii) C Z n+L is an identity of R, and gn is R-central with 1 E g,( R )R; (iv) C2n2 + is an identity of R, and g,(R)+ = Z ( R ) ; (v) R is Azurnaya ofrank n2.

,

Proof. (i) => (ii) By definition, R has PI-class n. Let I = g,(R)R. I f I # R then, taking a maximal ideal P 2 I of R , we see that gn is an identity of the simple ring RIP, which therefore has PI-class < n, contrary to the assumption that S2n-2 is not an identity of RIP. Thus g n ( R ) R= R.

(ii) * (iii) Trivial. (iii) * (iv) Proposition 1.7.10. (iv) * (iii) Obvious. (iii) (v) By Theorem 1.8.33. (v) => ( i ) By Corollary I .8.43 and Proposition 1.8.46. QED We shall now use the Artin-Procesi theorem to prove some results about Azumaya algebra; also cf. Exercises. Corollary 1.8.49. I f R is an An-ring and R , is a Z(R)-subalgebrci o f ' R which also is an .4,,-ring, then R, = R.

$1.8.1

Tensor Products

71

xf=,

Proof. Write gn(rilr.. .,rid) = 1 for suitable rij in R,, and note by Lemma 1.4.20 that R s C Z ( R ) r i jc R,. QED Theorem 1.8.50. R is proper maximally central of rank following two conditions hold:

it2 iff

the

,

(i) C2,12+is an identity r f R untl gn is R-centml; (ii) for suitable r , , . . . ,rnr in R the additive subgroup of Z ( R ) spanned bq1 gll((r,}x ... x {rl12} @ R x ... x R ) is ull q f ' Z ( R ) ,i.e., contains 1.

~17~Ucf,,for r , , . .., rrrJus in (ii), these elements,ji?rma base o f ' R over Z ( R ) . On theother hand, if'Ris proper ma.\-imallycentralof'rankn2 and R = C:!: Zri, then r,, . . . ,rlz?satisfj (ii) arid ure thus a base.

,

Proof. (-=) Immediate from Theorem 1.4.21 and Theorem 1.8.48; the rest is easy to see, using Theorem 1.8.48 and the fact gn is n2-normal. QED Corollary 1.8.51. I f R is proper maximally central of rank n2 and R' is a central extension of R , then R' is proper maximally central of rank n2 and R' 2 R O Z c R , Z ( R ' ) .

Proof. Apply Theorem 1.8.50 ( 5 ) to R , and then (-=) to R', to conclude that R' is proper maximally central of rank n 2 ; the last assertion follows from theassertionsat theend ofTheorem 1.8.50,whichshow that any baselifts from R to R'. QED

The Tensor Product Question: An Introduction Having seen tensor products enter naturally into PI-theory, we now ask the tensor product question: "If R , and R , are PI-rings, then is R , @ R , a PI-ring?" Here is an easy partial answer. Theorem 1.8.52 (Procesi-Small R2 6 Mn(Z[<]), then R i O L R 6~ M,tl(

[68]).

Zj R , ,< M,( Z[<])

and

z[iJ]).

Proof. Let A , = Zm{Y } and A , = Zl,{Y } , suitable rings of generic matrices. Since R iis a homomorphic image of Ai,we can write R , 0R , as a homomorphic image of A , @ A 2 . But it is easy to see that A , 0A , E ( A , 0 I Q,{ Y } O0On{Y } c M,,(H) for a suitable Qalgebra H , so R , 0R , d M , , ( H ) ,< M,,(Z[(]). QED Corollary 1.8.53. If R , and R , are semiprime, of respective PI-classes 172, then R , 0R 2 d Mn , . 2(Z [l ]).

n , and

One can extend this argument, to prove (cf. Exercise 2) that if R , is a PIring and R , < M n ( Z [ 5 ] ) ,then R , @ R , is PI. One might hope to finish the

72

[Ch. 1

THE STRUCTURE OF PI-RINGS

tensor product question by using Amitsur’s method ; however, this approach seems to fail. A successful approach was found by Regev [72], and his solution to the question is given in $6.1. The Brauer Group

Simple PI-algebras can be studied very well by means of the tensor product. Let F be a field and define an equivalence on the class of Falgebras by saying R , R, iff R , O FM , ( F ) z R , O FM J F ) for suitable m, n E Z t . Writing [R] for the equivalence class of R, we define Br(F) = ([R]IR is a simple PI-ring with center F ) .

-

Proposition 1.8.54. Br(F) is an abelian group under the operation [R,][R,] = [R, e F R z ] ; [ F ] is the neutral element. Proof.

Collect results from this section, noting that [ R I - ’

=

[R”’’]. QED

Br(F) is called the Brauer group of F , and is of utmost importance in the study of simple PI-rings in Chapter 3. We record one more result, which was really proved in the preliminaries to Kaplansky’s theorem. R

Definition 1.8.55. A commutative Z(R)-algebra H splits R if @ Z , R ) H z M , ( H ) for some n.

Theorem 1.8.56. splits R.

IfR

=

M , ( D ) is PI, then every maximal subfield of D

Proof. View R = End M,, where [ M :D] = t. Letting K be a maximal subfield of D and taking the closure of R with respect to K gives by Theorem 1.8.18, Theorem 1.5.18, and Lemma 1.5.13 that M , ( K ) 2 R @ Z , R ) K . QED

Many results about Br(F) can be generalized to a group Br(C), C an arbitrary ring, when we consider arbitrary Azumaya algebras (cf. Auslander-Goldman [60], as well as the exercises here); a very worthwhile project would be to apply PI-theoretic methods to obtain a systematic exposition of Br(C1.

91.9. The Prime Spectrum

Theorem 1.4.32 shows that if g,(R) is “large enough” for a ring R of PIclass n then R is very nice. Consequently one should expect prime ideals not

The Prime Spectrum

91.9.1

73

containing g,(R) to have similar, nice properties. In this section we examine the prime ideals of R in terms of Z(R), with special focus on the role of g,(R). At the end, we develop some facts about integral extensions for use in Chapters 1,4, and 5. Comparing Prime Ideals of Related Rings Define Spec(R) = {proper prime ideals of R}, partially ordered under set inclusion; given A c R, let Spec,(R) = { P ESpec(R)IA G P}. [There is a topology on Spec(R), whose closed sets are of the form Spec,(R)--cf. Procesi [73B]-but we shall not explore this topology.] Spec( ) plays an important role in commutative algebra, and we shall see that many theorems can be extended to PI-rings by means of comparing Spec of related algebras. Assume throughout that R , c R are algebras. We start with a very useful way of extracting ideals of R from R,. Definition 1.9.1. Suppose A ’ s R , and B d R , . An ideal B’ of R is (A’, B)-maximal if A’ c B’ and B’ is maximal among { I 0 R ) I n R 1 _c B). Remark 1.9.2. If A ’ d R , B d R , , and A‘n R, G B, then (A’,B)maximal ideals exist by Zorn’s lemma. Also, all ( A ’ ,B)-maximal ideals are also (0,B)-maximal. Proposition 1.9.3. I n the above notation, suppose B‘ is (A’,B)maximal. If B is a prime (resp. semiprime) ideal of R , , then B‘ is a prime (resp. semiprime) ideal of R.

Proof.

Suppose B;, B;

2

B’, with B‘, B;

s B‘. Then

( B , n R,)(B2 r,R , 1 E B’, B; r 8 R, c B’ n R , E B

If B is prime, or if B is semiprime with B; = B;, we get B; f i R , c B or B; r ,R , G B, so B; = B’ or B; = B’ [by ( A ’ ,B)-maximality]. Q E D Proposition 1.9.3 is fundamental and will be used implicitly. Definition 1.9.4. possible situations:

Given P I s P , in Spec(R,), define the following

(i) LO(P,), “lying over PI,” means that P‘, n R , = P , for some P‘, E Spec(R). (ii) GU(P P,), “going up from P to PZ,”means that for any P i in ) over P,, with P‘, s P2. Spec(R) lying over P there exists P; ~ S p e c ( Rlying

,,

,

,

(iii) INC(P,), “incomparability over PI,” means that for any P i , P; in Spec(R) lying over P we cannot have PI c P;.

,

74

[Ch. 1

THE STRUCTURE OF PI-RINGS

If LO(!',) [resp. GU(P,,P,), INC(P,)] holds for all P , c_ P , in Spec(R,), we say R, E R satisjes LO (resp. GU, INC). GU(-, P) means "GU(P,,P ) for all P , in Spec(R,)." Remark 1.9.5. If R is an extension of R, and P' is a prime (resp. semiprime) ideal of R, then R , n P' is a prime (resp. semiprime) ideal of R,. In particular, with P' = 0, if R is prime (resp. semiprime) then R, is prime (resp. semiprime). (Indeed, if A B E R , nP' for A, B a R , then (RA)(BR)c P', so RA c P' or BR c P', implying A E R , nP' or B c R , n P'.) Proposition 1.9.6. I n an e.utension R qf' R, GU(-, P) holds iJ'euery (0,P)-mauimal ideal OfR lies over P.

and we can Proof. ( a ) Let P' be (0,P)-maximal. Then P'~spec(R) define P , = P' n R, ESpec(R,). Hence there exists P i ~ S p e c ( R )lying , over P , such that P' c P,. By choice of P', we get P' = P,, so P' lies over P . (+) Suppose P I E P in Spec(R) and P', lies over P , . Let P be (I",,P ) maximal. Then P' is (0,P)-maximal so by hypothesis P' lies over P, proving GU(P,, P). QED Corollary 1.9.7. implies LO.

I n un extension GU(-, P ) implies LO(P), so GU

We can also treat maximal ideals, with GU. Proposition 1.9.8. Suppose R is an extension of R, satisfying GU. Then there is an onto map from {maximal ideals of R ) to {maximal ideals of R,), given by P -+ R, n P'. Proof. Suppose P', is maximal in R, and let PI = P ; n R , . Taking a maximal ideal P , 2 P , of R , and applying GU(P,,P 2 ) to P',,we see P', lies over P,, so P , = P, is maximal. On the other hand, if P is maximal in R , then take some P'eSpec(R) lying over P ; any maximal P" 2 P' also lies over P . QED

Corollary 1.9.9.

If R is an extension of R , satisfying GU, rhen = nR .

n(maximal ideals of R , I n{ maximal ideals of R)

,

Ranks of Prime Ideals

For P~Spec(R1,define rank(P) inductively as follows: If P is minimal in Spec(R) then rank(P) = 0; otherwise, rank(P) = 1 +max{rank(Q)IQ c P and QESpec(R)}. In other words, rank(P) is the largest possible number m in a chain P o c P , c ... c P , = P in Spec(R); rank(P) need not necessarily

$1.9.1

The Prime Spectrum

75

be finite, but when finite is often very useful. Localization ties in well with the above notions, as we now see. Lemma 1.9.10. Suppose S is a submonoid of Z(R), P$Spec(R) with P n S = @,and r s - l EPs,for some r in R and s in S. 7hen r E P . Proof. For some .Y in P and s, in S, we have rs-' = ss;', (rs, - ss)s2 = 0 for some s2 in S . Hence rsls E P , implying r E P. QED

so

Proposition 1.9.11. Suppose S is a submonoid of Z(R). Then there is a canonical inclusion-preserving, 1 : 1 corresponding from { P E Spec(R)I P n S = 0) onto Spec(Rs) given by P -+ P , ; the inverse correspondence is B v; ( B )for B E Spec(R,); in particular, jor all P E Spec(R) with P n S = 0 we have rank(P,) = rank(P). -+

Proof.

Clearly if BESpec(R,) then v;'(B)ESpec(R) and v;'(B)nS 1.9.10. QED Definition 1.9.12. Let rank(R) = max{rank(P)IPESpec(R)}

=

0.The rest follows easily from Lemma

Corollary 1.9.13.

For any submonoid S ofZ(R), rank(R) 2 rank(R,).

Here is a similar result for arbitrary extensions. Proposition 1.9.14. Suppose R is an extension qf R,. (i) IfINC holds and P E Spec(R,), then l o r every P' E Spec(R) lying over P , rank(P) < rank(P). (ii) If G U holds and rank(P) = k (in R,), there exists P' lying over P with rank(P') 2 k. (iii) I f G U and INC hold, then there exists P' lying over P , with rank(P') = rank(P). Proof. (i) Suppose Pb c P', c ... c P:,= P'. Letting P i = P i n R,, we get Po c ... c P , = P by INC; thus rank(P') < rank(P). (ii) Suppose Po c P , c ... c P , = P in Spec(R,). Take Polying over Po and inductively apply GU(P,, Pi+') to Pi to obtain P i + , 3 Pi lying over Pi+'.Then rank(P;) 2 k . (ii) Immediate from (ii) and (i). QED

Corollary 1.9.15. !f R , E R is an extension satkjjving G U and INC, then rank(R) = rank(R,). Spec, ( R )

We shall illustrate the ideas of the last few pages for the case R is a ring of PI-class n, with center Z , and R , = Z. (Obviously R is an extension of Z . ) Recall that 0 # g,(R)' a 2. Definition 1.9.16. SpecJR) = (PESPeW)lg,(R) $ P } .

=

(PESpec(R)lg,(R) $ P } and SpecJZ)

76

THE STRUCTLJRE OF PI-RINGS

[Ch. 1

There is a beautiful correspondence between Spec,( R ) and Spec,(Z). , and P'n Z Remark 1.9.17. Suppose P ~ s p e c ( Z )P'ESpec(R), P G P' iff P, s Pi,. (Immediate from Lemma 1.9.10.)

G

P.

Theorem 1.9.18. Suppose P ~ s p e c , ( Z ) . There is a unique P' in Spec(R) lying over P ; in ,fact, P' contains every ideal of' R whose intersection with Z is contained in P. Thus,for any P I , P , in Spec,(Z), we hare LO(P,), GU(P,, P z ) ,and INC(P,),,from Z to R. Proof. P , is maximal in Z , so by Theorem 1.7.11 there is a maximal ideal of R, lying over P,. Hence by Proposition 1.9.11 and Remark 1.9.17 there is some P' in Spec(R) lying over P. The rest of the theorem will follow immediately from the following claim: If Ad R and A n Z G P then A E P'. To prove the claim, take P to be ( A , P)-maximal. Then p,~Spec(R,) and P, c P'p by Theorem 1.7.11, implying P 2 P 3 A , as desired. QED

Before putting theorem 1.9.18 in final form, let us turn to maximal ideals. Proposition 1.9.19. (In this result we do not require R to have PIclass 11.) Suppose R is I I liomomorphic image of' R such that ,for a suitable multilinear, R-cerirral polyrwmial g, 1 E g ( R ) R . Tlieri Z = y(R)' = Z(R).

By Proposition 1.7.10, I EIJ(R)+. But g ( R ) + 4Z ( R ) , so Z ( R ) __ = y(R)+ = g(R)' E Z s Z ( R ) ; hence equality holds at each stage. QED Proof.

Corollary 1.9.20. Suppose P~spec,,(R)is a maximal ideal of R. Then Z(R/P) 2 Z / Z n P canonically, so Z n P is a maximal ideal of Z . Proof. Put R = R/P. Since 0 # g , ( R ) ' d Z ( R ) , a field, we get 1 Eg,(R)', so by Proposition 1.9.19 Z ( R ) = Z 2 Z / Z n P . QED

We are ready for a key result about maximal ideals. Theorem 1.9.21. Suppose P is a maximal ideal of Z and PeSpecJZ). Then PR is a ma.vima1 ideal of' R, and is the only proper ideal of R containing P. Proof. By Theorem 1.9.18 some P' in Spec(R) lies over P, so, letting A PR, we have L n A = P. Let - denote the canonical image in = R / A . Then Z z Z/P is a field, so 1E g , ( R ) + ; by Proposition 1.9.19 Z ( R ) = Z is a =

field. Hence R is simple by Theorem 1.4.26, so PR is maximal. QED Corollary 1-9.22. I f P' E Spec,(R) is a maximal ideal of R , then P' n Z is maximal in Z,irnd P' = ( P n Z)R. Proof.

Appl! Corollary 1.9.20 to Theorem 1.9.21. QED

$1 9 . 3

The Prime Spectrum

77

(These results do not hold for arbitrary maximal ideals; cf. Examples 5.2.28 and 5.1.19.) We shall now complete Theorem 1.9.18. Lemma 1 9.23. Suppose P E Spec(Z) and P' is (0, P)-maximal (in R). Jf P' E Spec,(R), then P E Spec,(Z) ( a d so P E P). Proof. P p is a maximal ideal of R , and is in Spec,(R,). Thus by Corollary 1.9.20 P', n Z(R,) is maximal in Z(R,), so 1.1 EP',+g(Rp).+ = ( P ' + g ( R ) + ) , , implying p + s = s for some p in P', .Y in g(R)+,and s in Z - P. Then p = s-.Y E P' n Z G P , implying .Y$ P , i.e., P E Spec,,(R). QED

Lemma 1.9.23 puts Theorem 1.9.18 into proper perspective, because we see that P' E Spec,(R) iff P' n Z E Spec,(Z). Before utilizing this result, we take a brief diversion I.o the Jacobson radical. Remark 1.9.24. Jac(R) P, 2 G Jac(Z). (For if 2 E Jac(R) n Z then ( 1 -I)-' E Z, proving Jac(R) n Z is a quasi-invertible ideal of Z.) Remark 1.9.25. For a semiprime PI-ring R, if Jac(Z) = 0 then Jac(R) = 0. (For Jac(R) n Z = 0, so apply Theorem 1.6.27.)

In general, Jac(R) n Z # Jac(Z); cf. Exercises 3, 5. However, the converse of Remark 1.9.25 is true. Theorem 1.9.26.

I f Jac(R) = 0 and Ann,(g,(R))

= 0,

then Jac(Z)

= 0.

Proof. Let I = g,(R) and J = Jac(Z). If ZJ # 0 then there is a maximal ideal P of R tzar containing ZJ. By Corollary 1.9.20 Z n P' is maximal in Z, so ZJ s J G Z n P', contrary to choice of P'. Thus IJ = 0, implying J = O . QED

In particular, if R is prime and Jac(R) = 0 then Jac(Z) = 0. This result will be extended to the better theorem that for every PI-ring R, if Jac(R) = 0 then Jac(Z) = 0; I can only prove this using rings without 1, and so defer the proof to $1.1 1. Prime Ideals Minimal over g,(R)

We continue to assume R has PI-class 11, and Z = Z(R). We want to push the results of the last subsection a bit further, to primes just a little too big for Spec,(R).

If P , P , P , ... in Spec(R) then (ni Pi)€Spec(R). Thus by Zorn's lemma Spec,(R) has minimal elements, which we call A-minimal primes; in fact, if P ~ s p e c , ( R )then P contains an Remark 1.9.27.

3

3

3

A-minimal prime. The (0)-minimal primes are the prime ideals of rank 0.

78

[Ch. 1

THE STRUCTURE OF PI-RINGS

If R is prime, “minimal prime” means prime ideal

Definition 1.9.28. having rank 1.

It may well be that all minimal primes of R contain g,(R) ‘, but even in this “bad” case we shall squeeze out some information. Theorem 1.9.29. I / P , c P , GU(P,, P 2 ) holds from 2 to R .

E

Spec(2) and P , is g,,(R)-minimal,then

Proof.

Obviously P , ~Spec,(Z).Take P‘, lying over P , , and take P, to ( P i , P,)-maximal. P‘$ Spec,(R) by Lemma 1.9.23. Thus P , ; since Z n P, E Spec(Z), we conclude Z n P; = P , . g,,( R ) s Z n 1’; QED be

Corollary 1.9.30.

If P ESpec(Z) is g,(R)-minimal, ther7 LO(P) holds.

(Unfortunately, INC cannot be extended to g,(R)-minimal primes.) Proposition 1.9.31. satisfies LO. Proof.

Zf R is prime, then erery minbnal prime of Z

Either PESpecJZ) or P is g,(R)-minimal.

QED

(The PI-hypothesis in Proposition 1.9.31 can be removed easily.)

integral Extensions of Commutative Rings

An important special case of the results on rank occurs when R is integral over a commutative subring C of Z(R) (viewing the extension C E R ) . Given r E R, write C [ r ] for the C-subalgebra of R generated by r ; obviously C [ r ] = { x r i c i l r E, R } is commutative. Remark 1.9.32. spanned by [rJIO< j

If r is integral of degree 1 ) over C.


t

over C then C [ r ] is

Lemma 1.9.33. 1 f R is irttegral over C and C is integral over C , then R is integral over C,.

2

C,

Proof. Pickanyrin R. Wecanwriter‘ = ~ ~ = : , c i r i f o r s u i t a b lEe ~C.. iLet C’ be the C,-subalgebra generated by co, . . . ,c,- Each ciis integral over C , of some degree ti, implying {c$...cj’::lO < j i < t i ) span C‘ over C,.Thus [ C ’ [ r ] :C,] < cx so, by Proposition 1.3.24, r is integral over C , . QED

,.

Modifying the usual commutative proofs (for example in Kaplansky [70B]), one can show that if R is a Pf-ring integral over C. then LO, GU, and INC hold ; the proofs are omitted here, because more general results are given in $4.1 (proved from scratch). (In fact, there are even more general

$1.9.1

The Prime Spectrum

79

results, due to Bergman, given in the exercises to $54.1 and 7.6.) Thus the earlier theorems of this section apply to integral extensions, and we shall have frequent occasion to use them. For the remainder of this section, we collect standard facts about integral extensions. Let H be a commutative C algebra. Proposition 1.9.34. H.

{ r E H l r is integral oiler C } is a C-subalgebra of

Proof. Suppose r l , r2 in H are integral over C. By Lemma 1.9.33, ( C [ r I ] ) [ r 2is] integral over C and contains r , + r , and r l r 2 . QED

(Later we shall extend this result to PI-rings, using Shirshov's theorem.) Definition 1.9.35. The integral closure o f C in H is the subalgebra { r c Hlr is integral over C} of R . If C is a domain, the derived normal ring of C is the integral closure of C in its field of fractions; C is normal if C is its own derived normal ring. Remark 1.9.36. The derived normal ring of any commutative domain is normal, by Lemma 1.9.33. Remark 1.9.37. If x : f = o c i r=i 0 for ci in C , r in H then for every c in C , cc,r is integral over C ; in fact, (cc,r)' = - ~ ~ ~ ~ e f ~ i c ~ ~ l ~ i c i ( c c r r ) i .

This simple computation instantly produces the next two results. Remark 1.9.38. If F is a field algebraic over C then F is the field of fractions of the integral closure of C in F . Proposition 1.9.39. I f C is integrally closed in H , then .for ererj' submonoid S OfC, C, is iritegrally closed in H,y. Proof. If r E H , s E S, and rs-' is integral over C , then for suitable s' in S by Remark 1.9.37, s'r is integral over C, so s'r E C and rs- = (s'r)(s's)-' E C,. QED

Corollary 1.9.40.

Every tocalization o f a normal ring is normal.

Proposition 1.9.41 (Gauss' lemma). Suppose C is normal with $field offractions F , a n d f ~C[A] is monic. ! f g , h E F[I] are monic a n d f = g h , then g, h E C[A]. Proof. There is an extension field K of F such that in K[I] we can write g = ni(A-xi) and h = n j ( I - y j ) for suitable x i , y j in K . But each xi and y j are roots of gh =,f and are thus integral over C ; hence the coefficients of g and h, being sums of products of the xi and y j , are elements of F integral over C, and are thus in C . Hence g . h E C[A]. QED

80

THE STRUCTURE OF PI-RINGS

[Ch. 1

We conclude with one more, very useful observation. Remark 1.9.42. If C E Cand c - ' in R is integral over C then c C 1 EC. (Indeed, ( c - I ) ' =: ~ : Z A C ~ ( C - ' for ) ~ suitable ci in C, so c - , = c ' - ~ ( c - ' ) '= cic' - I - E c.)

x;A

$1 .lo.Valuation Rings, ldempotent Lifting, and Their Applications

This section follows on the heels of $1.9, using many of the same techniques (and notations), but with two important new features-the use of algebras over valuation rings, and lifting of matric units. With these extra ideas we shall prove a major theorem of Bergman-Small [75] describing the PI-class of a prime ring in terms of the PI-classes of its homomorphic images. Along the route we shall also obtain a pretty theorem of Procesi-Marko, determining the rank of an affine, prime PI-ring in terms of the transcendence degree of its center. Valuation Rings Let C be a commutative domain with field of fractions F. Definition 1 .I 0.1. EC.

C is a oaluation ring of F if for every element x E F

either X E Cor x

In a ring R , say r , divides r , (written r , Ir,) if r , = rr, for some r in R . Obviously r , Ir, iff Rr, E R r , . Here is the key to valuation rings. Proposition 1.10.2. I n a valuation ring C of F , for m y Jinite set ( c , , . . . ,c,} of C, vome ci divides all c j , I < j < t .

if By induction on t . (Trivial for t = 1). Reordering (c,, . . . ,c,necessary, we may assume c, Icj for all j < t - 1. Consider x = c,c;' E F . If X E C then c, Ic, and we are done. Otherwise x - l EC, so c,Icl, implying c,\cj for all j d t . QED Proof.

Corollary 1.10.3. The ideals of a iialuation ring V are totally ordered; i.e., i f A , , A 2 Q V'. then A , E A , or A , G A , . Proof. Suppose A , $ A , . Pick an element a in A , - A , . For each x in A , we have V L9~ Vx, so V x s Va, i.e., A, = V A , E Va E A , . QED

Recall Remark 1.8.41.

$1.10.1

81

Valuation Rings

Corollary 1.10.4.

Every vuluatiorz ring C is quasi-local.

Proof. Let J = {noninvertible elements of C ) . If x , , x ~ E J ,then s,I (.Y, x,) or .Y,I(.Y~ +s,), implying (r,+s,) E J . Hence J 4C and C is

+

quasi-local. QED We say a ring R lies above a ring R , if R , c R and Jac(R,) c Jac(R). For example, by Corollary 1.9.9,if R is an extension of R satisfying GU, then R lies above R

,.

Corollary 1.10.5. I f V is a valuation ring of F and a quasi-local domain V, c F lies above V, then V, = V. Proof. Obviously V c V,. Now take any x in F . If .xY1EJac(V) G Jac(V,), so x # V,. Thus V = V , . QED

XI$V

then

One can use Corollary 1.10.5 to characterize all valuation rings between a principal ideal domain and its field of fractions; we shall demonstrate this idea for Z. Example 1 .I 0.6. If p is a prime number then for P = pZ ESpec(Z), L , is a valuation ring of Q. Every valuation ring of Q (other than Q itself) has the form Z, for suitable P = p Z 4 Z.Thus every valuation ring of Q has rank d 1. (The first assertion is immediate. To prove the second assertion, suppose V is a valuation ring of Q. Then some integer m is in Jac(V), implying, for some prime integer p dividing m, pEJac(V). But, letting P = pZ~Spec(L), we see that V lies above Z,,so by Corollary 1.10.5 V = E,, proving the second assertion. Finally, Z, has rank 1 because Z has rank 1.)

On the other hand, there are an abundance of valuation rings. Remark 1.10.7. If V is a valuation ring of F and V c C G F then C is a valuation ring of F. (Immediate from Definition 1.10.1.) In particular, every localization of V is a valuation ring of F. Remark 1.10.8. If V is a valuation ring of F and PESpec(V) then VIP is a valuation ring. (Indeed, let V = V/P, and let F , be the field of fractions of V . By Proposition 1.7.2 there is a canonical onto homomorphism $ : V , + F , . For any $(x) in F , we have X E V or X - I E V ; thus $(x) E V or $(x)- E V . )

There is a well-known method for finding a valuation ring containing a given commutative domain in a certain “nice” way, which Procesi generalized in the next theorem to PI-rings. Lemma 1.10.9. Let R be a torsion-jree algebra over a commutative ring C , and let S = C - { O } . Suppose P E Spec(R) and c is an arbitrary element Of’C,. Then ( i n R,) either R n PR[c] = P or R n PR[c-‘] = P.

82

[Ch. 1

THE STRUCTURE OF PI-RINGS

Proof. Suppose not. Then, letting c1 = c and c2 = c - l , put Pi R n PR[ci] 3 P, so that 0 # P J P Q RIP, a prime PI-ring. Hence there exists x ~ E Psuch , that 0 # x i + P ~ Z ( R / P )we ; choose .K, such that, writing x i = ~ ~ ~ ~forpsuitable , ~ pci j in { P , we can take mi minimal. Write m = m , , n = m2 ; by symmetry we may assume m 3 n. We shall now get a contradiction in the choice of Y ] . Let u’, = . Y ~ ( . Y ~ - ~ ~5~P), , Eand P ~ note P ~ that ( u 2 - p 2 , ) + P = Y , + P , so 0 # s‘,f P E Z ( R / P ) [since Z ( R / P )is a domain]. Then =

m

ni-

m- 1

m-l

1

+

=

C

~ l j ( ~ 2 - ~ 2 0 ) ~ < j= 0

n

~lm~2,m-jcil.

j=m-ii

Since p l j ( s 2 - p 2 , , ) E PI and p l m p 2 . m - jP~ I for all j , we have lowered m, contrary to choice of x,. QED Theorem 1.10.10 . Suppose R is a PI-ring that is torsion-iree oiwr a subring C c Z ( R1, and S = C - (0). Given any P E Spec(R), take an arbitrary subring V 2 C of C,, maximal with respect to R n PV c P . (Such V e.uists by Zorn’s lemma.) ‘Then V is a vuluation ring and R V s R,, ant1 Some Q E Spec(R V ) lies over P.

Proof. In view of Proposition 1.9.3 we need only prove that V is a valuation ring. If not, there is an element x of C, such that XI$ V and x - # V. But applying Lemma 1.10.9 to R V and the prime ideal Q we reach a contradiction to the maximality of V. QED

The Transcendence Degree, and Its Application to Rank Suppose C c H are commutative domains. “Algebraically C independent” will mean “algebraically independent over C.” By Zorn’s lemma, any set of algebraically C-independent elements of H can be expanded to a maximal such set, which we call a transcendence basis of H over C ; the smallest order of a transcendence basis is called the transcendence degree of H over C , written trdeg(H/C). Our goal IS to relate the transcendence degree to the rank of a PI-ring (in certain cases); the main step will involve the use of valuation rings. We begin by looking at transcendence bases, which are easy to investigate when one realizes that they behave analogously to bases of vector spaces (cf. Zariski-Samuel [58B, p. 96]), as we shall see now.

$1.10.1

83

Valuation Rings

Remark 1 .I 0.11. If F, F‘ are the respective fields of fractions of C, H, then tr deg(H/C) = tr deg(F‘/F), seen by “clearing denominators.”

For sl, . . . ,s,in H, define inductively C[.Y,,. . . ,r,] = ( C [ s , ,. . . , s,-,])[.Y,], the C-subalgebra of H generated by x I , .. , x,. ,

Remark 1 .I0.12. Suppose .s,,. . . , skare algebraically C-independent elements of H. Then for any x in C either .Y is algebraic over C[x,, ..., xk], or ( x I ,. . . ,x k ,.Y] are algebraically C-independent. Lemma 1.10.13. Suppose H is algebraic over C[x,, .. ., sk] and l , . . ., xu> 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~

<,

86

[Ch. 1

THF STRUCTLJRE OF PI-RINGS

independent. But rank(R) < rank(R); moreover c, $ P, so PESpec,(R) and __ R has PI-class ti. By induction, there is a chain 0 c c ... c P,- in Spec,(R) that lifts to 0 c P c P c ... c P,- in SpecJR). QED

,

,

In Exercise I this theorem is generalized to finitely generated extensions over arbitrary rings. We shall not enter into the subject of classifying PIrings through their ranks, an area with much research potential. Procesi [73B, pp. 117 -I??] discusses the rank 0 case for finitely generated PIalgebras over conimutative Noetherian rings; also cf. Braun [77P]. Affine Rings of Generic Matrices We shall use algebras of generic matrices to continue the study of the rank of affine PI-rings, as follows: Suppose R = F { x , , ..., x,} satisfies a polynomial identity of degree d. We plan to estimate rank(R) in terms of k and d . If k = 1 then R is commutative and rank(R) < 1 by Theorem 1.10.23, so we shall assume k >, 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[<]).Thus there is a homomorphism F,{ Y } R sending -+xi, 1 < i < k ; we restrict this to a homomorphism $: F,{ Y,, . . ., Y,} -+ R, which clearly is onto.

-

Proposition 1.10.24.

F o r k 3 2, F,{ Y , , . . ., Y,) is prime oj”P1-classn.

Proof. One can check that M , ( F ) is a homomorphic image of F,{ Y , , Y , } , so M , ( F ) < F,{ Y , , . . . Y,J G M , ( F ( < ) ) . But then F,{ Y , , . . . , Y,) F ( ( ) is a subalgebra of M , ( F ( ( ) ) having PI-class n, so by Theorem 1.4.8 Fa{ Y, ,..., Y,}F(<) = M , ( F ( { ) ) ; thus F,: Y , , ..., &} is prime of PI-class n. QED

.

(Procesi). I f F is u Jielrl und R = F i x , , . . . , x k } Theorem 1.10.25 satisfies a polynomial idrnrity of degree d , then .for n = [ d / 2 ] , rank(R) < trdeg(Z(F,{ Y , , . . ., Y k i ) / F ) ,equaliry holding iff R * F,( Y , , . . ., K}. Proof. Continuing the analysis preceding Proposition 1.10.24, we may assume k 2 2 and t,b: F,( Y , , . . . , Ykj R is onto. If $ is an isomorphism then rank(R) = rank(F,{ Y,, . . . , y k ) ) ; otherwise, any Po c ... c P, in SpecfR) In lifts to O c $ - ’ ( P , ) c . ~ ~ c $ - ‘ ( P , ) , so m + l drank(F,(Y,, either case. we conclude with Theorem 1.10.23. QED -+

...,&I).

Let t , ( k ) designate trdeg(Z(F,( Y,, . . . , Y , ) ) / F ) . To reach our goal of estimating rank(R), we must determine t , ( k ) for all n and k . Procesi proved t , ( k ) = ( k - I),? 1 ; we shall now show “6,”yielding the best possible bound for rank(R); Procesi’s theorem is given as part of Theorem 3.3.31. A -t

4 1.10.1

87

Valuation Rings

field extension generated (as a field) by a transcendence basis is called purely transcendental. Note first that by Proposition 1.3.15 we can replace Fn{Yl,..., Yk) by Fn{ Y;, Y 2 , .. . , Yk},where Y; = El=, <:,!'eiiis diagonal. Write Fk.nfor the field of fractions of Z(F,{ Y;, Y 2 , .. . , yk}); thus t,(k) = trdeg(F,,,/F). Let F;," be the subfield of F ( < ) generated by all and @, 1 6 i, j 6 ti, 2 6 u 6 k . Then FL,l,is a purely transcendental extension of F, with trdeg(Fk,,,/F)= 17 + ( k - l)n2. We view F,,: Y,', Y2, . . . , YkJs J V ~ ~ ( F canonically, ~ , ~ ~ ) putting CI = e 1 1 + ~ ~ = 2 ( \ : ' e iand i , let R = aFn{Y,', Y2,..., Y k ) a - ' = FCY;', ..., y,"), where Y," = aY;a-' = Y,' (since all diagonal matrices commute), and Y: = uY,u-' for all u 3 2.

clt)

Remark 1.10.26. Z ( R ) = Z ( F , ( Y ; , Y, ,..., yk)). Remark 1 .I 0.27.

Y~"= i;'Je,

For each u 3 2,

R

I1

n

j= 2

i= 2

i.j= 2

+ 1 t'$<\?))-'elj + C t\:)t$eil + 1 t\:)t$)(<(h))-'eij.

In particular, the coefficient of e , i i n YJ is 1 for allj 3 2. Theorem 1.1 0.28. Let Kk.llhe the subfield of' Fk,ngenerated (over F) by all the entries of the Y,", 1 d II d k . Then {(\:)12 < i 6 17} generate F;,n (as a ,field) o w Kk,nrand Kk,n is purely transcendental over F of transcendence degree ( k - l ) n 2+ 1. ( I n ,fact the entries q f the Y: other than 0, 1 .form a transcendence basis.)

Note first that by adjoining all the ti:), 2 6 i < n, we recover all the original ti:) from the entries of the Y,", and so {<\:)I2 6 i < t i ) generate FL,nover Kk,n.Call an entry nontrioial if it is distinct from 0, 1. Clearly Y," has ti nontrivial entries, Yg has n2 - (17 - 1) nontrivial entries (cf. Remark 1.10.27), and Y l , . . ., &'' each have n2 nontrivial entries. Letting t = trdeg(K,JF), we see that K k , n is generated by ti + ( n 2 - ( t i - 1)) + (k - 2 ) n 2 = ( k - 1 )n2 + 1 elements, so t < ( k - 1 ) n 2 + 1. But Proof.

(k - 1)n2 + 17

= tr deg

< ((k

+

(F;JF) < t tr deg(FL.ll/Kk,,l) + 1 ) + (11- 1) = ( k - l ) n z +PI.

- 1)ri2

Therefore equality holds at every step, proving the theorem all at once. QED Corollary 1 .I 0.29.

t,(k) d ( k - 1 )n2 + 1. Lifting ldempotents

In proving a PI-theorem, it is usually difficult to pass from the semiprime case to the general case. Here is one useful technique, which leads to an

88

THE STRUCTURE OF PI-RINGS

[Ch. 1

improved version of the Artin-Procesi theorem ; it is also crucial to the Bergman-Small theorem. Since there are good treatments in Jacobson [64B] and Lambek [66B], we omit some details. The underlying idea is that for A Q R and R I A z M J C ) we want to be able to conclude R 2 M , ( T ) for some subring T of R. T o d o this, we need to “lift” the matric units (eijl 1 < i,j < n) from RIA to R. Definition 1.10.30. An element e of R is idempotent if e2 = e ; e is nontrivial if e # 0, 1. Idempotents e l , e, are orthogonal if e l e 2 = r 2 i l = 0.

Given an idempotent e, clearly (1 - e ) is an idempotent orthogonal to e. One readily concludes that 0 is t he only idempotent ofJac(R).[Indeed. ife E Jac(R)is idempotent, then c = e(l - e ) ( l -e)-I = 0.1 Definition 1 .I0.31. An ideal A Q R is idempotent-lijiiny if each idempotent of RIA has the form x + A, for a suitable idempotent x of R.

(Note that 0, 1can be lifted respectively to 0, 1, so we need only check that nontrivial idempotents of RIA can be lifted.) Proposition 1.10.32. Suppose A Q R is idempotent-ljfiing and A E Jac(R). Then,for every idempotent r of‘R and everv idempotent e of R I A orthogonal to r A there is an idempotent x of R orthogonal to r, with x + A = e.

+

Proof. Write e = x,+A, where xo is idempotent; noting that x,rEA c Jac(R), let x1 = (1 -xor)-’xo(l - x o r ) , an idempotent of R . Clearly x1 = ( I - . ~ ~ r ) - ~- r .) ,~ so ~ ( xl l r = 0 ; we finish by putting .r = (1 - r ) x l . Q E D

Definition 1.10.33. (eiil1 6 i , j d n } is a set of ( n x n ) mcctric units (of R ) if the following conditions hold: (i) eii = 1 ; (ii) eijejk= eik for all i,j, k ; (iii) eijeuv= 0 whenever j # u. Proposition 1.10.34. I f R has a set of matric units {eijl1 d i,) < n), then R z M,,(T),fi)ra suitable ring T with respec1 to these eij.

Let T = {xi=] e u l r e l u ) r ~ RThen }. the eij and T Features of proof. generate a subring M , ( T ) of R . But for any r in R, letting rij = Et= euireju = xi=leut(elirejl let,, E T, we get rijeij= eiirejj, implying r = CL=]rijeij EMJT). Q E D Remark 1 .I0.35. Since T was constructed canonically, we see that if R = M , ( T ) and R .+ R is a homomorphism then R = M,(T). Theorem 1 .I0.36. Suppose A E Jac(R) and A is an idernpotent-lifting ideal of R. I f R;’A = M,,(T), then R z M,,(T,) for a suitable ring TI such that (Tl + A ) / A = T .

41.10.1

89

Valuation Rings

Proof. Let [eij(l< i,j < 11) be a set of matric units of RIA, .yo = 0, and inductively (on u ) lift eurrto an idempotent I,, of R orthogonal to .Y,- I = ~ y Z , ' , u i i ; for i < u, N,,,,.Y~~ = .Y,,s,_ sii= 0 = xiixUu. Take r l i and ril in R , such that r,,+A = e l i and ril + A = e i l ; define y l i = s l l r l i s i iand .y I. l = .y..r. rly . then.^,, -yli.sil E Jac(R)soforsomerinR,r(l - . y l l + y l i . u i l ) = I . Let s l i= .ullryli,and note .u1,+A = eLi.Moreover,.ulisil = . ~ ~ ~ r y , ~ s ~ = xllr(1- . ~ l l + ~ l i . ~ i l ) . = ~ lx l l , . It followsthat . ~ ~ ~ - . s ~ ~idempotent .~~~isan ofJac(R)and thus 0, so also sils1 = xi'. Define sij= y i l x1jfor all i j . The .yiiare a set of matric units of R , and the rest is immediate. QED

Here is the main tool for lifting idempotents.

x:=,,+

zirl jor suitable t 3 Lemma 1 .I0.37. Suppose r E R unrl r" = I I + 1, with ziE Z(R). Then e = z,"r')" is idempotent. Moreover,,fbr any homomorphic image R of R in which J is idempotent, we have J = F. and a = .xr. Then r"a = r"um for all m. In particular, e =-a" = x"r" = x"(r"a") = anan= e2. Now if F is idempotent we have J = (r")ll t-n --i+n n f - n -1 n - = Zi+,,r ) = ( X i = lzi+"r') - e. QED Proof.

Let

x

(x:,,:; z,+] +xi:; z n+i ri -l ,

=

x:I=n+l ziri r", so by induction r" =

=

Theorem 1.10.38. Any nil ideal A of R is idempotent-lifting. Proof. Suppose F = r + A is an idempotent of RIA. Then r2 - r e A , so (r2-r)n = 0 for some n. Expand and apply Lemma 1.10.37 to lift J to an idempotent of R. Q E D

Remark 1 .I0.39. By Lemma 1.10.37, if R is algebraic over a field, then every ideal is idempotent-lifting. This result is due to Levitzki.

The material developed in the remainder ofthis sectioii is to be used exclusively in proving the big Bergman-Smull theorem, although it has considerable independent interest.

The A-adic Completion Definition 1.10.40. Suppose Ad R . Define A-adic(R) to be the set of sequences {(ri+A') = ( r , + A , r 2 + A 2 , ...) En:, (R/A')[rj-riEAi for all i, all j > i). Definition 1.10.41. A'

=

R, and A"

=

(-)El A'.

Remark 1 .I0.42. A-adic(R) is a ring, under componentwise addition and multiplication of sequences. (Indeed, for (ri+ Ai),(ri + A ' ) E A-adic(R), and for j > i (rjr)-rirl) = ( r j - r i ) r ) + r i ( r ) - r i ) E A i . ) If A" = 0 then the

90

THE STRUCTURE OF RI-RINGS

[Ch. 1

canonical homomorphism R + A-adic(R), given by r + ( r + A'), is an injection, by which we view R L A-adic(R).

a

Theorem 1 .I 0.43. Siipposr A R is iiiemporrnt-lrffii~g, arid A' = ((a,+,4') E A-adic(R)lu, = O}. Theri A' R' is idernpotertt-lifriny.

a

let

Proof. Write K' = A-adic(R). Clearly A'd R' and A = R n A'. Thus we can view RIA c R'JA'. In fact, RIA = R'IA': indeed, if ( r i + A ' ) E R ' then (ri+ A ' ) - (rl + A ' ) EA'. N o w suppose e E R'IA' is idempotent. Then e = (el + '4, e l + A L , .. . ) + A ' for a suitable idempotent e , + A of R I A . We shall build a sequence (ei+Ai)E R' such that for each i, e , + A i is an idempotent of R,'A'. Well, inductively, suppose we have e l , . . .,P , Now RIA"-' :( R / A " ) / ( A U - l / A " ) , and (A"-1/,4")2 = 0, so by Theorem 1.10.38 we can lift e u - l + A " - ' to an idempotent e,+A" of RIA", as desired. Thus e has been lifted to the idempotent (ei A '). QED

+

Definition 1.10.44. R is A-aifically cornplere if for each ( r i + A ' ) in A-adic(R) there is some r in R such that r - ri GA' for all i. Remark 1.10.45. Suppose A" = 0, and view R G A-adic(R). If R is A-adically complete then R = A-adic(R). (Identify each sequence with its "limit" in R, easilj seen to be uniquely determined.)

The Completion of a Valuation Ring

Here is a different sort of completion, for valuation rings, which we need for the Bergman- Small theorem. There are many references (cf. Bourbaki [72B]) so we shall omit the routine details. A sequence ( c l , c 2 ,...) of a valuation ring C is called Cauchjl if for each C E C there is some i, (depending on c ) such that cI(c,-ci) for all j > 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)<

BE p(.P/)

d(B)+

d(B) BE+

d

Letting .?A' = (A- . d )u p ( . d ) , we see that { B n C ( BE 8 ) has o n l y < (rn - 1) distinct elements (since Po has been thrown out), so, by induction, X B E ,,d(B) < n. Thus x n E.d(B) < , . d ( B ) < 11. QED

xBG

This cute induction argument has a very important consequence.

Corollary 1.10.61 . Suppose R is rorsion free over a ruluution ring C, arid dim(R ;C) = 11. Theri R has at most 11 maximal i d w l s .

Proof.

{Maximal ideals of R J are incomparable. QED

One can also obtain good results concerning the rank. improving Theorem 1.10.18 in our special situation. Theorem 1 .I 0.62. Suppose R is u prime algebra oz'er u criluarion ring C , with dim(R ;CI < K . Also suppose we have a caluation ring C, E C with trdeg(C/C,) = t a.Then rank(R) < (rank(C,)+ I ) ( t + 1). c::

Proof. Clearly it suffices to show for any P ~ S p e c ( C , )and any chain Po c ... c P, in P -Spec(R) that necessarily k < t . Using lemma 1.10.57 and the trick of Remark 1.10.16,we may pass from R, C, C , , respectively, to RIP,, C/(C n Po). and C,/P. Thus we may assume that P o = P = 0. Now let S = C, - (0) and let F = (Co)s,a field ; P i n S = 0 , O < i < k , so we may pass to Rs, C,, and F, respectively. But then R is obviously algebraic over ( C , ) ; hence t = trdeg(C',/F) = trdeg(Z(R,)/F), so we are done by Theorem 1.10.18. QED

Corollary 1 .I 0.63. Suppose R is a prime algebra over a caluation ririg C , with dim(R ;C ) < co,and let F , be t h e j e l d of fractions of the suhriny o f R generated bj' I . ff'rrdeg(C/F,) = t < m , then rank ( R ) < 2 ( t + I ) .

$1.10.1

Valuation Rings

95

Proof. Let C , = C r , Fo, clearly a valuation ring; clearly either C , F , or F , = Q, in which case C, has rank < 1 (by Example 1.10.6)and we are done by Theorem 1.10.62. QED =

Corollary 1.10.64. I f R is a prinze algebra orer a valuation ring C, with dim(R ;C) < 00, and if the.field offractions OfC isjnitely generated as a j e l d , then rank(R) i s j n i t e . (Zn particular, taking R = C , we also get rank(C) isfinite under the given hypotheses on C.)

We also need some more information about central extensions. Remark 1.10.65. Suppose R , is a central extension of R, and P , ~ S p e c ( R , with ) P = R n P I . Then RIP and R,/P, have the same PIclass. (Indeed, view RJP, as a central extension of (R + P , ) / P , = R / ( RnP,), so that these rings are mult-equivalent.) Proposition 1 .I0.66. Suppose R is almost j n i t e dimensional and torsion free over a valuation ring C, cind let C, be a coinmutative C-algebra. , let P = C n P , . Letting $: R -+ R O c C , be the Suppose P , ~ S p e c ( C , )and canonical homomorphism for w e r y B E P-Spec(R), we have some B , E PI-Spec(R @( C , ) such that B = i k - ' ( B , ) .

Let R, R O c C 1 , R = R/B, C = (C+B)/B z C/P z ( C + P l ) / P l , C, = CI/P,, and R, = ROcC, ; $ induces a homomorphism $ : R -+ R , (by F - + f O l ) ,and it is enough to find B , ~ s p e c ( R , )such that B , n C , = P1 =_Oand $ - ' ( B , ) = 0. Let S = C - { O ) . C, is a field over which the prime ring R , is a finitedimensional algebra and is thus simple. Hence $ induces an injection R , -+ (12,)~, implying 8, is an extension of R, so $ is itself an injection. G take a (0,O)-maximal ideal of (Rl)s; this Moreover, viewing has the form (Bl)s for some B,ESpec(R,), and ($-l(Bl))s = 0, implying $-'(El) = 0. Clearly B , n = 0, so we are done. QED Proof.

c,

The "Little" Bergman-Small Theorem

We now turn to the two Bergman-Small theorems, which give us information about what happens to the PI-class of a suitable prime ring when we pass to subrings and homomorphic images. Both theorems are very difficult, but the theorem about subrings has an easy special case-all we need for our applications-which actually finds its roots in Jacobson [64B]. We recall from Theorem 1.5.22 that if D is a division ring of PI-class n then for any maximal subfield F of D we have [ F : Z ( D ) ] = n. Definition 1.10.67.

If R has PI-class n, we say deg(R) = 11.

96

[Ch. 1

THE STRUCTURE OF PI-RINGS

Definition 1.10.68. R is a (noncommutative) domain if R-[O) is a monoid. Theorem 1.10.69 (Little Bergman-Small [75]). R , E R, tlien degtR,) diiiides deg(R).

Zf’R is 11 domain and

Proof. Replacing R by Qz(R), we may assume R is a division ring. Let Z = Z(R), a field. Replacing R , by QZ(R,Z), we may assume R, is a division Z-algebra. Let F , be a maximal subfield of R , , and let F be a maximal subfield of R containing F , . Then deg(R,) = [ F , :Z(R,)], which divides [ F , : Z], which divides [ F :Z ] = deg(R). QED

There is a corresponding theorem for homomorphic images. Theorem 1 .I0.70 (Bergman-Small). If’ R is a quasi-locul prime ring of’PI-class n, then the PI-class ofR/Jac(R) divides n.

This theorem is (presently) very difficult and its proof will occupy the remainder of the section. The ”Big” BergmawSmall Theorem

We prove in fact the following important theorem, which will yield Theorem 1.10.70 as a corollary. Theorem 1.10.71 (Bergman-Small [75]). Suppose R is u prime PIring with a given prime ideal B. Then deg(R)-deg(R/B) = C:= nideg(R/Mi) for suitable k < deg(R), n , e Z + , and maximal ideals Mi ofR.

,

Proof. Let .Y’(R) be the assertion of the theorem, and define the following weaker assertion:

Y O ( R ) :IfR is 11 prime PI-ring and B€Spec(R), then deg(R)-deg(R/B) =

I:=, mideg(R/Bi)for suitable t < deg(R), m , € Z + ,and B i ~ S p e c ( R ) .

Reduction 1. Y ( R ) holds if Y o @holds ) for all homomorphic images R of R. Indeed, suppose cYo(R)holds for all homomorphic images R . In parinideg(R/Bi) ticular Y0(R) holds, so we write d = deg(R)-deg(R/B) = in such a way that the subsum d’, taken over those Bi which happen to be maximal ideals, is as large as possible; reordering the Bi, we may then assume for some u < t that the Biare maximal for 1 < i < u and that the Biare not maximal for u < i < t . We claim that u = r, i.e., that .SP(R) holds. Indeed, otherwise B, is not maximal; let M be a maximal ideal containing B,. Writing - for the canonical image in R = R / B , , we then have B,, . . ,B,, in Spec@) for suitable t’ and m,, . . ,m,. in L’‘ such that

xi=,

,,.

,,.

$1.10.1

Valuation Rings

97

xi‘=t+ ,

deg(R)-deg(R/M) = mideg(R/Bi).Taking the preimage Biof Bi (in Spec(R)), t < i d t’, we have RIB, RIB,, so deg(R/B,) = deg(R/M)+ m,deg(R/B,); therefore d = d ’ + d e g ( R / M ) + x ~ l ~m,deg(R/B,)+ + m,deg(R/B,), contrary to the assumed maximality of d‘. Thus u = t after all, proving the claim. Thus it suffices to prove .Y,(R) for all prime PI-rings R ; this is very good because we can now change rings without keeping track of the maximal ideals. We shall first justify a series of additional assumptions. Since R is prime PI, R is torsion free and almost finite dimensional over a central subring C (actually we take C = Z(R)). Applying Theorem 1.10.10 and Remark 1.10.65, we may assume furthermore that C is a valuation ring (although we no longer assume that C is all of the center). Thus we have arrived at Reduction 2. In proving .Y’,(R) for all R, we may assume R is a prime algebra over a valuation ring C and for 17 = deg(R), dim(R ; C ) = 172. Now let F be the field of fractions of C, let K be the algebraic closure of F , and let C, be a valuation ring of K lying above the integral closure of C in K (cf. Theorems 1.10.53, 1.10.54). The simple ring Q , ( R ) O FK = R BCF O FK = R BCK is a central extension of R BCC,, implying R OcC, is prime. In view of the results about integral extensions and Proposition 1.10.66, we may pass from R, C, respectively, to R B C C , , C , , yielding Reduction 3. In proving .Y’,(R) for all R, we may also assume that C is a valuation ring of an algebraically closed field.

x:l,+,

xi:,+,

Note. In view of Corollary 1.9.15, in the process of passing from C to C, given above we have rank(C) 2 rank(C,). This fact will be needed soon. Reduction 4. In proving cY,(R) for all R, we may also assume B n C (Indeed, letting S = C-B, just pass from R, C , respectively, to R,7,CJ. Also note that we have trot increased rank(C). Now, instead of proving .Y,(R) under these additional assumptions, let us aim to prove the stronger sentence Y’(R). We summarize the permissible assumptions on R : = Jac(C).

R is a prime algebra over a valuation ring C of an algebraically closed field, dim(R ;C) = n2 = deg(R)2,and B r ,C = Jac(C).

Let us now collect information to be deduced from these assumptions. (i) C = Z(R). Indeed, Z ( R ) is algebraic over C, thus has the same field of fractions (since the field of fractions of C is algebraically closed). Hence Z(R) is a valuation ring; since Jac(C) E B n Z(R), we see Z ( R )lies above C . Thus by Corollary 1.10.5 C = Z ( R ) .

98

THE STRUCTURE OF PI-RINGS

[Ch. 1

(ii) B is maximal, and there are only a finite number of maximal ideals of R (by Proposition 1.10.59(i)and Corollary 1.10.61, respectively). ( i i i ) R/Jac(R) is a finite direct sum R, @... @ R,, where R , = R I B and each R , is simple (by (ii) and Proposition 1.7.16). (iv) Each R i is almost finite dimensional over a prime homomorphic F, is algebraically closed. image Ci of C. Let Fi be the field of fractions of Ci. (Indeed, if .Y is algebraic over F, then for some c in C,, C.K is integral over C,, and so is a root of some monic irreducible ,/'EC,[A]. Lift ,/ to a monic 9 E C[A] ; g is irreducible in C[A], and by Gauss' lemma is linear. Thus,/'= i. -cx, proving s E F i . ) ( v ) By (iv). [ R , :F,] is finite. But Ri = M , , ( D i )for some division algebra D,. Since [ D i : F - i ]< K , every maximal subfield of D , is an algebraic extension of F, and is thus F, itself, implying D i= F,: thus t i , = deg(R,). At this point, we claim that it suffices to show that Jac(R) is idempotentlifting. Indeed, if so, then we can find orthogonal (rj4,1 1 6 i < I , I 6 u < nil such that for i fixed the images of cia#in R iare orthogonal idempotents in M , , ( F , ) , 1 d u d I ? , , and x,,ueiza = 1. It follows easily from Theorem 1.10.36 that for i fixed the {Rei,,[1 < u 6 ni) are isomorphic as left R-modules; in fact, for R' = Q,(R), the jR'eiz,l1 6 u < H,) are isomorphic as left R'modules. For R'e,,, a sum of ci minimal left ideals of R', w e get R' = R'e,,, is a sun] of cini minimal left ideals. But it is easy to see that this implies 1;= c,n, = deg(R') = i i , proving the theorem. So it remains to show Jac(R) is idempotent-lifting. I think this might be so, but cannot prove it. We shall therefore follow the method of Bergman-Small "751, which is, basically, to apply several more reductions in order to pile on enough additional assumptions to force the proof to a conclusion. The procedure is quite complicated, so we list it in a number of steps, which we shall then verify. Step 1. Passing to a certain subring of R, we may assume that C is a ring having finite rank, as well as the following information: R is prime, dim(R :C) < x , ( ' is a valuation ring, B is maximal, and R/Jac(R) is a finite direct product of matrix algebras over fields. Step 2. It suffices to prove .Yo(R)whenever R is a prime algebra over a valuation ring C of rank I , with Jac(C) s B. Step 3. We mcty also assume (in Step 2) that C is a valuation ring of an algebraically closed field. Thus C = Z ( R ) . Step 4. In step 3 we may replace R by R OcC', where C' is the completion of ('; then Jac(R) is idempotent-lifting. Verificatiritz of'Step 1. Let P,, ..., P, be the maximal ideals of R, assuming B = P : ; and write R,'Jac(R) = R, @ ... @ R,. where each R i = R/P, 2 M n , ( F i ) ,1 d i d k. Also let t,bi:R + Ri be the canonical homo-

,

,

4 1 . 10.1

Valuation Rings

99

morphism, 1 d i < k . Now for each i we take a set E j of representatives (in R) of a set of I ? , x / i i matric units i n R,, i.e., :t,hi(.y)l.yE E i ) are a set of matric units in Rj. Also, let S = C- {O), and let Q = R,, a prime, finite-dimensional algebra over the field F = C,s:thus Q is simple, and we can take elements r l , ..., r, in R , which are an F-basis of Q. (Here t = n’.) Let E = i r l , . . . , r , )LJ E l w ” - E,; E is a finite set, whose elements we write as { r l , . . . , r,,,} for some m > t . Now for each i, j < m we write ri = ajUr,and rirj = I aiiur, for suitable aiura j j u in F . Let F’ be the subfield of F generated by { a i u , a i j u ) 1 < i, j < m, 1 < u < t } , a finite set, and let Q’ = F‘r,, an m-dimensional F‘-algebra. Q is a central extension of Q’. implying Q‘ is prime, and thus simple. Now let R’ = R P, Q’ and C’ = C r ,F‘. Clearly F‘ is the field of fractions of C’, implying C’ is a valuation ring of F‘, and deg(R’) = deg(Q’) = I ? . Moreover, C’ has finite rank by Corollary 1.10.63. Let J = R‘ n Jac(R),so we can view R’/J c R/Jac(R). By choice of E , we can get R’/J :R’, 0... 0R;, whereeach R; :M,,,(K,)foracommutativedomain K , , and RI = R‘/(P, r ,R‘). Suppose r + J is not a zero-divisor i n R’/J. Since R/Jac(R) is semisimple, we seethatthereexistsyinRsuchthat (.yy- 1 ) E Jac(R)and(y.y- 1 ) E Jac(R).Thus xyand psare invertiblein R, proving .x is invertible in R c Q ;moreover, since Q’ is finite dimensional over F’, .Y- I E Q’. Therefore s-’E R n Q’ = R‘, implying ( s + J ) - l ER’/J. It follows instantly that each K i is a field, so B n R’ is maximal in R’, verifying Step 1. Verification of’ Step 2 . Applying Reduction 1 to Step I , it suffices to prove .Y0(R), whenever R is a prime algebra over a valuation ring C of finite rank, with Jac(C) c B. If we know this in the rank 1 case, we shall prove it in general, by induction on rank(C). Indeed, if 1 < rank(C) = r a n k ( B n C), we have by Proposition 1.10.59(ii) nonzero B’ESpec(R) properly contained in B. Let S = C-B’. Then passing to R, we have “blown up” B n C, so rank(C,) < rank(C) and, by induction (working in Rs), deg(R)-deg(R/B’) = deg(R,y)-deg(R,,/Bk) = lnzjdeg(Rs/(Bi),) = mideg(R/Bi) for suitable Bi in Spec(R), suitable t, suitable mi.Now passing to R = RIB, we again apply induction to get deg(R)-deg(R/B) - i.i--f+ Y U m, deg(R/B,) for suitable u, suitable m,, suitable B , E Spec(R), and so deg ( R ) - deg (R/B) = (deg ( R ) - deg (R/B’)) (deg ( R )- deg ( R / B ) ) = m,deg(R/B,), proving Y 0 ( R ) . Verijicatioii ? f S t e p 3. Apply Reductions 3 and 4, keeping in view the note that rank(C) was not altered. Verifcation of Step 4. Let C‘ be the completion of C. By Theorem 1.10.48, Remark 1.10.49, and Proposition 1.10.51 C’ is a rank 1 valuation ring of an algebraically closed field, lying over C; thus we may replace R by R @<.C‘ and assume C is in fact complete (cf. Proposition 1.10.66).Now we L)

-:”=

xi=

xy=,

x:=

+

100

THE STRUCTURE OF PI-RINGS

[Ch. I

shall show Jac(R) is idempotent-lifting. Let P = Jac(C) and J = Jiic(R). Since C = Z(R). every maximal ideal of R intersects C nontrivially at a nonzero prime ideal which is thus P, showing P c J ; hence RP c J . Pick any p # 0 in P. We shall show J/RP is idempotent-lifting in RIRP, then (RP/Rp) is idempotent-lifting in R/Rp, and finally Rp is idempotentlifting in R : putting these pieces together will show J is idempotent-lifting. First note that by Lemma 1.10.57 R/RP is a finite-dimensional algebra over the field C/P; thus by Remark 1.10.39 J / R P E Jac(R'RP) is idempotent-lifting. Secondly, note that (P+ Rp)/Rp 2 P/(C n Rp) is the only prime ideal of C/(C n Rp) and is thus nil, so RP/Rp is a nil ideal of R/Rp and is thus idempotent-lifting. Having discarded the first two stages, we need only show Rp is idempotent-lifting to finish the verification of Step 4, and thus the proof of the Bergman-Small theorem. First note by Theorem 1.4.21 that for some element co of C, Rc, is riC for suitable ri in R , 1 Q i Q t contained in a free finite C-module (where r = nZ). 'Thus, for any c in t, we have Rc,c c I:=, riCc. Now Chi then for each k , bZklc,c,, so by n Z l Cb'E Spec(C) (for if clcz E , is thus 0. Therefore 0 = Rc,b' Remark 1.10.46 hk(c, or b k J c z ) and = Rhi)c,, implying (Rh)' = 0. Since C is complete, one sees easily (through a similar argument) that R is (Rb)-adically complete. Hence, by Remark 1.10.45 and Theorem 1.10.43 Rb is idempotent-lifting. QED

x:=

(n,P= ,

0: ,

ni..=,

Having spent so many pages on this one theorem, let me add a few lines of comments. First of all, as noted in the proof, a major simplification might be possible if there were a better theorem about lifting idempotents; the whole process of reducing to finite-ranked, and then rank 1, and then complete, valuation rings was introduced merely to maneuver into a position to lift idempotents up Jac(R). To underline this fact. we have reversed the order of the proof of Bergman-Small [75]. As current knowledge stands, this proof is extremely delicate, connected by a fine thread woven back and forth between two statements .Y'(R) and .Yo(R),and almost seems on the verge of coming apart at many places. 51.11. identities of Rings without 1

Occasionally it is expedient to consider algebras without 1, in order to treat nil algebras. For example, there is the Nagata-Higman theorem, which says that if X ; is an identity of a Q-algebra A (without 1 ) then X , " ' X , is also an identity of A ford = 2"- 1. A short, self-contained proof is found in Jacobson [64B, p. 2741, and is summarized in Exercise 2.3.1 1. Another reason to consider algebras without 1 is that at times one needs

51.11.1

Identities of Rings without 1

101

to transform a given algebra R into a smaller algebra, in order to use induction (say on the PI-degree), and the smaller algebra may lack the element 1. This is the reason that motivates the discussion below. Example 1 .I1 .I. If R IS semiprime of PI-class 17, and 2 E Z ( R )then :R is a semiprime algebra without 1, i.e., zR has no nilpotent ideals. Indeed, if B a zR and B2 = 0 then for any zr in B we have zR(zr)zR G B , so (RTR)c ~ R(zRzrzR)2= 0, implying zr = 0 ; thus B = 0. Q E D We shall need Example 1.11.1 later in this section. In what follows, R , denotes an algebra without 1 (over 4). Definition 1.11.2. $ : X i o term 0).

=

[all elements of # { X I having constant

One can easily show that $ { X ) , is an algebra without 1, and is in fact the free algebra without 1. Note that # { X ; , d $ [ X } . Definition 1 .11.3. An element f of 4 { X ) , is an identity of R , i f f is in the kernel of every homomorphism from q b ( X ) , to R,.

There is a problem with this definition-If R , just happens to have 1 after all, then we have two definitions of identity (one using d { X ) , and one using $ ( X } ) . Fortunately, these definitions are consistent, because the restriction of a homomorphism $ : 4 ( X ) + R , to $ : d { X } , + R , is a homomorphism, whereas, if we are given a homomorphism $: 4 ( X ) , -+ R , and if 1 ER,, then the map $: X + R , extends uniquely to a homomorphism t,b:4{X} -+ R,. Having made this link between identities for algebras with 1 and algebras without 1, we should like to find a way of “adding 1” to R,. Definition 1 .11.4. Define Rb, the ulgebra with 1 ,formally udjoined to R,, to have the &module structure 4 0R,, with multiplication given by ( a 1 , r l ) ( a 2 , r= 2 ) (a,cc2,cc1r2+a2r,+ r , r 2 ) for CI, in 4, r, in R,. Remark 1 .I1.5. Rb is an algebra with multiplicative unit (1,O). There is a canonical injection (as algebras without 1 ) from R , -+ Rb, given by r + (0, I ) , under which every ideal of R, is identified as a n ideal of Rb. Also there is a projection n : Rb --t R,, given by ~ ( c Ir,) = r, of algebras without 1. Remark 1.11.6. If

/([XI, X , ] , . . . , [ X 2 , _

f ( X , ,..., X , )

is a n

,,X 2 , ] ) is an identity of Rb.

identity of

R,,

then

Theorem 1 .I1.7. Every nil PI-algebra without 1 is locullj1 riilpotenr. Proof.

Apply Remark 1. I 1.6 to Proposition 1.6.34. Q E D

Nevertheless, Rb is usually too big, and we shall modify it somewhat.

102

[Ch. 1

THE STRUCTURE OF PI-RINGS

R = Rb/AnnR$, is the reduced algebra with 1 of denotes the canonical image of Rb in R.

Definition 1 .11.8. -

R,. Proposition 1.11.9. Suppose AnnRoRo = 0. Then there is an in__ jection from R, ro R, given b y r -+ (0, r), under which we view R o d R. Ann,R, = 0; moreover, ifR, has 1, then Ro = R . __

Clearlv r -P (0, r ) is a homomorphism Ro --* R, whose kernel is Ro n Ann,; R, = AnnRoR, = 0.__ Since R, a Rb, we get R, R under the new identification. Now if (a, r )__ R , = 0 then (a, r)Rb E AnnRbR,, so 0 = (a, r)RbR, = [a, r)Ro, implying (a, r ) = 0. Thus __ Ann, R, = 0. Finally, if Ro has 1 already, then ((1,O)- l ) ~ A n n , R , = 0,so (1,O) = 1 E R,,implying R,= R. QED Proof.

a

~

One instance where AnnRuR, = 0 is when R, is semiprime, in which case we have even more information. Remark 1.ll.lo. Suppose R, is semiprime. Then R is semiprime, and Z(R,) G Z ( R ) .[Indeed, if A d R and A’ = 0 then (AR,)’ = 0. implying AR, = 0, since A R o a R , so A = 0. Thus R is semiprime. Obviously Z(R0) E Z(RI.1 Here is a useful decomposition result. Proposition 1.11.11. Let R , be a subdirect product of {R,,, I-}, arid for each y let R ; be the reduced algebra with 1 qf R, . Then R is a subdirect product of { R;ly E r}. = R,/B;.(yE

For any r in R,, let r ; denote the image of r in R,,. Then define $,.:Rb + R;. by $;(a,r ) = Clearly ker$; = { ( a ,r ) E RbI(a, r ) R , L B:,}, so ker 4;= [(a,r ) E Rbl(a, r ) R , c_ = 0 ) = AnnRb(Ro).Now each $y induces a homomorphism R = Rb/Ann(Ro) -P R.:, and 0 k e t - K = O . QED Proof.

nYEr

m.

6:

OB;,

We are now ready to prove a theorem stated in $1.9, improving Theorem 1.9.26, to illustrate these methods. Theorem 1.11.12. Jac(Z(R,)) = 0.

If R, is a semiprimitiue PI-algebra with 1, then

Proof. Otherwise choose a counterexample R with smallest possible PI-class. Thus Jac(R,) = 0 and there exists nonzero z in Jac(Z(R,)). Let Ro = zR, and let 9 = {maximal ideal of R1 not containing 2). Then O = ( z R , ) n ( n { P E 9 } ) = n c R o n P J P E ~ } , s o , w r i t i n g . 9 ={ P , J y E r } a n d Roy= R,/(R, n P; 1 for each y in r, we see that R, is a subdirect product of the R,.,. Let R be the reduced algebra with 1 of R , . Now R,,? 2 (R, +P,.)/P,,

531.11.3

Identities of Rings without 1

103

= R J P , , a simple PI-ring with 1 ; by Proposition 1.11.9 each Ro7is its own reduced algebra with 1, so by Proposition 1.11.11 R is a subdirect product of the {Ro71y€r}. In particular, Jac(R) = 0. Moreover, since z $ P 7 we must have P ; , n Z ( R , ) not maximal in Z(R,), implying by Corollary 1.9.22 gn(Ro7)= 0 for each y. Thus g n is an identity of R, implying R has PI-class < n. We shall conclude the proof by showing that Jac(Z(R)) # 0, contrary to the assumption about the counterexample R,. In fact, we shall show Z(Ro) is a quasi-invertible ideal of Z(R). Indeed, for any zr in Z(R,) and for each r‘ in R , we have 0 = [zr, zr’] = z2[r, r’], implying (R,z[r, r’]R,)’ = 0; since R , is semiprime, 0 = z[r, r‘] = [zr, r’], proving zrEZ(R1). Thus, for some z1 in Z ( R , ) , (1-z,) = (1 -zr)-’. Then 1 = (1-z,)(l-zr), implying z, = z,zr-zr€Z(R,). But this same equation holds in Z(R), proving (1 - z r ) - l ~ Z ( R ) ; thus Z(R,) is indeed a quasiinvertible ideal of Z(R), so 0 # Z(R,) G Jac(Z(R)), as desired. QED

EXERCISEST $1.I 1. The homomorphism +{X} + 4 { X I . X , ) given by X i w X I X 2 is an injection. Hence any PI-ring satisfies an identity in two indeterminates having all coefficients 1. 2. For any r in &Jx} and all xi,x, in E we have xirxj = - -xjrxi 3. [XI, X,Iz is an identity of 4 E ( x ) .

$1.2 Iffis t-primitive then S, ~ / ( d { X1. j Write formulas for the Capelli polynomial, analogous to 1.2.13(i),(ii). 3. I f t isodd thenS,(l,X, ,... , X , ) = S , _ , ( X , ,... , XI);iftiseven then S i ( l , X , , . .. , XI) = 0. Thus S,, is an identity of R iff SZr+ I is an identity of R . 1. 2.

$1.3 1. (Procesi [73B, p. 941) The characteristic values of the generic matrix Y, are algebraically independent. ( H i n t : Using the theory of elementary symmetric functions in Lang [65B, pp. 132-1341, it suffices to prove the coefficients of the characteristic polynomial of Y, are algebraically independent; show this by sending ~ ~ ~ for l all w i O# j . )

$1.4 1. If 1 / 2 ~ then 4 each identity of M , ( 4 ) of degree < 2 n has the form p S Z nfor some p 4. ( H i n t : Look at the last step of the multilinearization procedure.)

in

2. The upper triangular matrices over Q do not have a central polynomial of degree > 0. ( H i n t : When can eii be a value of a multilinear polynomial?) ?Throughout the book, an asterisk before an exercise number denotes an open question.

104

THE STRUCTURE OF PI-RINGS

[Ch. 1

Here are some exercises on identities of matrices over dE(.xj,leading to Rosset's proof of the Arnitsur-Levitzki theorem. The notation is as in 1.1.41 IT.

3. M n ( 4 , { . x ) : ) Mn(4!:{.v;,"J 0 M n ( d f ~ { as . ~+modules. ),) 4. S , ( s , , ,...,.Y , < I = r!.x,, ....Y;, for all xi,,in B. 5. If j = ~ : u u , ~ x , ,where U . E Z ( M , ( ~ ~ ~ : , (and . Y ) ) )x,EB, then for any t , y' = x.S,(a",,.... au,)Xu,".X,,,thesum beingiakenoverallt-tuples(u,.. ..,u,)withu, < ... < u,. Hence tr(y2') = 0. 6. I f y ~ M ~ ( @ , { . x } , then ) y z " = 0. [ H i r i r : We may assume 4 = O ( 0 . Since .yz E Mn(Z(4&v])), it suffices to show tr(j.'") = 0 for all m d n, by Newton's formulas.] 7. Taking j = xi: I uU.xudeduce the Amitsur-Levitzki theorem from Exercises 5 . 6 . 8. Calculate g n explicitly. 9. (Amitsur [77]) In the notation of Theorem 1.4.14. f?Zn2-l is M,(H)-central for every , ( M , , ( H ) )has an element of trace # 0.1 i is rt2-normaI. commutative ring 11. [ H i n t : cZn-. 10. (Saltman [74] If g is M,(Q)-central and F is a field of characteristic p , then some multilinearization of g p is M,(Q)-central but an identity of M , ( F ) . [ H i n t : g p - 4 IS an identity of .%f,(Z/pE): multilinearize.] I t . (Formanek [ 7 6 ] ) S,+,(X;X,, X;-'X2,...,X2) = S,([X;, X , ] . [XI;-',X,], ..., ~

CXi~X,1)X2.

12. (Formanek [ 7 6 ] ) If , / ( X , , X , ) is a proper identity of M,(Q), with d e g ' j s deg2,1; then deg',f> ~ 7 : moreover, if deg'j'= n then d e g ' f g n(t7+ 1)/2. (Hint: Linearize the part in X I; substitute diagonal matrices for X,, and matric units for the indeterminates obtained from x i

.)

What is the minimal possible degree of an identity f ( X , , X , ) of M , ( Q ) ? (Use Exercises 1 1 and 12 for the case d e g 2 f = n ; for n 2 7 look at S 2 , ( X , , XI, X:. X,X,, *13.

x,x,.xt,...1.) *14. What is the minimal positive degree of an M,(Q)-central polynomial? (The smallest known is n', from Formanek's polynomial; cf. Appendix A,) *15. What identity of M J Q ) has the smallest number of monomials? (Partial results are given in @7.2and its exercises, and in Exercise 8.4.3).

$1.5 1. Define the following topology on End M,: For y , . . . . .y, in M and B in End M,, let ECy, ,...,_ v , ; B ) = { / ~ ~ E n d M , I ~ ~ ( ~ , ) = B ( y , ) fi.o 1r adl il < t } . L e t & = { B ( y ,,...,y,;/I)II ,< r < m, y , , ...,v , E M , / I € End M , ; . .Q is a base for the open sets of a topology called the finite topology. 2. If M is a faithful, irreducible R-module and D = End,M then R is dense in End M D with respect to the finite topology. 3. D e f i n e a n o p e r a t i o n o o n R by r , o r 2 = r l + r 2 - r , r , = 1 - ( 1 - r , ) ( l - r 2 ) . T h e n o i s associative, and r , 0 0 = Oor, = r , . Jac(R) is a group under 0, and there is an injective group homomorphism Jac(R I -+ {invertible elements of R, under multiplication). given by r - ( 1 - r).

$1.6 1. (Amitsur [ 5 5 ~ ] ) If R is Lie solvable of characteristic # 2 or if R is Lie nilpotent, then R/N(R) is commutative.

A polynomial is R-c-orrect if, for each r # 0 there is a coefficient of the polynomial which does not annihilate r. The next few exercises, due to Amitsur, analyze R-correct polynomials. 2. Suppose R is prime. R has PI-class d [d/2], iff R has a n R-correct identity ofdegree d d iff R has an R-proper identity of degree d d.

Ch. I]

Exercises

105

3. Ifj’is R-correct then for each nonnilpotent element r of R there is a coefficient rl of fsuch that ar is not nilpotent. 4. If Nil(R) = 0 and ,f is R-correct. then R is a subdirect product of prime algebras ( R 1 ; ’ ~ rl such that for each y,fis R -correct. Nil(R ) = 0. and (nonzero ideals of R , ) # 0. 5. If R satisfies an R-correct identity / of degree d, then for some k in H C S 5 , d z , is an identity of R: moreover, if R is semiprime. we can take k = I . 6. A subset A of a ring R is called weuk/!, niulriplicariw if for every r , , r2 in R there is some n i ( r l , r 2 )in H such that r , r 2 + m ( r , , r z ) r 2 r E, A . If R satisfies a polynomial identity of degree d and has a weakly multiplicative, nil subset A. then G L,(R). [Hint: First prove it for matrices-cf. Jacobson [64B. VIIl.S] and then apply injections and the proof of Theorem 1.6.36(i).]

n

$1.7 1.

Every completely homogeneous identity of R is an identity of R,

Here are a number of examples concerning rings of quotients of semiprime PI-rings. R satisfies the I& Ore cotidition if for any u. h in R with a regular there exist a’, h’ in R with a’ regular, such that b u = a‘b. The right Ore condition is defined analogously.

2. (Rowen [74a]) A semiprimitive PI-ring R satisfying the left and right Ore conditions, with Qz(R) = R, but having a regular, noninvertible element. Let H = a$(<), the fraction field using indeterminates #, I < i , j < 2. 1 < k < m. For each u, take H, = H, and let R be the Qsubalgebra ofII,, :. M , ( H , ) generated by I, t he (direct)sum ofthe M , ( H , ) , and the element .Y whose uth component is the generic matrix Y,, i.e., Y = ( Y , , Y,, . . .). 3. (Rowen [74a]) A semiprimitive PI-ring R’ which is neither left nor right Ore. Let H , H,, and R be as in Exercise 2, and let R’ be the Q-subalgebra of M , ( H , ) generated by R and the element ( Y,, 0, Y4, 0,. . .). (Thus we see that the central quotient construction is not as “good” as the classical quotient ring construction, which is not good enough to analyze PIrings. For an alternate example to Exercise 3, cf. Bergman [74].) 4. (Fisher [73]; Martindale [73]) A left ideal J of R is large (in R) if J n L # 0 for all nonzero left ideals L of R. Show AnnkJ = 0 for every large left ideal J of a semiprime ring R. [ H i n f : { u ~ R I A n n , nis large) R, and if nonzero intersects Z(R) nontrivially.]

nusE.

a

$1.8 If R is simple with center C and A is a C-algebra, then every ideal of R O r A has the form R @ 1 for suitable 14 A . 2. (Procesi-Small [68]) If R is arbitrary and R , is semiprime of PI-class n, then R BzR, C M , ( R ) . This is a funny result; although by Amitsur’s method MJR) is PI whenever R is PI, much remains to be learned about the actual identities of M,(R). (Some results are given by Berman [71], [72], [75].) 3. I f R is quasi-local then Z ( R ) is quasi-local. 1.

Here are several exercises leading t o the Brauer group of an arbitrary commutative ring. 4. (Azumaya [51]) Suppose R is an algebra over a subring C of Z(R). R is proper maximally central iff there is a C-base r , , . . . ,rm such that the m x m matrix (rl,) is invertible, where r,j = r , r j , 1 < i, j < m. (In fact, such a property would hold for each base.) 5. M,(C) is proper maximally central. for every commutative ring C. 6. If R , , R, are two proper maximally central algebras over C then R, O r R , is proper maximally central over C .

106

[Ch. 1

THE STRUCTURE OF PI-RINGS

7. If R,, R2 are Azumaya algebras over C then R, O r R , is Azumaya over C. Thus 0 makes (Azumaya algebras over C ) into a commutative semigroup. To study Azumaya algebras further, see Demeyer-Ingraham [71B] and Knus-Ojanguren [74B].

61.9 View R as an extension of C = Z(R). If p ~ S p e c ( C and ) P' is (0,P)-maximal, then Pp is maximal in R,, and the ring of central quotients of RIP' is canonically isomorphic t o R,/P,. 2. Find a ring R of Pl-class 2 such that Z(R) is prime and R is not prime. 3. Let (6 be a class o f prime PI-rings, such that for any R in V, and any P in Spec(Z(R))we have R,E%. Consider the following sentences, each for all R in '6: ( i ) Jac(Z(R)) = J a c ( R ) n Z ( R ) : (ii) J x ( Z ( R ) ) L Jac(R); (iii) G U from Z(R) to R ; (iv) for every maximal ideal P' of R, P' n Z ( R ) is a maximal ideal of Z(R): (v) P = PR n Z(R) for all maximal P q Z ( R ) : (vi) L O froin %(R) to R. Then (i). (ii). (iii). (iv) are equivalent and imply f v) . (vi): also (v) and (vi) are equivalent. Thus (i)-(vi) hold when R is integral over Z(R) and prime. (For the sharpness of this exercise, cf. Exercise 1.10.4 below.) 4. GU holds from Z(R) to R if R has PI-class n a n d rank(Z(R)/g,(R)') = 0. This is true in particular if R = F,{ Y ] for a suitable field F. 5. A counterexample to assertion (v) of Exercise 3 (and thus implies that all the assertions fail in various rings; another example is given in Example 5.1.19). Let F be a field. Viewing FJY) c M n ( F ( < ) )let , p,,.. . , p n be the characteristic values of Y; and define x , = pi, a2 = Cicjpipj... . , a, = p , . . . f i n . Let R be the subring of M , ( F ( ( ) ) generated by Fn{ Y] and q... ., a,. Putting A = a i Z ( R ) dZ(R), prove that (v) fails, by showing I EAR but A # Z(R). [Hint: Define Z , = F [ a , , . . . , xn] G Z(R). and let R , be the Z,-subalgebra of R generated by Y,. Kote that Y;'ER,. It suffices to prove Z ( R ) n R , = Z , . Take c = zL=,Jk(a)Y:e Z ( R ) n R , where 4, r are integers; by induction o n t . show we can take r < 0, and conclude C E Z , . You need the fact (cf. Lang [65B. pp. 133-1341) that all polynomials symmetric in the p i lie in Z,.) 6. G U for prime ideals implies G U for semiprime ideals. [Him:Suppose R c R' is an extension. B , c B z are semiprime ideals of R. and B; is a semiprime ideal of R' lying o\er B , . Let S; = (P'/P' is I E , .P)-maximal for some P E Spec(R)containing B z ) , ] 7. Every unique factorization domain is normal. 1.

'.

x;=,

,

n

I. (Procesi [73B]) IfR = W i r , , . . . , r A j has PI-class r ~ a l rl j ~ C R ( W ) . a nQeSpec(R) d lies over PsSpec(W), then rank(Q) < ( ( k - l)uZ+2)(rank(P)+1).(Hirirr It is enough to show f(>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 ~<~ ds .s Proof. Let { P , l y E r } = {maximal ideals of R } . By Kaplansky’s theorem nysrPy= 0. Let B y = P , n P: for each y in r. Then each B , d ( R , *), nyerB,,= 0, and so R is a subdirect product of ( R , , * ) = (R/B,,*),each of which is simple by Lemma 2.2.10. The rest is immediate from Proposition 2.2.12. QED

This result is quite decisive, but leaves open the question of precisely which (*)-identities R can satisfy; this is postponed until $2.5. Note that it is false in general that if ( R , *) is simple then Z ( R ) is a field. Hence we are led to a different definition of “center” for a (*)-ring. Definition 2.2.14. { z ~ Z ( R ) l z= * z}.

The

center

of

( R , * ) , written

Z ( R , * ) , is

Proposition 2.2.1 5. Z ( R , *) is a subring of R fixed by (*). If ( R , *) is simple, then Z ( R , *) is afield. Proof. The first assertion is immediate; the second follows from Remark 2.2.6. QED

Definition 2.2.16. (*) has the first kind on R if Z ( R ) = Z ( R , * ) ; otherwise (*) has the second kind. We say ( R , *) has the first (resp. second) kind if (*) has the first (resp. second) kind on R.

Actually, involutions of the second kind are not very interesting for us, because of the following proposition. Definition 2.2.17. A, B of ( R , *).

( R , *) is prime if A B # 0 for all nonzero ideals

122

THE GENERAL THEORY OF IDENTITIES

[Ch. 2

Remark2.2.18. If ( R . * ) is prime and O # z = + z * € Z ( R ) , then Ann,z = 0; in particular, Z ( R , * ) is a domain. [Indeed, 0 # z R d ( R , * ) , so we are done by Remark 2.2.7.1 I f ( R ,*) is prime of the second kind, rhen every Proposition 2.2.19. multilinear GI of ( R , *) w i t h coeflcients i n R is special. Proof. Otherwise, take a multilinear G I , f ( X , , X : , . . . , X , .X : ) of ( R , *) of minimal degree such that j ’ is nor special. Let .fl (resp. ,ji) be the sum of those monomials of .f in which X , (resp. X : ) does not appear; clearly f ’ = , f l +fZ. By hypothesis, we have some z # z* in Z ( R ) . Then ( R , *) has the following two GI’s: zf’ = zf, +& and f (XI, X:, .. . , z X , , ( z X , ) * ) = z*J; +zfz. Hence ( z - z * ) , j ; is a GI of ( R , *), s o f l ( R , *) L Ann,@-z*) = 0, by Remark 2.2.18. implying .fi is a GI of ( R , *); thus ,fz is also a GI of ( R , * ) . Hence, for each r in R , J , ( X , , X ; ,..., X r - , , X : - , , r * ) and J Z ( X , ,X y , . .. , X,-,, XT-,, r ) are GI’s of ( R ,*) of degree t - 1 and, by induction, are special. Thus f l ( X , , X : , . . ., X,-1 , X f - l , X:) and f 2 ( X l , X :,..., X ,-,,X , * _ , , X , ) are special GI’s of ( R , * ) , implying j ’ is special. QED

This result is the most straightforward example I know of a fact that has an easy proof in the GI-theory but, as we shall see later, is considered harder to prove without generalized identities. Corollary 2.2.20. If ( R , * ) is simple of the second kind, rhen ezery multilinear identity Of(R,*) is special. Example 2.2.21. Here is a simple (*)-ring of the second kind that is not special. Let F be the field of four elements (i.e,, F = ( 0 , 1, u, h ) , where a + b = 1 = ab, and s + x = 0 for all .Y in F ) . F has an automorphism II/ given by $ ( a ) = b and cl/l,b)= a , having degree 2; since F is commutative, $ is an involution (*). Now the symmetric elements of F are 0 and 1, so ( X , + X : ) ’ ( X I+ X : ) is an identity of ( F , *), which is not special. This example might be considered artificial but in fact is representative of the only kind of example of ( R , *) simple of the second kind and not special, such that R is a PI-ring (cf. Exercise 3.1.11). Let us continue to look at prime and semiprime (*)-rings. Proposition 2.2.22. # 0, then R is semiprime.

[f,for every nonzero ideal B of ( R , * ) we haae B Z

Proof. Suppose A a R and A’ = 0. Then ( A n A*)’ = 0. implying A n A* = 0. Hence ( A + A * ) ’ = A Z + ( A * ) 2 + A A * + A * A = ( A * ) 2 = (A’)* = 0, so 0 = A + .4* 2 A . This proves R is semiprime. QED

Corollary 2.2.23.

I f ( R ,*) i s prime. then R is semiprime.

82.3.1

123

Sets of Identities of Related Rings

Theorem 2.2.24. Suppose R is u semiprime PI-ring, and R is also a (*)-ring. Then every nonzero ideal qf R intersects Z ( R ,*) nontrivially. Proof. Suppose 0 # A a ( R ,*). By Theorem 1.6.27 we have a nonzero element z in A n Z ( R ) .Then z +z* E A n Z ( R ,*), so we are done unless z = -z*. But then 0 # -z2 = zz* E A n Z ( R ,*). QED

Corollary 2.2.25. simple.

I f Z ( R ,* ) is afield and R is semiprime, then ( R ,* ) is

If R is a (*)-ring and S is a submonoid of Z ( R , * ) , Remark 2.2.26. then (*) extends to R , by the action (rs-l)* = r*s-I. [Indeed, if rs-’ = 0, then for some s1 in S , 0 = s l r = (sir)* = sly*, so r*s-l = b, proving (*) is well defined on R,; it is immediate that (*) is an involution.] Theorem 2.2.27. [f ( R , * ) is prime und S PI, then (Rs,* ) is simple.

=

Z ( R , * ) - { 0 } , and

if R is

By Proposition 1.7.4 Z(R,) = Z(R),, implying Z(R,, *) the field of fractions of the domain Z ( R ,*). But R is semiprime, implying R, is semiprime, so by Corollary 2.2.25 (R,y. *) is simple. QED Proof.

= Z ( R ,*),,

Let us throw in one general observation describing “(*)-prime” in terms of generalized (*)-identities. Remark 2.2.28. If ( R ,*) is prime and 0 # A a ( R , *), then by Remark 2.2.1 and Corollary 2.2.23, Ann, A = 0. Proposition 2.2.29. ( R ,* ) is prime fi wheneuer r1 E R and 0 # r2 E R such that r I Rr, = r,Rr: = 0 we have rl = 0. Proof. Suppose (R,*)is prime, and r 2 # 0, rl satisfy rlRr2 = r,Rr$ = 0. Letting B = Rr2R RrZR, we have rl E Ann,B = 0. Conversely, suppose A # 0 and B are ideals of ( R ,* ) with B A = 0. Then, taking any rl E B and r 2 e A we have rlRr2 = rlRr: = 0. Hence r1 = 0 for each rl in B, so B = O . QED

+

$2.3. Sets of Identities of Related Rings ( w i t h Involution)

Motivated by Chapter 1, we want to study the concept of passing identities from one ring to another ring of a simpler structure; as in Chapter 1, we are most successful when the identities are multilinear, but some important general results are obtained by considering the multilinearization process. With a very little extra work we can treat simultaneously the PI, GI, (*)-PI,and (*)-GI theories. For the reader’s convenience, and also to

124

THE GENERAL THEORY OF IDENTITIES

[Ch. 2

show that GI’s are really no more difficult than polynomial identities, we do not refer to W after the following definition; throughout R will be a W-ring, and “identity” really will denote a GI of R with coefficients in W. (Similarly for (*).IThe reader may take the special case W = 4, and merely view R as an algebra, thereby reading this section PI-theoretically. Definition 2.3.1. We shall generalize certain important notions of Chapter 1, for those who wish to read this section in full generality. Write R, < R, if Y ( R , ;W) E 9 ( R , ;W); R , <multR2 if every multilinear element of 9’(R2; W) is a GI of R,. R, and R, are equivalent if R , < R, and R , < R, ; R 1 and R 2 are mult-equivalent if R 1 GmmultR2 and R , <multR1. Likewise, for rings with involution, write (R,, *) < (R,, *) if . 9 ( ( R 2 ,*); W) c_ Y ( ( R , ,a); W); (R,, *) <mult(R2,*) if every multilinear element of 9’((R2,*); W) is a GI of ( R , , *). “Equivalent” and “mult-equivalent” are defined analogously. Remark 2.3.2. Given ( R , , *) is equivalent (resp. mult-equivalent) to (R,, *), we have R , is equivalent (resp. mult-equivalent) t o R,.

The converse is not true, as evidenced by ( R , , * ) = M , ( F ) with the transpose involution, and (R2,*) = ( M 2 ( F )0 I L ~ , ( F ) ”0~),. [Look at (XI - X : ) 2 . ] However, if ( R , , *) and (R,, *) are special and if R , and R, are equivalent (resp.mult-equiva1ent)thenclearly (R,, *)and ( R , , *)areequivalent (resp. m ult-equivalent ). We shall want also to treat central polynomials. Definition 2.3.3. Given a polynomial [resp. (*)-polynomial] f ( X , , . . .,X,,,), write f(R)[resp. f ( R ,*)] for { f ( r , , ... ,r,,,)lriE R ) . Say f is R-central if 0 # f ( R ) E Z ( R ) ; respectively, f is ( R , *)-central if 0 # f ( R , *) E Z ( R ; *). Remark 2.3.4. A (*)-polynomial f ( X , , . .., XI) is (R, *)-central iff [ X , + , , f ] and$-$*, but mf, are identities of (R, *). Thus, two equivalent rings (resp. (*)-rings) satisfy the same set of central polynomials. A subring of ( R , * ) is a subring R, such that R: G R,. We denote this situation by (R,, *) E (R, *). Also, note that if (R,(yE r>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

Sets of Identities of Related Rings

127

(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 .

128

THE GENERAL THEORY OF IDENTITIES

[Ch. 2

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

Sets of Identities of Related Rings

129

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

130

THE GENERAL THEORY OF IDENTITIES

[Ch. 2

For example,

Proposition 2.3.26

(Vandermonde determinant).

I ~ . l . . . . - c r= I

nl


Proof. We may assume C is the free commutative ring Z [ i ] and each ci is an indeterminate ti.Now write p ( l l , . . . ,tt)= It1,.. . ,<,I. By inspection, we see p is a polynomial (in commuting indeterminates) of total degree t ( t - 1)/2. But, specializing t j --+ ti identifies two rows of the matrix (Ci-') and thus sends p t o 0; therefore ( t j - t i ) l p ( ( l , . . . , tr)for all j > i, implying ( I T i s i < j s r ( ~ j - ~ i ) ) lBut p . d e g ( I l l s i < j d t j - t i ) ) = (i)= t ( f - 1)/2, SO p - nnl,i<j6r(<j--ti) for some ~ E Z Checking . the contribution of the diagonal to ltl....,trl (namely, 1t2t:-..<:-'), which also occurs in s i < j s t ( < j - & )by always choosing ''
n,

There are shorl, computational proofs of the Vandermonde determinant, but the generic proof is beautifully transparent. One uses the Vandermonde determinant as follows. Proposition 2.3.27 (Vandermonde argument). Suppose we are yioen in R such that,for each i, 1 < i elements a,, . . . , a d + in Z ( R ) and v l , . . ., < d + l , we have ~ ~ _ + : a j - ' u j = OThen . (a,, . . . , a , + l ~ v j = O , ~all o r j . In particular, ifAnn,(a,, ..., a,+,I = 0 rhen each uj = 0. Proof.

v=

1'1

:

.

ud t 1

Writing the hypothesis in matrix form. we have A V = 0. But A E M , ( Z ( R ) ) , so left multiplication by the adjoint matrix of A yields 0 = det(A)V = Jal,...rad+llV. QED WriteAkij)for E(monomia1s h offldeg, h = j } , . Corollary 2.3.28. Iff(Xl, . . . ,X , ) is an identity o j R [resp. ( R , *)I and deg"(f) = d , t h m j b r all a,, . . .,ad in Z ( R ) [resp. Z ( R , *)I,andfor a l l j 6 d , la,, . . . , a d + I fik,j) i y an identity O f R [resp. ( R , *)I. Proof. For convenience of notation, assume k = 1. Take any r l , . . .,rm in R . For all i, 1 < i < d + 1 , 0 = , f ( a i r l ,... , r m ) = ~ ~ = , , ~ ~ , .f. .i,r,,,). ~ ; ~ ~ ( r ~ ,

$2.3.1

Sets o f Identities of Related Rings

131

Thus, by the Vandermonde argument, la,,. . .,a d + l , ~ l ; j ~ ( .r.,,r,) , . = 0 for allj, implying I f f l , . . .,ad+I lji, : j , is an identity of R [resp. (R, * ) I . QED Theorem 2.3.29. Suppose there exist a , , . . . , a d + in Z ( R ) [resp. Z ( R , * ) ]suchtlzat A n n R ~ a , , . . . , u d= + lO.I/'degj(f') ~ < d,/oralljarid i/:/'isan identity qf R [resp. Of'(R,* ) I ,then ,/'is R-stuble [resp. ( R ,*)-stable].

By Corollary 2.3.28 the hypothesis o f Corollary 2.3.24 is satisfied. QED Proof.

Corollary 2.3.30. If Z(R) contains a subring of >(d+ 1 ) elements ouer which R is torsion ,free, then every irleiitity (of R ) o f degree < d in each indeterminate is R-stable. [Likewise,/br (*).I Corollary 2.3.31. Zf R is uii algebra over afield of 2 ( d + 1) elements, then every identity of degree < r i in each indeterminate is R-stable. [Likewise .lbr (*).I

(Corollary 2.3.31 is sharp, in view of Example 2.3.7.) Corollary 2.3.32. I f R is an algebra over an infinite ,field, then R is equivalent to every central extensiori Of'R. Likewisefor (*). Corollary 2.3.33. I f R is prime and Z(R) is injnite, then R is equivalent to every central extension. Remark 2.3.34. Every finite commutative domain is a field. (Rudimentary application of the pigeonhole principle.) Corollary 2.3.35. If R is semiprime and Ann,(Z(R) nJac(R)) = 0, then R is equivalent to everjl central extension of R. Proof. It is enough to show that R is equivalent to R[A] by Theorem 2.3.23. Let ./p = {prime ideals of R not containing Z(R) nJac(R)}. Then n ( P E : Y } E Ann,(Z(R) nJac(R)) = 0; since R[A] is a subdirect product of { ( R / P ) [ A ]I P E .P}it,suffices to show for each P in 2 that R is equivalent to R[A]. (Here we write - for the canonical image in R/P.) But Jac(Z(R)) 2 Z ( R )n Jac(R) 2 Z ( R ) n Jac(R) # 0, so Z(R) is not a field, and thus must be infinite (cf. Remark 2.3.34). Thus ?f is equivalent to R[A] by Corollary 2.3.33. QED

For R prime, there is a very decisive result. Corollary 2.3.36. I f R is a prime PI-ring [resp. (*)-ring],then either R is equivalent to every central e.utension [resp. (*)-e.utension] of R, or R is simple [resp. (*)-simple] and a j n i t e set. Proof.

By Corollary 2.3.33 we are done unless Z ( R ) [resp. Z ( R , *)I is a

132

[Ch. 2

THE GENERAL THEORY OF IDENTITIES

finite domain and thus a field. But then R [resp. ( R , *)] is simple and finitedimensional over a finite field, proving the assertion. QED (We shall see in $3.1 that every finite simple ring is a matrix algebra over its center.) It follows immediately from Corollary 2.3.36 that every prime PI-ring is equivalent to each of its central localizations. Here is a much stronger result, which could be used to improve several results in Chapter 1. Theorem 2.3.37.

For ertery submonoid S of Z ( R ) , R ,

< R.

Proof. Since S is central, any set of m elements of R , can be put in the form r l s - ',. . . ,r,s- I for suitable r l , . . . ,rm in R and s in S . Let f ( X , , . . . ,X,) be an identity of R . It suffices to prove that f ( r l s - . . . ,r,s) = 0 for all rj in R , s in S . Let j i be the sum of all monomials off' having total degree k, and let uk =X(r,, . . .,r,). Then, letting deg(f) = d, we have (for each i 3 0) 0 = , f ( s ' r l ,... ,s'r, t = x kd = O sik vk, implying, by the Vandermonde argument, . . 11,s ,..., s d J v k = O f o r e a c hk . Now, writingI1,s,..., s d ( = r I o 5 i < j s d ( s J - s ' ) , we see that this expression can be written as s"(1 - s p ( s ) ) , where u = l(d- 1)+2(d-2)+3(d-3)+ ... and p is some polynomial with integral for coefficients. Thus, multiplying by s-', we get c k l = sp(s)v,l each k , implying rli1 = (sp(s))"v,1 for all I? in a+.Hence

'

',

-'

d

=

d

d

d

k=O

k=O

1 p(s)ku,lrl = 1 f ( ~ ( s ) r l , . . . , p ( . ~ ~ r m ) l - ~

=O.

QED

Corollary 2.3.38. I f S is a subrnonoid o f Z ( R ) and ifevery element of S R , then R and R,s are equivalent. [Likewisejor (*).I

is regular in

Thus, any ring is equivalent to its ring of central quotients. This fact is very important in the study of relatively free rings. Now we can decide when some (*)-rings are special. I f R is PI arid ( R , *) is prime of'the second kind, Proposition 2.3.39. then every ( R , *)-stableidentity oj'(R,*) is special. Proof. Taking central quotients, we may assume Z = Z ( R , *) is a field. Also we may assume Z ( R ) is not a field, for otherwise we can pass to ( R @ , Z ( R ) , * ) , which is simple [and Z ( R ) @ , Z ( R ) is not a field]. We

conclude with Proposition 2.2.12. QED

$2.4.1

Relatively Free PI-Rings and T-Ideals

133

$2.4. Relatively Free PI-Rings and T-Ideals

The generic matrix ring has appeared in the proof of several important results. In this section we examine the structure of relatively free PI-rings in general. Of course, their structure is intrinsically connected to T-ideals, so we shall also look at T-ideals. In particular, we determine all semiprime Tideals of $ { X } when 4 is a field. We shall also discuss Specht's problem (and related questions), leading to the theory of "trace identities."

Building T-Ideals

Relatively free W-rings are not particularly useful in this generality, so we shall focus on relatively free algebras and (*)-algebras (over 4). As was seen in $2.1, these correspond naturally to the T-ideals of 4 { X > [resp. ( q 5 { X } , * ) ] ;moreover, if A a , R and B 2 A with B / A a T R / A , then clearly B a , R [and likewise with (*)I, so, in view of Theorem 2.1.8, we can find new relatively free rings by finding new T-ideals in a given relatively free ring. For convenience, we adopt the following notation: U [resp. ( U , *)] is a relatively free ring; the images of the X i in the canonical homomorphism 4 { X } + U [resp. ( + { X } ,*) + ( U , *)] will be written as X i . We shall still speak of elements of U as polynomials [resp. (*)-polynomials], by slight abuse of language. Definition 2.4.1. morphism.

An

epimorphism

is

an

onto algebra

endo-

Lemma 2.4.2. Suppose f , , ...,,/; E U . I f $ is an algebra endomorphism of' U , theii there is an epimorphism $ I : U -+ U such that $'(fJ = $(fi), 1 i t , arid $'(T) = T f o r every T-ideal of U . [Likewise,for (*).I

<

<

Suppose the indeterminates XI, . . .,X , occur in f i , . . . ,f;. Define $' by its action on X, as follows: $'(Xi) = $(Xi), 1 < i < m, and $'(Xi) = Xi-, for all i > m. Clearly $'(ji)= $(,L) for all i ; moreover, for every ,/'(XI,.. ., X , ) in T, we have,/'= $'(,/'(Xm+l, . . ., X , , , ) ) , so $'(T) = T. QE D Proof.

Thus, to check that an ideal A of U [resp. ( U , *)] is a T-ideal, it suffices to check that $ ( A ) c A for every epimorphism $ of U . But, in this case, $ ( A ) is an ideal. So, heuristically, if A is the "largest" ideal having a certain property expressible in the ring [resp. (*)-ring] operations, then $ ( A ) will also be an ideal having this property, and we may conclude that $ ( A ) c A and A is a T-ideal. Thus we have the following remark : Remark 2.4.3.

Jac(U) and Nil(U) are T-ideals.

134

=

THE GFNERAL THEORY OF IDENTITIES

[Ch. 2

Here is a deeper result. Given ideals A , B of R , define ( B : A ) ( r G R ( r . 4c B : .

Proposition 2.4.4. !f A, B are 7‘-ideals of U , then ( B : A ) is also a Tideal o f u . [LikeH ise.for (*).I Proof. By Lemma 2.4.2 we need only show $ ( B : A ) c ( B :A ) for every epimorphism t)j such that $ ( A ) = A and $ ( B ) = B. For f~ ( B :,4), $( f ) A = t ) j ( . f ) $ ( A ) = $(.fa)E $ ( B ) = B, so t ) j ( , f ’ ) ~( B : A ) . QED

The Jacobson Radical of a Relatively Free PI-Ring Is Nil Given a ring [resp. (*)-ring] R, let # ( R ) [resp. /N(R,*)] denote the , *), *)I. relatively free ring [resp. (*)-ring] 4{X}/.f(R) [resp. ( ( 4 ( X } *)/.U(R, Proposition 2.4.5.

I f Nil(R) = 0, theri Nil(lN(R)) = 0. !f Jac(R) = 0

rkeri Jac(#(R)) = 0.

Proof. Let U = 4 Y ( R ) and A = N i l ( U ) a , U . If 0#f’{X,, ..., X , ) e A then f ( X , , . . . , X , ) is not an identity of R, so for suitable r l , .. .,r, in R we have f ( r , , . .. , r k ) is some nonzero element r of R. Moreover. for any elements r i l , r i 2of R , 1 < i < j, for any j, we see that rilrri2is a suitable image of Xlr...rXlr)Xk+j+i, so lrilrri2 is nilpotent. Thus RrR E Nil(R) = 0. contrary to r # 0, and we must conclude that A = 0. An analogous proof works for Jac( ). QED

x~=l~k+if(

The principle that oa(R) is “at least as nice as R” is extremely important, and we use it to prove the next theorem. Lemma 2.4.6. Jac( # ( R ) )= 0.

I f NiI(R) = 0 and R is quitlalent to R[A], rheri

Proof. /N(R)= ,ilC(R[A]); since Jac(R[A]) are done by Proposition 2.4.5. QED

Theorem 2.4.7. Zf

=0

by Amitsur’s theorem, we

U is a relatioely jree PI-ring, thrri Nil(U)

= Jac( U ) .

Proof. We may pass to the relatively free PI-ring U,/Nil(U), and thereby assume that Nil(U) = 0. Note that if 4 contains an infinite field then U is equivalent to U[A] by Corollary 2.3.32, so we are done by Lemma 2.4.6. In general, let J = Jac(U) and let U = U/Ann,J, a relatively free PI-ring, by Remark 2.4.3 and Proposition 2.4.4 (taking A = J and B = 0). Let denote the canonical image of a subset of U in 0. We claim that t^‘ is semiprime. Indeed, if A a O and A 2 = 0, then A Z J = 0 in U , so (AJ)’ = 0. implying AJ = 0 and A = 0. A similar argument shows Ann0 J = 0.

$2.4.1

Relatively Free PI-Rings and T-Ideals

135

Let B = Annu(Z(0)n Jac(0)). Then ( Z ( 0 )n BJ)' G B ( Z ( 0 )n J) = 0, implying Z ( 0 )n SJ = 0. Since 0 is a semiprime PI-ring we have BJ = 0, implying B E A n n o J = 0. But then 0 is equivalent to by Corollary 2.3.35, so by Lemma 2.4.6 0 = J ac ( 0 ) 2 J , implying J 2 = 0. Therefore J = O . QED

o[A]

Relatively Free PI-Rings of Prime and Semiprime Rings

Example 2.4.8. Let F be the finite field Z / p Z , p prime. Then Jac(&(/(F))= 0 by Proposition 2.4.5. Nevertheless, % ( F ) is not prime because X,(Xp-'-l)=o.

Thus information about '@(F)can be obtained only with some restriction on F. We turn to Corollary 2.3.36 to obtain positive results.

4

Theorem 2.4.9. I f R is an infinite prime algebra of PI-class n, and if :4.1 G R , then +Y(R) = 4,,{Y ) .

Proof. By Corollary 2.3.36 R is equivalent to Q,(R), which is simple with center an infinite field K . By Lemma 1.5.13 Q , ( R ) has a central extension of the form M,,(F) [where F is a maximal subfield of Q , ( R ) ] , and M,,(F) is equivalent to Q z ( R ) .Also M , ( F ) is equivalent to M,(F[(]), which is equivalent to M,,(4[5]). Thus # ( R ) = -t#(M,,(4[(]))= 4,,{ Y ) by Pro-

position 1.3.10. QED Corollary 2.4.10. I f $c, is m i injiniteheld, then for every semiprime PIalgebra R of'PI-class n, 'U(R)= 4,,{ Y). Proof. R is a subdirect product of prime algebras of PI-class apply Corollary 1.4.19 to Theorem 2.4.9. Q E D

< n, so

Thus the 4 , { Y ) are very important, and have wonderful application in Chapter 3. Let us make one simple observation here. Proposition 2.4.11. Suppose 4 is a domain. esterision of&,{ Y) ; consequently 4,,{Y) is prime.

M , ( 4 ( i ) ) is

a cenrral

Proof. Let A be the 4(t)-subalgebra of M , , ( 4 ( ( ) )generated by +,,{ Y } . C Z n 2 - is not an identity of M,,($(()),and hence is not an identity of +,,{ Y ) , nor of its central extension A . Therefore [ A : 4 ( ( ) ] 2 n2, implying A

,

= M,,(d(()), proving the first assertion. The second assertion is now immediate (cf. Remark 1.9.5). QED

Corollary 2.4.12. domain 4.

4,1{Y

IS

u relutiuely free PI-algebra ,for euery

136

THE GENERAL THEORY OF IDENTITIES

Proof.

d,,{ Y > is an infinite prime ring, so apply Theorem 2.4.9.

[Ch. 2

QED

Relatively Free Prime Rings with Involution

Theorem 2.4.1 3. Suppose ( R , *) is iifinite, arid prime of PI-class n. Either ( R ,*) is special or else there is a siritablefield F and suitable irivolutiori (*) qf'thejirst kind on M , , ( F )such that ( R , *) is equivalent to ( A 4 J F ) . *). Proof. Assume ( R , * ) is not special. Then ( Q , ( R ) , * ) is equivalent to ( R , *) and by Proposition 2.3.39 is of the first kind. Thus Q , ( R ) is simple with some infinite center C . Take F to be a maximal subfield. Then ( Q , ( R ) OCF , *) z (M,,(F),*) is of the first kind by Proposition 2.3.39, and is clearly equivalent to ( R , *). QED

At this point, we can say the following about a T-ideal I of ( d { X l ,*) arising from an infinite, prime, faithful algebra with involution ( R , *) having PI-class t i . Either ( R , * ) is special, and then I is the (*)-ideal generated by .g(4,,{Y}),or ( R , * ) is not special, so that I = / ( M , ( E ) , * ) for a suitable algebraically closed field F, and (*) of the first kind.

Conversely, we have the following result. Proposition 2.4.14. Suppose F is afield and ( M , ( F ) , * ) is qfthefirst kind. Let V = (.Y E M,(F)Js* = x}. Then V is an F-subspace cf A f , , ( F ) ; letting t = [ V : F ] , we have t < n2. c2,+I V I

+ x:,. . .. x,+I +x:+ 1 , x,+2t.' . ,x2,+1 )

is a rionspecial identity q f ' ( M , , ( F )*). , Proof. If a, bE V then (ab)* = ba; since M , ( F ) is not commutative we see V # M , ( F ) , so t < n 2 . The last assertion follows immediately from the properties of the Capelli polynomial. QED Also, one may wonder whether in fact there is a prime algebra R of PIclass n, having an involution (*) such that ( R , *) is special.

Proposition 2.4.15. d n i

I f d, is a domain, then +,,{ Y j is equivalent to

Yt"".

Proof.

4,,[Y}"' is prime of PI-class n, so we use Theorem 2.4.9. QED

Theorem 2.4.1 6. If d, is a dotnain, the reversal involution ( # ) and canonical inrolutiou (*) 011 4 { X } induce involutions on d , , [ Y } .(4,#!Y ] , * ) is special. Proof.

d,.{Y)

t

4 { X ] / $ ( q 5 , , { Y ) ) ;by Proposition 2.4.15 Y(qh,{ Y J ) is

52.4.1

Relatively Free PI-Rings and .T-Ideals

137

invariant under the involutions ( # ) and (*), which therefore induce involutions on #,,{ Y } . If f ( X , , Xf, . . . ,X,, X : ) is an identity of (#,,{ Y } , *) then f(Y,, Y,, . . ., YZt- Y,,) = 0, implying f is special. Thus ($,,{ Y } , *) is special. QED

Specht’s Conjecture and Related Considerations

Returning to the noninvolutory case, we shall now look more closely at the structure of T-ideals. In Corollary 2.4.10 we determined all semiprime, relatively free PI-algebras when 4 is an infinite field. Equivalently we have found all semiprime T-ideals of 4 { X ) ; namely, there is one [S(M,(4)] for each n, as well as “0.” Writing I , = Y(M,(cj)),we have I , 2 I , =I ..., so the semiprime T-ideals of $ { X } are linearly ordered. In particular, every ascending chain of semiprime T-ideals of $ ( X } is finite.

0;

Remark 2.4.17. Ii is a semiprime T-ideal of be 0. (It cannot be any I,, obviously.)

4 ( X } and so must

This last observation leads to another question, best presented after some new definitions. Definition 2.4.18. A collection 8 of subsets of R is closed under intersection if for each family ( E , E F ( ~ E Twe } , have n { E , ( y E r } E & . If & is closed under intersection, we say R satisfies ACC(€) if there is no infinite chain El c E , c E , c ... of members of 8. Example 2.4.19. A = (Annihilator ideals (of R ) } is closed under intersection, because n,Ann B,.= Ann(C,,B,). Similarly, 8 is Zorn. We discussed PI-rings with ACC (annihilator ideals) in $1.7. Example 2.4.20. { T-ideals} is closed under intersection and is Zorn. If To is a given T-ideal of R, {T-ideals containing To} is closed under intersection and is Zorn.

Definition 2.4.21. If S is a subset of R and if 8 is closed under intersection, the member of 8 generated by S , denoted &(S), is n{E E E I S G E } . (Intuitively, this is the “smallest” member of 6 containing S . ) A member of 8 isfinitely generated if it is generated by a finite set.

In general, a finitely generated T-ideal is not finitely generated as an ideal. The point of these definitions is in the following easy result: Proposition 2.4.22. Suppose F is closed under intersection and is Zorn. R satisfies ACC(8) i$every member of 8 isfinitely generated.

138

[Ch. 2

THE GENERAL THEORY OF IDENTITIES

Proof. (=) Suppose R satisfies ACC((S) and we have E E C . Take E , = 8(@) and, inductively, if E,-l # E choose x i E E- Ei-, and let E , = 8 ( E i - , u {xi)) = A({xl, ..., x i ) ) . Clearly El c E , c ..., so, by hypothesis, for some k, E = E, = R ( { . Y ~.., . , x k ) ) . (-=) If we had a chain El c E, c ... then, letting E = U , E , , we would have E = G({xl... , , x k ) )for suitable .xj in E , suitable k. But then each xi is in some EiIIso, letting u = max(ij(1 j 6 k}, we have E = E,, implying the

<

chain is finite. QED We are now ready for a famous conjecture of Specht [SO]. Specht's Conjecture.

Q{ X ) satisfies ACC( T-ideals). The point of Specht's conjecture, if true, would be that every T-ideal of Q { X ) was finitely generated, so that there are certain "nice" identities. Of course, we already know what the nice identities are. Namely, from Chapter 1, C2,++, and [ g n . X 4 n 2 + 2 ]are the only identities which are needed to develop virtually the entire structural PI-theory; if one wants we can throw in Sid for all d , k , to make sure that we have an identity for every PI-ring. So Specht's conjecture has limited use, even if true. Nevertheless. it is the oldest and most famous PI-theoretic question, having attracted considerable attention from the Russian school. Some of the best results to date are , showed two major facts: (1) a ( M , ( Q ) )is from Razmyslov [73b, 7 4 ~ 1who finitely generated (as T-ideal); ( 2 ) Q,( Y ) satisfies ACC(T-ideals). We shall prove some positive results on finite generation of T-ideals in Chapter 6. For the time being, we give a negative result of Amitsur [74bJ which adds some flavor. Since the standard polynomial S,, is the minimal identity of M,(F), one might expect that S,, generates . Y ( M , ( F ) ) ,but this is not so.

Proposition 2.4.23.

For each n > 1, S,, does not generate . Y ( M n ( F ) ) .

Proof. Consider f ( X 1 , X 2 )= S , ( [ X l , X , ] , [ X ; - ' , X 2 ] . .. . ,[ X I ,X , ] ) E . Y ( M , ( F ) ) by Proposition 1.1.37. Then deg'(f) = n(n+ 1)/2 and deg2(f ) = n ; we shall conclude by showing that for any h , , ..., h,, in F { X , , X , ) , S2,(hl,.. .,h2,) has no monomial h for which deg,h d n(n 1)/2 and deg,h < n. Indeed, since S,, is normal, we may assume that h l , . . . . h z , are monomials and deg,h, < deg,h,+ 1 6 i 6 2n - 1. Assume deg, h < n(n + 1)/2 and deg,h < n. Then X , appears in at most n of the h,, so deg2h,= 0 for i < n. Moreover, if any hi = a for a in F , then S,,(h, ,..., h,,) = 0, and again we are done. Thus h , , . . .,h, are monomials in X , , of positive degree, and must be F-independent. The most efficient way of doing this is to put hi = X i , 1 < i < n. But then we already have deg,(h, ,..., h,) = n ( n + 1)/2, so we must have deg,hi = 0 for all i > n. Thus hi = a i X , for a, in F, all i > n. But then S2,,(hl,....h2,) = 0 if n > 1. QED

+

$2.4.1

Relatively Free PI-Rings and T-Ideals

139

This argument can be refined a bit; cf. Exercise 6.3.5. A question which, to my taste, is more interesting than Specht’s problem, is, “What is the kernel of the canonical map Z’,; Y } + (Z/pZ),{ Y ) for p prime?” The obvious conjecture is pZ,{ Y } , which is obviously true for n = 1. Specht’s question has some bearing on this problem.

Introduction to Trace Identities

Razmyslov [74a] and Procesi [76] independently gave satisfying responses to the question whether .f(M,,(Q))is finitely generated, actually by changing the question, broadening the scope from “identities” to trace identities. These are formal expressions in the Xi which also involve a symbol t r ; for example, tr(X,X,-X,X,) is a trace identity of M , ( F ) for any n, and, more generally, tr(S,,(X,, . . .,X2,J) is also a trace identity of M,(Q) for all n by Remark 1.3.4. These trace identities are called “trivial” because they work for all n. Rather than go deeply into the theory of trace identities, we shall just give some very important examples and state the main theorems of Razmyslov and Procesi. We work in characteristic 0, for simplicity.

xy=

Definition 2.4.24. Define inductively f o = 1 and j ; = ,((- l)’-’. k-lji-i)tr(X\), and 3 = X; +xi= (( - l)kjJX;-k. For example, T, = X:-(trX,)x, +((trX,)2-tr(X:))/2. Proposition 2.4.25. Proof. theorem.

T, i s a trace identity o j M , ( Q ) .

Immediate from Newton’s formulas and the Cayley-Hamilton QED

The Razmyslov-Procesi theorem says that every trace identity of M J Q ) is a “consequence” of T, and the trivial trace identities; just knowing this fact facilitates simple proofs of PI-t heorems, such as Razmyslov’s proof of the Amitsur-Levitzki theorem (which we presented). Since tr is a linear function, we can apply multilinearization to trace identities and get new trace identities. Multilinearizing X,T, yields X,X,X,+X,X,X, -tr(X,)X,X, -tr(X,)X,X, + tr ( X , ) tr(X,)X, tr(X,X, X,X, )x,.

-+

+

Applying tr again and using the trivial trace identity tr(X,X, -X,X,), we get another trace identity:

+

tr(X,X,X,) tr(X,X,X, ) - tr(X,)tr(X,X,)- tr(X,) tr(X,X,) -tr(X,) tr(X,X,)+ tr(X,)tr(X,)tr(X,).

1 40

THE GENERAL THEORY OF IDENTITIES

[Ch. 2

This is the “fundamental trace identity” of Procesi for 2 x 2 matrices; likewise, multilinearizing X , , T, and applying tr gives Procesi’s fundamental trace identity for n x n matrices. (Procesi proved that every trace identity in fact is a consequence of this trace identity and the trivial trace identities.) So Specht’s question, suitably generalized, is very neatly answered. The theory of trace identities is fascinating, and leads to new PI-theorems (such as an improvement of the Nagata-Higman theorem in Razmyslov P a l ). $2.5. Identities of Matrix Rings with Involution In this section we examine two specific involutions on M,(C). and use the results for a precise study of minimal polynomials of symmetric and antisymmetric elements ;consequences in the (*)-PI-theory are given. [Some interesting central (*)-polynomials are given in the exercises of Appendix A.] Although we shall only consider two involutions here, we show in $3.1 that these two involutions (the “transpose” and the “canonical symplectic involution”) yield all possible prime T-ideals of ( $ ( X } , *). Throughout this section, C is a commutative ring, F is a field, and {eijl 1 ,< i , j ,< n } is the usual set of matric units of M,(C). Let (t) denote the transpose involution. If n = 2m is even, and I + 1 # 0 in C, there is another involution which will turn out for us to have even greater interest, called the canonical symplectic involution (s), defined by xs = ax‘a for all x in M , ( C ) , where a = X‘‘’(ei,itm-ei+m,,). In other words, partitioning a 2m x Zm matrix A into m x m matrices A,, 1 d i d 4, we have A, A , A\ (A3 A ) = ( - A :

-A; A:)’

where t is the transpose o n M,(F). Definition 2.5.1. An element x of ( R , * ) is symmetric (resp. antisymmetric) if .Y* = x jresp. x* = - x). We shall write R + (resp. R - ) to denote the set of elements of R of the form x +x* (resp. x - x*).

Of course, the definition is dependent on the particular involution we use, and we must be careful that there is no ambiguity. Remark 2.5.2. Let Z = Z ( R , * ) . Then R + and R - are Z submodules of R . If ~ E Rthen R 2 R + O R - [as Z-modules, the correspondence given by r -+ ( ( r + r * ) / 2 , ( r - r * ) / 2 ) ] , and R’ (resp. R - ) are the set of symmetric (resp. antisymmetric) elements of (R, *); in particular, if 2 is a field then [ R :Z ] = [R’ :Z ] + [ R - :21.

$2.5.1

Identities of Matrix Rings with Involution

141

Remark 2.5.3. If f E R and ( R , *) = ( M , ( F ) , t ) then, as F-vector space, R + has basis {eij+ejilI < i < j < n } u {eijl < i < n } , and R has basis {eij-eji(l 6 i < j d n ) . In particular, [ R + : F ] = n(n+ 1)/2 and [ R - : F ] = i l ( t 7 - 1)/2. Remark 2.5.4.

If $ E R and ( R , *) = ( M z m ( F )s), , then R + has basis

{ e i j + e j + m , i t m1I

< i, j < m } u {ej,j+,,,-ej,i+ml1 < i < j < m} < i <j < m } ,

u {ei+m,,-ej+m.iI 1

and R - has basis 6 i , j ~ ~ ~ j ~ { e i . j t , + e j ,
{eij-ejtm,itmtI

Thus [ R : F ] = m(2m - 1 ) and [ R - : F ] = m(2m + 1 ). +

Proposition 2.5.5. Suppose n is even. .f(M,,(F),s) Q .f(M,(F), t) and . f ( M , , ( F ) t, ) Q .f(M,,(F),s). Proof. Let u = (n(n- I)/?) + 1 . In view of Theorem . 1.4.34, C z u - l ( X 1 +X:, . . ., X , + X : , X u , I , . .., X z u - ,) is an identity of ( M , ( F ) ,s ) butnotof(M,(F),t),whereasC,,_,(X,-XT ,..., X , - X : , X , + , ,..., X z u - l ) is an identity of ( M , ( F ) , t ) but not of ( M , ( F ) ,s). QED In view of Theorem 2.4.16, we now have three distinct T-ideals corresponding to prime F-algebras with involution, having PI-class 11. As we shall see in Chapter 3, these are all the T-ideals. Minimal Polynomials of Symmetric and Antisymmetric Matrices

A deeper analysis of identities of (M,(F),t) and ( M , ( F ) , s ) can be developed from the minimal polynomials of symmetric and antisymmetric matrices. Of course we shall utilize the corresponding involution (t) and (s) on M n ( E [ < ]). Proposition 2.5.6. Suppose (*) = (t) or (s) on M,(C) for any commutative ring C with 4. I f x* = - x and ,Yy=o~i.3‘ i s the characteristic = a,= ... = 0. polynomial of x, then a,- = a,-

,

Proof.

Every antisymmetric matrix a has trace 0 since tr(a) = tr(a*)

= tr(-a) = -tr(a). Now assume C = Z [ ( ] and conclude inductively, using

Newton’s formulas. Q E D Further results are available through consideration of an old notion, the “Pfaffian.”

142

T H E GENERAL THEORY OF IDENTITIES

[Ch. 2

Lemma 2.5.7. Let r = ( r i j )E (M2,,,(C),t), with r‘ = - r . Writing r i for the (2m,i) minor of r (rhe determinant of the matrix formed from r by discarding row Sm and column i) and r‘ = xf!y=-:rijeij, we hace (a2,,,= det(r’)det(r). Proof. Note that uZm= 0, being the determinant of a (2m- 1) x (2m antisymmetric matrix. Also, for all i, we have det(r) = x;21(- 1 )ir2m,ia, = xfZ)=m;( - l)ir2m.iai; likewise, for all k ,< 2m - 2, we have 0 = x:Z)=m;( - 1 )‘rkiui.This gives (2m- 1 ) equations for the 2m - 1 quantities { ( - l)iaill 6 i < 2m- 1 ) in terms of {rkilk # 2m- 1,i # 2m). Thus, solving for ~ 1 by ~Cramer’s ~ - rule, ~ we have (- 1)2m-1a2mlfl = det(r’)det(r),where B is the (2m-1.2m)-minor of r ; since r ‘ = - r we have f l = - a 2 m - 1 . so ( a 2 m - 1 )= 2 det(rOdet(r). QED - 1)

’

Proposition 2.5.8. For any antisymmetric element r of (M2,,,(C),t), det(r) is a perfect square in C. Proof.

We may assume C = Z[<] and r = x l s i < j s n ( t i j - c j i ) e i jBy . induction on m, notation as in Lemma 2.5.7, det(r’) is a perfect square so, by unique factorization in Z[<], so is det(r). QED Definition 2.5.9. If r E (M2,,,(C),t) and r‘ = - r , define the Pfufian o f r to be a square root of det(r).

[When C is a domain, det(r) has exactly two square roots, so there is a bit of ambiguity in the definition. This is really irrelevant, but could also be overcome by considering generic matrices.] For us, the point of the Pfaffian is really with (s). Theorem 2.5.10. Suppose n = 2m and xs = X E M,(C), C arbitrary commutative; let a = (ei,i+m-ei+n,i)and, viewing M , ( C ) c M,(C[E-]), note that (1-.u)a is antisymmetric with respect to (t). Define p(1) = P f ( (A- .u)a)E C[A]. Then deg(p) = m, and p 2 is the characteristic polynomial 0f.u. Hence the characteristic values of x are double roots o f ’ p 2 ,and p ( x ) = 0.

xy=

Proof. (Passing to Z [ < ] , we may assume C is a domain; work in a field F containing C as well as all characteristic values of x.) Obviously (so that p is defined), deg(p) = m , and p 2 = ((1-x)a)‘= -(A-.u)a det((A-x)a) = det(l-.u)det(a) = det(l-.u), the characteristic polynomial of .Y. Every characteristic value is thus a root of p , so is a double root of p2. It remains to show p(.u) = 0. Take r = x : l s i < j s n ( < i j - < j i ) eand i j using notation of Lemma 2.5.7, write p for the Pfaffian of r. Then p 2 = det(r) divides (u2,,- 1)2; thus 11 divides u ~ , , , -and ~ , likewise divides each entry of adj(r), the adjoint of r. Thus adj(r) = pr’ for some r’ in M , ( Z [ ( ] ) . Then det(r) = r(adj(r)) = prr’, sop = rr‘. Specializing the indeterminates so as to send r to (1- .u)a,we

42.5.1

Identities of Matrix Rings with Involution

getp(2) = @ - s ) a b , w h e r e b ~M , , ( F [ i l ] )Writingb . = and writing p(2) = C k i l k , we get t < m and co = -.uabo,

c,

= ab,-,,

143

bkilkforbkEM J F )

ck = -..uabk+abk-,

1
for

Thus

x.k"=

Ck.Yk =

xrz

.ykCk

+ xab, - .xmab, + .xmab, = 0.

= -. ~ a b o

QED

Some Identities of ( M n ( C )t), and (Mn(C1,S)

Let us translate these results into identities. Proposition 2.5.11. polynomial. (i)

Zfn

Suppose

f ( X , , . . .,X d ) is

an

m-normal

= 2m, then

f(r(xl-x:w21,[(x, - X : ) ~ - ~ J ~ I ~ . . . , ~-XT)~,X~I,X~+~~..., (X~ is an identity of'(M,(.Z), *),ji)r (*) = (s) or (*) = (t).

(ii) Z f n f((xl

=

2m- 1 then -x:)"-2x2,.

-x:)"x2,

'.

9

(xl

-x:)x.2,

x m + 1,

...

1

xd)

is an identity of (M,(Z),t). (iii) Ifn = 2m then f([(xl

+X$)m, x2],

[(xl

+x:)m-

' 9

x2]7...

>

+x:),X2]rXm+

1,

...

9

xd)

is an identity of (M,(Z),s). Proof. Use the minimal polynomials of antisymmetric and symmetric matrices. Q E D Corollary 2.5.1 2.

( M 2 , ( F ) ,s) 2 ( M , ( F ) ,t)for all n > m.

Proof. Take Proposition 2.5.1l(iii) in conjunction with Theorem 1.4.34, noting that every diagonal matrix is symmetric with respect to (t). QED Remark 2.5.13. Corollary 2.5.12 is sharp, because we have a 0 ) + ( M 2 , ( F ) ,s), given by canonical injection of (M,(F) 0 Mm(F)UP,

Since (M,(F) 0M,(F)"P ,0 ) is special, we get ,f(M,(F), t )

3

. f ( M , ( F ) 0 M,(F)"P, 0 ) 3 *f(M,,(F), s ) .

144

THE GENERAL THEORY OF IDENTITIES

[Ch. 2

Minimal Identities

Let us turn now to the following question: “What is the identity (resp. central polynomial) of minimal possible degree of (M,,(H), *), where (*) = (t) or (s)?” This question is largely open, and we shall summarize the known results. First, an easy negative fact. Let C be an arbitrary commutative ring. Neither Remark 2.5.14. (M,(C),t) nor (M,,(C),s) satisfies any identity of degree < n . [Indeed, otherwise we would have a multilinear identity of degree < n - 1, which is impossible in view of the staircases {ei,i+lll< i < n} for (M,(C),t), and ~ ~ l l , ~ 1 2 , ~ 2 2 , . . . , ~ m - l , ~ (where , ~ m m m } = n / 2 ) for (M,,(C),s).] Remark 2.5.1 5.

xl-x:€.f(M1(C),t); [x,-x:,x,-x:]E.f(Mz(C),t); [ X , + x:,X,] E .f(M,(C), s); [ S 3 ( X , - X ? , X , - X f , x3-x3, X,]€.P(M3(C), t ) ; “x1 + x:,x,- xrI2,X,] € .f(M,(C), s); S , ( X , -x: ,...) X,-X~)E.~((M4(C), t). (Each of these verifications is immediate from Remarks 2.5.3 and 2.5.4, and Theorem 2.5.10.) As far as I know, all minimal identities of (M,(Z), *) (for n < 4) are implied by these identities; even for n = 3, 4, Remark 2.5.14 is not sharp in characteristic 0. Also note that minimal identities of (M,(Z), *) need not resemble standard polynomials, although the situation for n > 6 is completely unknown in this respect. For a time, it was even unknown whether ( M , ( h ) ,t ) had an identity of degree <2n, for n =- 4. (The parallel assertion for (s) is still unknown for n > 8.) In a brilliant paper relating PI-theory to other branches of mathematics, Kostant [58] proved S2n-2(X1 -Xr,...,X2n-2-Xfn-2) is an identity of (M,(Z), t) for all n eoen. Accordingly, interest has focused on the standard polynomial after all. Definition 2.5.16. Y ( n , d, k, (t)) means

&(XI

+x:,...) x , , + x : , x , + ~ - x ,...) : + ~X , - X Z )

is an identity of (M,(E), t) [and thus of (M,(C),t) for any commutative ring C). Y ( n ,d, k, (s)) is defined analogously. Note that Y ( n ,d, k, (t)) and Y ( n ,d, k, (s)) hold whenever d 2 2n, by Amitsur-Levitzki. Rowen [74a] gave a complete classification of U ( n ,d , k,@)), as follows:

42.6.1

Sentences of Algebraic Systems

145

Theorem 2.5.17 (Kostant [58]-Rowen [74a]). Y ( n , 2n-2, 0, (t)), Y’(n,2n-l,O,(t)), and Y(i1,2n-I, I , ( t ) ) hold for all t i ; Y’(ii,211-2,l,(t)) holds for all odd n. All other Y ( n ,rl, k , ( t ) ) are.fa1.w wheneuer d < 2n.

The negative results are given in Exercises 8,9. The proof of the positive results combine easy algebraic methods (given in Exercises 1-6) with an intricate graph-theoretic argument which we omit here because of its length. (J. Hutchinson and F. Owens also announced graph-theoretic proofs of Kostant’s theorem, but their proofs fell. Hutchinson’s proof was resurrected later, after she read a preprint of Rowen [74a].) Procesi [76, $81 proves that every trace identity for (M,(Z),t) is a “consequence” of a finite number of trace (t)-identities (which he calls orthogonal trace identities), and he writes these down. Thus, one should be able to produce an easy algebraic proof of the Kostant-Rowen results, although I do not know of such a proof. The identities of ( M , ( L ) , s ) are mysterious, and I do not know of any general theorems of the form Y ’ ( ~ i , dk,, (s)).Of course for n very small one has some results, such as Y(2,2, I , (s)) and Y’(2,2,2, (s)). A reasonable conjecture is Y ’ ( 4 2 ~ - 1,2n- 1, (s)). Procesi [76] proved that the trace identities of ( M , ( Z ) , s )come from a certain explicit, finite set. $2.6. Elementary Sentences of Algebraic Systems

Polynomial identities (and GI’s) fit very naturally into formal logic, being a very fundamental type of “elementary sentence.” We discuss elementary sentences briefly (and pseudorigorously) in this section, to indicate useful generalizations of the PI- and GI-theories, and to lay the groundwork for a very pretty application of the “ultraproduct” to ring theory in $7.5. A fine treatment is given in Maltsev [73B]. Intuitively, an elementary sentence should be a sentence about elements ; e.g., for an R-module M we want

( V r e R ) ( V y , E M ) ( ~ y 2 E M ) ( r ( Y , + Y 2 ) - ~ Y , - r y=, 0 ) to be elementary. Accordingly, we consider the following set-up: Let S , , ..., S , be sets, and let F , , . .., Fk be functions defined on the S , , .. . , S , and taking values in these sets; i.e., we have for 1 d j d k F j : S j , x ... x S j 8 ,-P Sj,, for suitable j , , . . .,j,,+) in { 1,..., m } , where u depends on j . S j , x ... x Sj,, is called the domain of F , and S j , , . is called the codomain of F j . . . ,S,; F , , . . . ,Fk)iS called an (algebraic)system. ,

I

I

Definition 2.6.1.

An atomic,formula (1 for a system =

( S , , .. .,s,; F , , . . . )Fk)

146

THE GENERAL THEORY OF IDENTITIES

[Ch. 2

is an expression of the form ( f ( x , , .. ., x t ) = g ( x l , ..., x t ) ) , where j ; g are compositionsoftheF j , and xidenoteseither an indeterminate X ior an element (i.e. “constant”) of the appropriate component S , . Inotherwords,westart withthexiand build theformula upthroughrepeated applications of the given functions. To stress the role of the indeterminates, we often shall write down only the indeterminates (ignoring those .xi which are constants), writing $ as $ ( X l , . ., x d ) , or ( f ( X l , . . .,x d ) = g ( X , , . . ., x d ) ) . We say $ holds in Y when ,f(sl,.. . ,sd) = g(s,, . . . , s d ) for all sl,. . . ,s,, (in their appropriate S j , ) , whenever defined, stipulating that some f(s,,. . . ,s d ) and g ( s , , . . .,sd)must be defined in the same S , to ensure that makes sense. In case 9’ has the structure of additive group (perhaps in addition to other structure),wecanreplace$by(f(X,, ..., x d ) + ( - g ( X l , .. ., x d ) ) = O),sothat we may assume without loss of generality that g = 0. Then atomic formulas closely resemble generalized identities; also cf. $8.1. holding in 9’

.

Definition 2.6.2. Define an elementary formula for a system Y (along with whether it holds in Y ,and the rank), inductively as follows.

(i) Every atomic formula is elementary, of rank 1. (ii) If $ is elementary of rank u, then (I$) is elementary of rank (u 1); ( 1$) holds in Y iff $ does not hold in Y . (iii) If G1, tJ2 are elementary, of ranks ul, u,, respectively, then ($1 A $J is elementary of rank max(u,, u,)+ 1 ; (+hl A $,) holds in .Y iff $,, +h2 both hold in Y. (iv) If $(X,,. . . , X d ) is elementary of rank u, then ((3xi)$(Xl,..., X i - l , x i , X i + l ,... ,X,))iselementaryofrank(u+ I),and holds in Y iff $ ( X l , .. .,X i - xi, X i + . . .,X d ) holds for suitable .xi in the appropriate component Sji ; to emphasize this, we often write (3xi E Sj,)($(xl,.. , xi- 1, xi, xi+1, .. x d ) ) .

+

,,

9

,,

’ 9

An elementary sentence (or sentence of the j r s t order) is a formula in which no X i occurs, i.e., each X i has been changed to xi [accompanied by a previous @xi),according to Definition 2.6.2(iv)]. We call 3 a quantijer. It is convenient to introduce another quantifier V (meaning “for all”) as well as the symbol v (meaning “or, inclusively”) which are defined in terms of 1,A , and 3; there are certain very well-known rules for simplifying sentences, which match intuition. We leave it for the reader to justify that every sentence can be put in (proper) normal form (Qlxl). .. (Q,x,l (d A .. . A du) where each Qiis a quantifier and where each cdihastheformll/i,v ” . v $i,v ( l $ i vt“+ - vl() l~)~,,);hereeach$~~isan atomic formula, and t , u depend on i. We say .dihas type t, and cotype (u - t ) ;if each .dihas type 0 or 1, we call the sentence Horn. We say a sentence (Qlxl)*.-(Q,x,)(.dl A ..* A d u is ) ‘‘Q1...Qrn,”and,

$2.6.1

Sentences of Algebraic Systems

147

in fact, delete consecutive appearances of the same quantifier. Thus sentences using only the quantifier V (in normal form) are called “V sentences.” Definition 2.6.3. A sentence in normal form is atomic if, in the above notation, u = t = u = 1. Example 2.6.4. Suppose R = S , and M = S,. One can describe the algebraic system denoting that R is a ring and M is an R-module by nine functions F , through F , (corresponding to three constants OR, l,, and O M , additive inverses for R and for M , addition R x R -+ R and M x M + M , and multiplication R x R -+ R and R x M + M ) and seventeen sentences, all but two of which are atomic (these two being the existence of additive inverses). Details are left for the reader. Example 2.6.5. R,M as in Example 2.6.4, M is R-faithful iff the sentence = (Vx, E R ) (3x2E M ) ((x, = 0) v (xlxz # 0)) holds. M is Rirreducible iff the sentence $2 = (Vx, E M)(Vx, E M)(3x, E R ) ( ( x , = 0) v (~3x1 = x2)) holds. Note that $, is Horn, but $z is not Horn. Example 2.6.6.

R is a domain iff the sentence $ = (Vx,)(Vx,)

((xI = 0) v (x2 = 0) v (x1x2# 0)) holds in R ; t,b is not Horn.

Example 2.6.7. R is prime iff the sentence $ = (Vxl)(Vxz)(3x3) ((x, = 0) v (x2 = 0) v (xIx3x2 # 0)) holds in R ; $ is not horn. Example 2.6.8.

R is semiprime iff the sentence $ = (Vx,)(3xz)

((x, = 0) v (xIx2xI # 0)) holds in R ; $ is Horn.

Example 2.6.9. Let us consider the system of rings with involution (*). (*) is of the first kind, iff the sentence = (Vx1)(3x2)((x: = x,) v (1 (x1x2-x2xI = 0))) holds in (R, *); $ is Horn.

We return to discussing systems. Two systems ( S , , . . ., S m ; F , , . . .,Fk) and (Si,... ,S&,; F , , . . .,Fk,) are similar if m = m’, k = k‘, and the correspondence Si+ Si, 1 < i < m also corresponds the domains and codomains of F j , Fj for each j, 1 < j < k . For example if R’ is a ring with module M‘ then we can define F,,. . ., F, analogously to Example 2.6.4, and get ( R ,M ; F , , . . . ,F,) is similar to (R’, M‘, F; ,. . .,F,).Similarity is an equivalence relation on systems; we call any such equivalence class a class (ofsystems). Suppose % is a class of systems, and (S,,. . ., S m ; F, , . . .,Fk)E%. For any sets ..., Sm such that there are onto maps S i + S i , 1 d i d rn,-we can _define functions F j , by setting F,(s,,. . ., s,) = F,(s,, .. ., su), for all si in S,,, 1 d i ,< u. (Here - denotes the canonical image of the onto maps.) Clearly

G,

148

THE GENERAL THEORY OF IDENTITIES

-

_ _

[Ch. 2

.

-

(Sl,...,S, ; F , , . .. ,F,.) E%, and is called a hornomorphic image of (Sl,.. . S,; F,, .. ., Fk). Remark 2.6.10. Every atomic sentence of a system holds for all of its homomorphic images. Definition 2.6.11 Form the direct product of systems {(Sly,. . . ,S,,,, ; F l y , .. .,F k , ) ( y E r}in a class of systems %‘ as follows: Let Si= , and define F j componentwise on the Si (i.e. the y component of F j is F,?). Then

clearly (Sl,.. .,S,: F , , . . . ,F , ) E 59.

Remark 2.6.12. Every atomic sentence holding in each component also holds in the direct product.

We leave it to the reader to define a “subsystem” and to show that every V sentence holding in a system also holds in each subsystem. This observation, as well as Remarks 2.6.10 and 2.6.12, are not used further, but indicate the importance of atomic sentences and their connections with direct products and homomorphic images. Polynomial identities are atomic V sentences, and thus are the most fundamental type of sentence. The generalized identity is an atomic 3V sentence, and if we expand the system so that the “constants” include the coefficient set of the generalized polynomial, the GI actually becomes an V sentence. Thus, logically, the PIand GI-theories are the most natural theories of rings. If one is to attempt a systematic classification of rings by elementary sentences, the next step is atomic V3 sentences. One class of these sentences is treated in Appendix B, with mostly satisfactory results. The general classification problem, although very interesting in my opinion, promises to be extremely difficult. For example, one V3 sentence is “von Neumann regularity,” given by (Vx,)(3x2)(x,x2x, = x,); only very recently has an example been given of a prime von Neumann regular ring which is not primitive.

EXERCISES

52.1 1. Suppose W s R, and R satisfies every GI of W . (This must be stated appropriately.) Then R is a W-ring. 2. Suppose G is a finite group of automorphisms and antiautomorphismsof W . Then G can be extended to a group of automorphisms and antiautomorphismsof W ( X } .Use this fact to define a “generalized G-identity.”

g2.2 1. If S = {regular elements of Z ( R , *)}, then R , = Qz(R).

Exercises

Ch. 21

149

2. (Martindale [69a]) (R, *) is prime iff P n P* = 0 for some P E Spec(R). 3. If (R, *) is semiprime and satisfies ACC((*)-ideals) then Qz(R, *) is a finite direct product of (*)-simple rings.

52.3 1. Y(nyerR7)= n & ( R ? ) for any algebras R,,. 2. Let R be a class of rings [or (*)-rings] closed under taking of homomorphic images, subrings, and direct products. Then .M is a variety. ( H i n t : Take I , = n{Ia4{X}I4{X}/IE8}. is relatively free and is in a.) cpfX}/lo The next two exercises show how to bypass the entire discussion on stability when q5 is an infinite field. 3. Suppose 4 is an infinite field and p ( [ 1 , . . . . & ) ~ q 5 [ t ] such that p ( q , . . . , a , . ) = 0 for all ai in 4.Then p = 0. (Hint: Vandermonde argument or, alternatively, repeated application of Herstein [64B, Lemma 5.2)). 4. (Amitsur [53]) If H is a commutative algebra and 4 is an infinite field then every identity of R is an identity of R @+ H. [ H i n t : Take a basis {r;Iy E r} of R over 4; given f(X,, . . .,XJ,and xl,. . .,x, in R @ H , write x i= ci,.r;,ci,.in H , andf(x,, . . .,xk)= x,.p:(c)r.;, where each ~,EI$[C] and p , ( c ) is evaluated at various cil Using Exercise 3, conclude that each p; = 0.1 5. Redo Exercise 4 in the (*)case. 6. If g is a generalized monomial and the multilinearization Ag is R-improper, then g is R-stable. Subtracting out all such generalized monomials from a GI yields a GI whose multilinearization is R-proper. 7. Every identity is the sum of blended identities. 8. Every identity of R[A] is R[A]-stable. 9. (Procesi [73b, p. 151) Find a ring R with a completely homogeneous identity which is not R-stable. 10. If R is as in Corollary 2.3.24 and ,f is a completely homogeneous identity of R with deg'(f) < d + 1 for all i then f is R-stable. In particular, every completely homogeneous polynomial of degree < 2 in each indeterminate is stable. 11. (Nagata [53]-Higman [56]) If A is a Q-algebra without 1, satisfying the identity X;, then X,...X,is an identity of A, where m = 2"- 1. [Extensive hint due t o Higgins: Write h ( X l , X 2 )= x:::XiX2XT-'-'. By a Vandermonde argument applied to ( X I faX2r,h is an identity of A. But then we have the identity

x,

~ ( x , 3,xj)x;-j-I x2 = x?:I x\x xjxn-i-Ix"-j-l 3 2 1

~ " - 1

J=o

hJ'0

=

nX;-1X,X2-1 + ~ ~ : ~ X ~ X 3 hXT-i-'), (XZ,

so X;-'X,X;-' is an identity of A. Conclude by induction.] to prove that every division PI-ring is 12. (Amitsur [54]) Use Sn+,(X;X2,...,X2) generated over its center by two elements.

52.4 1.

(Amitsur [55b])

Suppose R is relatively free. Then N(R) is a T-ideal of R, and N(R)

= NiI(R). (Hint: Use Theorem 1.6.21.)If R is a Q-algebra, N(R) = Ll(R).

Let R = &{ Yl, Y2}andf(Y,, YJER. 0 #JEZ(R) ifffis Mn(4[A])-central. . .,X,)g(X,+ I , . . .,X,)is M,(F)-central 3. (Regev) If F is an infinite field and f(X,,. 2.

t h en j g are both M,(F)-central. [Hint: If a, b # 0 in M,(F) and a'b is scalar for every conjugate a' of a. then a, b are scalar.] 4. (Bergman [76P]) Suppose F , , F , are finite fields. 9 ( M , ( F , ) ) c_ 9 ( M , ( F , ) ) iff (F21 divides IF, I.

150

THE GENERAL THEORY OF IDENTITIES

[Ch. 21

5. Assume Theorem 3.1.3 (due to Wedderburn), that any finite simple ring R has the form M , ( Z ( R ) ) for some k. Every semiprime T-ideal of Z{X} has the form F ( B ) , where B is a direct product of matrix rings over fields. Hence Z{X} satisfies ACC(semipr1me T-ideals). 6. If (R, *) IS prime and special, then (R, *) is of the second kind. 7. The reversal involution on +"{ Y} has the second kind.

$2.5 The algebraic parts of Rowen [74b] are in Exercises 1-6. 1. tr((X, +X:)(X,-X;))is a trace identity of (M,(Q),*). 2. If re M,(Q)' and tr(r.r) = 0 for all .Y in M,(Q)+ then r = 0. Likewise for M,(Q)3. Suppose .Y ,,...,. ~ , E M . ( & P ) +and .rk+ ,....,x d e M,(Q)-. Then S,(u, ,..., yd)* = (-

l ) l d / Z I + d - k s d (-y 1 ,

..

7

.d.

_..._ yd in M J F ) , tr(Sd(xl,.. . , . y d ) ) = tr(Sd-l(xl,.. ., xd- l)xd)d. 5. Suppose tr(Sd(X,+X: ,..., Xk+X:,X,+l-X:+I,..., X , - X : ) ) is a trace identity of (mod 4), then (mod 4) or if k is odd and d = 1 (M,(Q),*). If k is even and d = 3 Y ( n ,d - 1, k , (I)) and U ( n ,d - I, k - I , (*)) hold. 6. To verify the Kostant-Rowen theorem, it suffices to prove that for all odd n tr(S,,_ l ( X l + X:, X , - X:. . . . ,X z n - - X:,- ,)) is a trace identity of ( M , ( F ) .t ) . *7. What is the minimal possible degree d of an identity of ( M , ( Z ) ,t)? (Note that n < d < 2n-2.) What ahout (M,(Z),s)? Here are some negative examples. (See Rowen [74b] for more.) 8. Let x1 = P~~ --el, and, for u 2 I , let xIu = e l , u + 2 - e u + z and . l x 2 u + l= e,+Z.Z-e2.,+2. Then S z n - b ( x z , . . . ,Y , . - ~ ) = ( - I)'n+1)12(n-2)!(e12-e21) for n odd, and ( - l ) ' n + 2 ) i Z . ( ( n - 2)!(elI + e , , ) 21n- 3 ) ! ( e , ,+... +en")) for n even. 9. With xi as in Exercise 8, S2n-3(xI,...,_ u,,_,)= ( - l ~ ' z ( n - l ) ! ( e l z - e ~ ,for ) n even, *lo. Is SZn-,(XI-X: ,..., Xzm-3-X;n-3) an identity of (M,( ZlpiZ). t ) when divides 2[n/2] - I ? (For n 7 the answer is, "Yes".) 11. Let F be a tield, and consider Yl + Y:EF,(Y, Y * ) E ( M , ( F ( t ) ) ,s). Then for some suitable finite extension field K of F(r), Yl + Y: is diagonalizable in M , ( K ) . 12. Suppose n = 2m and x* = XEMJC). Letting p ( I ) be the polynomial obtained from Theorem 2.5.10, and writing p ( A ) = l ) k p k I m - kfor suitable p, in C, po = I, show 2kpk = (- l y - ' p k ~i tr(x') for all k , 1 < k < m. ( H i n t : Mimic the proof of Newton's formulas.) 4.

F o r d odd and all xl.

+

xr=o(-

xt=

$2.6 1. Call R identity-separated if, whenever the sentence (V.w,)--~(Vx,)((fl(x,,...xf) ... v (ft(x,,. . , x f ) = 0)) holds in R then someA is an identity of R. If R is torsion-free over an infinite suhring of Z ( R ) ,then R is identity-separated. R [ I ] is identity-separated. 2. (Rowen [78b. part 111) A disjunctive identity is a sentence = 0) v

(Vw,)

,...).u,) = 0)

~~(VX,)((fl(S1

v

".

v

(/k(Xl

,..., w,) = 0 ) ) .

R[A] satisfies thesamedisjunctiveidentitiesasR iffR isidentity-separatedand every identityofR is R-stable. This happens when 4 is an infinite field. *3. What is the largest class of elementary sentences passing from R to R[d], when 4 is an infinite field?

CHAPTER 3

CENTRAL SIMPLE ALGEBRAS This chapter develops the important theory of simple PI-algebras (often called finite-dimensional central simple algebras), with heavy emphasis on the algebras of generic matrices.

$3.1. Fundamental Results Any simple PI-algebra R has some PI-class, which we call deg(R),the degree of R. Remark 3.1.1. further reference.

We shall use the following results, mostly without

(i) (Theorem 1.5.16 and Corollary 1.5.11) If R is simple PI, then R z M , ( D ) for some division PI-ring D determined up to isomorphism, where t = deg(R)/deg(D); also (deg(R))’ = [ R : Z ( R ) ] . (ii) (Theorem 1.8.25 and Corollary 1.8.26) If R , is simple with center Z and R is an extension of R,, then R z R, O z C R ( R , ) ;if C R ( R , )is simple, then R is simple. (iii) Suppose R,, R , are simple PI-algebras over Z = Z ( R , ) , and R = R , OirR,. Then C,(R,) = R,. [Indeed, CR(R,) 2 R,, and equality follows from a dimension count and (ii).] (iv) If D , and D, are division PI-rings with center F, and with relatively prime degrees, then D ,O F D zis a division ring. [Indeed, write R = D, OFD2= M,(D,), and let 1 7 ~= deg(D,). Then 0’;’’O r R z M,,,(D,), so t l r t , ;likewise tln,, implying t = 1.1 (v) (Theorem 1.8.56) Suppose R = M,(D), deg(R) = n, and K is a maximal subfield of D. Then R OZ(R,K = M,(K). (vi) (Theorem 1.5.23) If D is a division ring with maximal subfield K such that [ D : K ] = n, then deg(D) = n and [ K : Z ( D ) ]= n. (vii) For any field F there is the Brauer group Br(F) of equivalence classes of simple PI-algebras with center F (discussed in $1.8). In this section we continue with some basic structure theorems of central 151

152

C E N T R A L SIMPLE ALGEBRAS

[Ch. 3

simple algebras. The starting point is the Skolem-Noether theorem, which leads to Wedderburn's theorem, and also later to characterization of the involutions (of first kind) of central simple algebras. We continue with a careful investigation of the Brauer group in terms of crossed products, following Albert [39B], Herstein [68B], Jacobson [75B], and Reiner [75B], and also discuss cyclic algebras as a special case. The Skolem-Noether Theorem and Wedderburn's Theorem

If Y is an invertible element of R, the automorphism x + + r - ' . ~(of r R ) is called inner. Theorem 3.1.2 (Skolem-Noether). Suppose R is a simple PI-algebra with center F , and A , , A 2 are isomorphic, simple F-subalgebras of R . Then every isomorphism $ : A , 4 A , (of F-algebras) can be extended tv an inner automorphism of R .

Proof. Always, 0 denotes tensor product over F. Take a minimal left ideal L of R, let D = End&, a division PI-algebra with center F , and work in R 0 D O p . Note that L is an R@DoP-moduleunder the action ( C , r i @ d i ) y = x ( r i y ) d ifor ri E R, y E L, di E D ; we let M, denote the set L as A, 0 D O p module under this action (restricted to A l ODop).Also, let M , denote the set L, as A, @WP-moduleunder the action x ( a i @ d i ) y= xi($(ai)y)difor all a i E A , , y E L, di E Dop. Let S = A, @ D O p . a simple PI-ring ;writing S' M , ( D , ) for a suitable division algebra D, and suitable f, we have 1 = xi=,e j j . Thus for i = 1, 2, M i = C(Syly E Mi} = xi= x { S e j j y l yE Mi}, clearly written as a sum of irreducible submodules. Since [Mi : Dop] < co,any minimal such sum is finite and direct. By proposition 1.5.9 all irreducible S-modules are isomorphic. Since [M, :D O p ] = EM2 :DOp], we see MI and M 2 are finite direct sums of the same length of (isomorphic) irreducible S-modules, proving there is a module isomorphism Q: M , -+ M 2 . In other words, o : L -+ L is a 1 : 1 map with a((a 0 d ) y ) = ($(a) @ d)a(y) for all u E A,, y E L, d E D . Taking u = 1, we see that a(yd) = u(y)d for all d in D,so u E End L, = R. Thus we can write D = r for some invertible r in R and get r-'ay = $(a)r-'y for all a E A , , y in L ; thus r - ' a - $ ( u ) r - * E Ann, L = 0, implying $ ( a )= r - l a r for all a E A , . QED The Skolem-Noether theorem is probably the most important tool in our study of division algebras, to follow; one application is a result that enables us to fill in a rough spot in Chapter 2. Theorem 3.1.3 (Wedderburn's theorem). Every Jinite simple ring has the form M,(F).ior a suitableJeld F, and suitable n.

53.1 .]

Fundamental Results

153

Proof. Since every simple PI-ring has the form M , ( D ) , it suffices to prove that every finite division ring D is a field ; i.e., letting F = Z(D), we claim that F = D.Otherwise, [ D : F ] = mz for some rn > 1. F is a finite field with, say, k elements. Let G = D - {0},a multiplicative group, and, taking a maximal subfield K of D,let H = K - {0}, a subgroup of G. The number of subgroups of G conjugate to H is at most lGl/lHl. But 1 E each subgroup of G , so by counting we see that some element x of G is not a conjugate of any element of H. On the other hand, clearly F [ x ] is contained in a maximal subfield K' of D.Now [ K ' : F ] = [ K : F ] = m, so K' and K each have mk elements and are thus isomorphic (since all finite fields of order ink are isomorphic). By the Skolem-Noether theorem, K ' - (0) and H are conjugate subgroups of G, which is absurd. QED

Corollary 3.1.4. Any prime PI-ring with jinite center has the form M , ( F )for a suitablefield F. Proof.

The center is a field ; use Corollary I .6.28. QED

Corollary 3.1.5. .for a suitablejield F.

Any prime ring of PI-class n is equivalent t o M , ( F )

Proof. We are done unless the center is infinite, in which case use Corollary 2.3.36. QED

Splitting Fields We shall now look at splitting rings with special properties. The first result, which is quite easy, completes the proof of the Artin-Procesi theorem (cf. Fact 1.8.45). Write 4 , ( Y ) for Qz($,( Y ) ) .By Proposition 2.4.11 4 , ( Y ) is simple.

Proposition 3.1.6. Suppose R is simple and deg(R) = n. Then there exists a basis of R [over Z(R)] of the form {yj-'xiI 1 < i , j < n} for suitable x,y in R . In fact, ifR = 4,(Y), we can take xi = and y = Y,+ 1 .

x

Proof. First consider the case R = M , ( C ) , where C is any commutative xi = eii and y = en,+ x ; : : e i , i + , , one sees that y - ' = y"-' and ring. Taking . . e 13. . = y J - ' x j ,so we are done. Let t = n z ; consider next $,,( Y ) . By Theorem 1.4.34,

czt- ~

~

~

l

~

~

x,,x,+ l , lX ~ n~ , .. .,+X",;:X,,X,+l,. . ., XZ,-l)

" 1x1 + 7.l. . 1 ~

is not an identity of M,(4[5]), and by Theorem 1.3.11 is not an identity of bn{Y}. Thus { Y:;/ rill < i , j < n} are Z(4,(Y))-independent. Finally, consider the general case. Let F = Z(R). If F is finite, then R is

154

CENTRAL SIMPLE ALGEBRAS

[Ch. 3

also finite, so bq Wedderburn’s theorem R 2 M , ( F ) and we are already done. If F is infinite, then by Theorem 2.4.9, R is equivalent to F , ( Y ) ; from the second paragraph, we conclude there are elements .Y I , . . .,x,, v , and rl+ . . ,r 2 r - in R such that C , ( . K ~ , J ~ 2-X Y II ,, .J..~,x,, JIX,,, . . .,V, n- 1.K”,rr + 1 , . . .,r2r- 1 ) # 0, implying {yj-’xil 1 < i,j 6 n ) are a basis of R over F . Q E D Actually, for the theory of central simple algebras, we need the splitting field to be a separable extension of the center. There are two pretty ways of obtaining such a splitting field. We give here the “classical” Albert-Noether-Jacobson approach; an alternate approach, due to Amitsur [77] and applicable to Azumaya algebras, is given in Exercises 9, 10. We start with a theorem of independent interest, the “double centralizer theorem;” although there are slick proofs using the density theorem (cf. Jacobson [75B]),I prefer an older proof given in Albert [61B] (also largely due to Jacobson!). If R is a ring, a Z(R)-Jieldmeans a field which is a Z ( R ) algebra (i.e.. contains Z ( R ) . ) Lemma 3.1.7. Suppose R = M , ( F ) , and K is an F-subfield of R . I f [ K : F ] = t , then C‘,(K) 2 M n I r ( K ) . Proof. Viewing R = End V, for some vector space V over F, and letting x , , . . . ,x, be an F-basis of K , we take a maximal set / I , , . . .,urn in I/ such that fx.u.11 J 6 i < r, 1 < j < rnl. is F-independent. Note that tor any u # 0 in V , (.xjuI 1 < i 6 t ) is F-independent. Let V’ = Kuj. If u E V then V’ P, K u # 0, so xu E V’ for some .K in K , implying u E V ’ ; thus V = V’, so 17 = tm. Moreover, we have identified V as a K-vector space of dimension m, and obviously C R ( K ) = End,- V 2 M , ( K ) . QED Proposition 3.1.8. Suppose R is simple of degree n, and K is a Z ( R ) subjield of R with [ K : Z ( R ) ]= t . Then C R ( K ) is simple of degree nlt. with center K . I

xy=,

Proof. Let F = Z ( R ) and T = R‘lPOrR z M,,(F). C , ( K ) is simple with center K by Lemma 3.1.7, so C , ( K ) = ( R o P @ F K ) @ K Bfor some simple B having center K.-Also [ B : K ] = [ C , . ( K ) : K ] / n 2= (n2/t)’/n2= ( n / t ) 2 ,so we need only show B = C , ( K ) . Identify R with 1 O R 5 T ; B is the commutator of R“@FK in C , ( K ) , so clearly C , ( K ) E B. On the other ) R, so B G R n C,(K) c CR(K). Hence B = CR(K),as hand, B G C , ( R D P L desired. QED

Theorem 3.1.9 (Double centralizer theorem). Suppose R-’is a simple PI-ring with simpk Z(R)-subalgebra A. Then (i) C,(A) is simple with ceriter Z ( A ) ;(ii) C,(C,(A)) = A ; (iii) [ R : Z ( R ) ]= [ A : Z ( R ) ][ C , ( A ) : Z ( R ) ] .

43. I .]

Fundamental Results

155

Proof. Let F = Z ( R ) ,K = Z ( A )2 F, T = C R ( K ) ,and t = [ K : F ] .Then T and C R ( A )c C R ( K ) = T , SO C R ( A )= C , ( A ) ; since T is simple with center K , we have T z A @ . C , ( A ) = A @ , C R ( A ) . Now (i) is obvious. Moreover, CR(CR(A))E C R ( K )= T , so CR(CR(A))= C , ( C , ( A ) ) = A , yielding (ii). Finally, (n/t)' = [ T :K ] = [ A : K ] [ C , ( A ) : K ] = LA : F ] [ C R ( A ) :F]/t', A

yielding (iii). QED We can now prove existence of a separable splitting subfield. (Noether-Jacobson). Suppose D is a division PITheorem 3.1.I 0 ring with center F . Any subfield K of D that is a separable extension of F is contained in some maximal subjield Of0 that also is separable ozier F. Proof. We may assume that no subfield of D properly containing K is separable over F ; we want to prove that K is a maximal subfield. Otherwise, by the double centralizer theorem C,(K) is a noncommutative division algebra whose center is K . Let D , = C,(K). If some element d E D , were separable over K then K ( d ) would be a field separable over K , and thus over F, contrary to assumption. Hence we may assume every element of D , is purely inseparable over K . Thus K has some characteristic p # 0, and for any d in D , , the minimal polynomial of d (over K ) is /1P'-tl for suitable tl in K , suitable i E Z+. In particular, we have some d in D , such that d $ K and d P E K . Define $ : D l + D , by $(x) = [ d , x ] ; inductively, put $' = $ and for m > 1, $" = Using a version of Leibniz's rule, one sees easily that I/P(x) = [ x , d P ] = 0 [since d P E Z ( D , ) ] Taking . d , not commuting with d , we have for some m < p , $"'(d,) # 0 and @'""'(d,) = 0. Write x 1 = $"-'(d1), xz = ~)(x,) # 0, and y = x,.u;'d. Then $(x2) = 0, so x;' commutes with d , and [ d , y ] = [ d , x , ] x ; ' d = a, implying dyd-' = y + 1. In other words, K ( y ) has a nontrivial automorphism (induced by conjugation by d ) and so K ( y ) cannot be purely inseparable over K , contrary to previous assumption. QED

I+V""$.

Corollary 3.1.I1 . Every division PI-ring has a maximal subjield which is a separable extension of the center.

Proof.

Take K

=

F in Theorem 3.1.10. QED

Corollary 3.1.12. Every simple PI-ring has a splitting subjield that is separable over the center.

The Index of a Simple PI-Ring Definition 3.1.I3. If R 2 M , ( D ) , where D is a division ring, then index@), the index o f R , is by definition deg(D).

156

[Ch. 3

CENTRAL SIMPLE ALGEBRAS

Remark 3.1.14. Index(R) is well defined; moreover, if [ R , ] Br(F), then index(R,) = index(R,).

=

[R,] E

Remark 3.1.16. If R is simple PI and if K is a Z(R)-field then index(R @ , , R , K )divides index(R)).[Indeed, if F = Z(R) and R = M,(D), then

ROFK

2

M,(F)OF(DOFK)

2

M,(F)@F(M,(K)@KD1)

(for some t , some division ring 0 , )2 M , , ( K ) @ k D 1 , so index(R@,K) = deg(D, ) = index(R)/t.) Definition 3.1.16. If R is simple PI and K is a Z(R)-field then the index reductionfal*tor of K (with respect to R ) is index(R)/index(R @ Z ( R , K ) . Theorem 3.1.17. (i) For any simple PI-ring R and anj- Z(R)-field K the index reducrioii.factor of K diuides [ K :Z(R)]. (ii) I f in (i) R = D, a division ring, [ K :Z(D)] = t, and k i5 the index reduction factor oj K, then K is isomorphic to a subfield ofM,,,(D). Proof. (i) First observe that it suffices to prove this when R is a division ring D. Let F = Z ( D ) , t = [ K : F ] , R , = D@FK, n = deg(D) = deg(R,), and let k be the index reduction factor of K ; we must prove k ( t . Clearly R, 2 M , ( K ) O K D , for some division ring D , [since n = deg(R,)], so R , = M k ( F ) O F ( K O R D , implying ), Mk(F) E R1.By the regular representation, K E M , ( F ) , implying R , c D@FM,(F); thus M , ( F ) is a central simple subalgebra of D@FM,(F), implying D m F M , ( F )z A O F M k ( F )for some simple F-algebra A . Obviously k J t . (ii) Continue the proof of (i). A z M,,k(D). But A is the centralizer [in D@FM,(F)] of kfk(F), and thus contains an isomorphic copy of D,. In particular, A has an isomorphic copy of K . QED

[Albert [61B, Theorem IV.221 squeezes out even more information, noting that D, t C,(K).J Here are some extremely important consequences. Corollary 3.1.18. I f D is a division ring of degree 11. and K is a splitting jield f . r D such that [K:Z(D)] = un, then K is isomorphic to a maximal commutative subring oj'M,(D). Proof. In Theorem 3.1.17(ii) take k = n and t = un and count dimensions. QED

Corollary 3.1.19. Suppose R is central simple and K is a Z(R)-jield. T h e index reduction factor of K divides the greatest common divisor of deg(R) and [ K :Z(R)] ; if' these are relatively prime, then index(R @ z ( R I K ) = index(R).

93.1.]

Fundamental Results

157

By definition, the index reduction factor divides index(R) and thus deg(R), so we are done by Theorem 3.1.17(i). QED Proof.

Corollary 3.1.20. The index of a central simple algebra divides the degree of each splitting field.

Using the existence of separable splitting fields, one can apply Galois theory decisively to the above results. Theorem 3.1.21. Let R be central simple of index m = pjq, where p is a prime number not dividing q. Theri there is u Z(R)-field K with pl[K:Z(R)] such that index(R @ z ( R ) K= ) pj. Proof. Let F = Z(R). By Noether-Jacobson, R has a splitting field L , separable over F , with [L:F] = m. Hence L is contained in a Galois extension L: of F (with [C:F]< m!).Write [ C : F ] = pkq’; clearly j 6 k and 414‘. Let G be the Galois group of L: over F . By Sylow’s theorem G has a subgroup H of order p k , and so the fixed subfield K of L: under H has the property [C:K] = p k . Thus [ K : F ] = q’, so by Theorem 3.1.17 the index , get reduction factor of K is nor divisible by p ; writing i = index(R O F K )we i = p’q” for some q” not divisible by p . On the other hand, L: splits R @ F K , so z([C:K] = pk, implying q” = I . QED

Albert [61B] uses Theorem 3.1.21 and Exercises 5,6 to reduce the further study of the Brauer group to the special case of cyclic algebras of prime degree (to be defined shortly). Nowadays it is customary to develop the Brauer group properties by means of crossed products and their connection with cohomology ; although this method is considerably more complicated than Albert’s, it is very beautiful, and so we shall turn now to a discussion of crossed products. We want to record a cancellation result for further use. Corollary 3.1.22. Suppose K is a field extension of F with [ K : F ] relatively prime to n. ij‘ [R], [R‘] E Br(F), deg(R) = n, arid R K : R’ O fK , then R z R’.

of

mf

Proof. (R”” OFR’) K :R ” ” @ f R O FK :M , , ? ( K ) ; by Theorem 3.1.17(i) the index reduction factor of K with respect t o RoPOFR‘divides both n2 and [ K : F ] , and thus is 1. Hence RoP@FR’% Mn2(F),proving R‘ 2 (RopyP= R. QED Crossed Products

In what follows, we adopt the convention that maximal subfield means a subfield that is a maximal commutative subring.

158

CENTRAL SIMPLE ALGEBRAS

[Ch. 3

Definition 3.1.23. A central simple algebra R is a crossed product if R has a maximal subfield K which is Galois over the center; R is an abelian crossed product (resp. R is cyclic) if K can be taken such that the Galois group is abelian (resp. cyclic). To emphasize K , we shall often call R a K-crossed product. Definition 3.1.24.

If K is an extension field of F ,

Br,(F) = { [ R ] E Br(F)IK is a splitting field for R ) . Remark 3.1.25. Br,(F) is well defined (i.e., independent of R ) , and is a subgroup of Br(F).

There is a profound connection between crossed products and Br,(F), which will be uncovered in the next few pages. Let us first clear a little sand. Proposition 3.1.26. If K is a Galois extension of F arid i s a splitting field of D. then [D] has a representative which is a K-crossed product. Consequently, every member of Br(F) has a representative which is a crossed product. Proof. The first assertion is a corollary of Corollary 3.1.18. The second assertion follows easily, in view of Corollary 3.1.1 1. QED

One of the features of K-crossed products is the description of the algebra in terms of field properties of K . Theorem 3.1.27. Suppose R is a K-crossed product, where G is the Galois group of K over Z ( R ) . Then there are invertible elements { r , ( o ~ Gof) R such that R = XnsGr,K. a direct sum of right K-vector spaces. where@ all x in K , xr, = r,a(x). Moreover, dejning x,,~ = r;'r,,rr for each G, 7 in G, we have xu.r E K, and the following equations hold for all a, 7 , p in G : X0.7pXT,p

Proof.

= -yUT,Vp(xrJ,7)'

(1)

By Skolem-Noether, we have invertible r, in R such that r, ' x r ,

= a ( x ) for all x in K , so xr, = r,,a(x). We claim that x:nPGr,Kis a direct

sum. Suppose CnsGroxa= 0 for various x , in K , not all 0, the expression taken such that the number of nonzero x , is minimal. Suppose x , # 0. For each x in K we have

o =x

c r,x, (c r,x,)7(.x) c r,.x,(u(.x)-r(x)). -

nEG

=

nsG

nsG

Now the coefficient of r, is X , ( T ( X ) - T ( X ) ) = 0, so by minimality of the length of the original sum we get .~,(~(x)-a(x))= 0 for all u, implying 7 = u when x , # 0. This is absurd, so the claim is proved. But now, counting dimensions, we get R = C n s C r a K .

$3.1. ]

Fundamental Results

159

Note that for any .Y in K , XX,.,

= .ur;'r,r,

= r;r'(ot)-'(.x)r,,rT = r;lr,,rr.Y,

implying x,,,~E C , ( K ) = K . Finally, r,,(rTr,,)= (r,,rr)r,, implies - rnr,,.uY"r.,,p(.u,,T), and (1 1 follows. QED

r,,7,,.Y,,,T,,.xr,,,

Defining P(a, 7 ) = x,, we have associated to the K-crossed product R the map p : G x G 4 K - ( 0 ) satisfying (1). We now obtain the converse. Definition 3.1.28. Suppose K is a field and G is a group of automorphisms on K . A ,facror set is a function 8 : G x G + K - ( 0 ) satisfying (1) of Theorem 3.1.27. I n general, e will denote the identity element sf G.

Note that K is Galois over the fixed subfield under G, with Galois group G, by Herstein [64B, Theorem 5.T]. Theorem 3.1.29. Suppose p : G x G + K - (0) is a.fbctor set. There is an algebra, denoted ( K ,G, p), constructed as ,follows: T a k e formal symbols [ r, I IJ E G ) and define r, K as a ,formal ( 1 -dimensional) right vector space over K , with basis r,; write xOtGr,,K as the direct sum of these vector spaces, and define ( K ,G , p ) t o be x,,tGrOK,with multiplication given by (x,,r,.xC)(xTrTx,) = r,,(xnr=,,p(u, t)t(x,).u,~forall.u,,.u,in K . (K,G,p)issimpleOfdegree)GI a n d Z ( K ,G,p)isthe~xedsubJieldof'Kunder G. Moreover,K isama.uimalsubJeld q / ( K , G, p), r,P(a, e)- = 1 .for all IJ in G, and s r , = rr?(.Y),forall t in G, .Y in K .

x,,FC

'

Proof. We rewrite the multiplication in ( K , G , p ) in the more intuitive form (x,,r,,x,,)(z,rr.ur) = x,,,rr,,rD(~, T)T(.Y,,).x~. In particular, rnrr = r,,,p(a, t) so, writing e for the identity element of G, we have rnre = r,,p(o,e); thus r,p(o, e ) - ' = 1. Also, ( I ) yields O(e, p)' = p (e, p)p(P(e, e)) (with IJ = t = e), so P(e, p ) = p(p(e, e)). Thus, putting IJ = e and x, = p(e, e)- '.Y, we get

xr,

=

( r e x e )( r , 1) = r,p(e,.).(.Y,)

=

r , ~ ( e , ? ) ? ( ~ ( e , e ) ) -= ' t rrr(.u). (~~)

Now (1) readily implies rC(rrrp)= (r,,r,)rp,and it is now clear ( K , G , B ) is a ring. Write R = ( K ,G, p). To prove simplicity, suppose l a R and 0 # r = C,r,,x,EI with the minimal number of nonzero x,, (in K ) . Suppose x, # 0. Then for all .Y in K , x r - r t ( x ) = x,,r,.u,(u(x)-?(x)), which has shorter length and thus must be 0. This is impossible if there is any nonzero x, for IJ # z. Hence r,.q = r E I , so 1 E I , proving R is simple. Now clearly Z ( R ) is the fixed field of K under G. Then [ R : Z ( R ) ] = IGI2, so deg(R) = IGI. Therefore K is a maximal subfield of R. QED Let us now determine when two factor sets give rise to the same

160

[Ch. 3

CENTRAL SIMPLE ALGEBRAS

crossed product. Suppose p,p': G x G .-+ K - ( 0 ) are factor sets and (K,G,B) z ( K , G , P ' ) . Taking R = (K,G,p') and letting +(K,G,p)-+R be the isomorphism, we have K z $(K); using Skolem-Noether we get an so, applying inner automorphism on R whose restriction to $(K) is I,-': this automorphism after $, we may assume $(x) = x for all x in K. Now suppose { r o 1 c r ~ G (resp.{r;loEG}) } are as in Theorem 3.1.27, with respect to B (resp. to p'). Then for all c in G, all x in K, +(ro)-'x+(ru) = $(r;'.xrO) = $(o(x)) = o(x), implying r;$(r,,-' E C , ( K ) = K, i.e., r; = +(r,,)x,, for suitable xu in K. Then B'(o,7) =

(+(roT)xor)-

'+(ro)xo$(rr)xr

=

II/(r,'rnrr)~nr'~(~o)x,

= p ( C , ?)XLIT(.Xo)Xr

for all o , in~ G. Conversely, given such an equation for each o,?,we can reverse the steps and build an isomorphism (K, G, p ) + (K, G. /Y), by r,Hrbx;' and s w x for all .X in K. This motivates the following definition and proposition. Definition 3.1.30. Two factor sets p,p': G x G -+ K -{O} are associated if there is some function ?: G + K - ( 0 ) such that, writing xu for y(o), we have ~ ' ( ( T , T )= fl(o,~)x;~'z(.~,)x, for all 0,'s in G . Proposition 3.1.31 . Two factor sets p,p': G x G + K - (0) are associated i$ ( K .G . p) = (K, G, p), in which case the isomorphism can be given as above. Proof.

Given before Definition 3.1.30. QED

We have thus identified all K-crossed products (having Galois group G), with the equivalence classes of associated factor sets 8: G x G -+ K - {O}. This new interpretation is very important and we shall make a brief diversion to illustrate its significance. Let G, H be groups, H abelian, written additively such that G acts as a group of distinct (group) autamarphisms on H. Let Cn(G,H) be the set of all functions f : 0") -+ H , [where G'O' is by definition the trivial group (e)] and define a coboundary 6": C"(G,H) + Cn+'(G,H) by

(W) ( g l ** . ., g n + 11 = f ( g 2 , ..' g n + 1) + (- 'Y+ + C (- l H ( g l , . . .,g i g i + i= 1 9

'gn+ 1

( f ( g 1 , . .* Y g n ) )

n

1,.

.

*

9

g n + 1 ).

Clearly 6"6"+' = 0 for all n, so we really have a coboundary; also note that C"(G,H) is an abelian group (under addition of functions). Let W ( G , H) = { S " - ' ~ ~ ~ E C " - ' ( C , and H ) } A!"(G,H)= ( f ~ C " ( G , H ) l d " f = 0}, both abelian subgroups, and let . V ( G ,H ) = 2 " ( G ,H ) / W ( G ,H ) .

Fundamental Results

43.1.1

161

Example 3.1.32. G is finite, K is a field, and H = K - ( 0 ) . The multiplicative group H is abelian, so we could have written H additively and applied the above definitions (e.g., writing 0 instead of 1, and + instead of .). Retranslating the definitions to the multiplicative notation, and writing a, T , p for typical elements of G, we have the following special cases:

.'A'(G, H ) = { f : G -,H l , f ( a )= a(x)-'x for some given X E H ,all a} 9 '(G, H ) = { . f :G + HI ~ ( G T=) r ( f ( o ) ) f ( for ~ ) all a, 71 d 2 ( G ,H ) = { f : G"' -+HJf(o,T ) = . Y ; ~ ' T ( ~ , ) x , for some function a -+ s,} 2 ' ( G , H )= { , f :Gt2' -, Hlf(a, t p ) J ( T , p ) = f ( o t , p)p(.f(o, ~))fora11fl,t,p}.

Thus Y '(G,H ) is precisely the set ofequivalence classes of factor sets @: G x G -+ K - ( 0 ) . Theorem 3.1.33.

With the notation as in Example 3.1.32, X1(G,H)

= 0.

Proof. Suppose f E I '(G, H ) . By Herstein [64B, Theorem 5.Q], there exists some x in H such that CgEc.f(o)a(x)# 0. Let a = CnEcf(a)a(x). Then ) )C a~ m(ExG ) f ( a r ) o t= ( xa) ; hence for all T in G , f ( t ) r ( a )= ~ . ( T E ( i f ( f ) t ( . f ( a = f ( t )= z ( a ) - ' a for all T in G,implyingfE.d'(G,H). QED This proof was quite slick, and so we would like to interpret the theorem in one very important case, where G is cyclic. Corollary 3.1.34 (Hilbert'sTheorem 90). SupposeG isacyclicgroup of automorphisms { e , a , a 2 ,. . . , a n - ' ; on a field K . For x in K , xo(x)...a"-'(x)= 1 rffx = a ( a ) - ' a f o r s o m e a E K .

Proof. (*) Defineinductively f ( e )= 1, f(a) = x,andf(a') = xo(x)... o'-'(x). Clearly ~ E Z ' ( G , H )so , IE # ' ( G , H ) , Le., for some U E K ,a ( a ) - ' a = .f (a) = x. The converse

is obvious. QED

Having disposed of H',we return to N 2 , with notation as in Example 3.1.32. There is a set correspondence between X 2 ( G ,H ) and isomorphism classes of crossed products. But H '(G, H ) is an abelian group, so somehow this structure should pass naturally to crossed products. We bring in the following operation: Given factor sets p1,p2:G x G K -{O) we take PI . p 2 to be elementwise multiplication of functions. -+

Theorem 3.1.35. Suppose K is a Galois extension O f F , ofdimension n, with Galois group G. For an)' factor sets B1,b2:G x G K -{O}, we haw ( K , G, P I ) O F ( K , G, P2) 2 M n ( F ) O F ( K G, PI . P 2 ) ; i.e., [ K G, P,][K G, DZI = [ K G, P I . B 2 1 in Br(F). -+

162

[Ch. 3

CENTRAL S1MPL.E ALGEBRAS

Proof. First note that if r is an idempotent of R , with [ R ] E Br(F), then rRr is also central simple, and (identifying Fr with F) we have [rRr] = [ R ] . [Indeed, write R = M , ( D ) = End V,, for some division ring D and some vector space 1’. “Changing the basis” of V to a basis built up from the subspace r V , we may assume that r = ,eii for some u < 1. Then rRr 2 M,(D), so [rRr] = [ R ] . ] So, letting R = ( K , G , I J , ) @ , ( K , G , ~ , ) , we shall conclude by finding an independent r of KBFK such that [rRr]

xy=

=

[KG,Pl+P,].

By Wedderburn’s theorem, there is nothing to prove if F is finite, so assume F is infinite. Since K is Galois over F, by the primitive root theorem (cf. Herstein [64B, theorem 5 . P ] ) we can write K = F ( x ) for some x in K. Let p = (1-cr(x)), the minimal polynomial of x over F, and define r= ((?I0 1 - 1 @ a(.u))/(x -a(x)) @ 1). (This makes sense since K 0K is commutative.) Clearly .u - a(x) # 0 and (x 01 - 1 0a(x)) # 0 in K @ K (seen through a dimension count) for all cr # e in G, so r # 0. Also, since the @ 1 - 1 @a(x)) coefficients of p are in F z 1 @ F 2 F @ 1, we have = p ( x @ l ) = p ( x ) @ l = O ; thus ( . r @ l - l @ x ) r = O , implying ( x O l ) r = (1 @ x)r. Inductively, we get for all i, ( x i 0 I)r = (1 @ xL)r. implying (a@ 1)r = ( 1 @ u ) r for all U E K. This was the crucial step in the proof, and we shall use this equality often. I n particular,

nntG

no,,

nOEG(.x

n r n ((.u

r2 = r

1-I

((x

crx)/((x-ax)

1))

O#P

=

@ 1 -ax 0 l)/((x-ax) @ 1)) = I

.

n?e

Hence r is idempotent. Write K‘ = (K @ I)r = (1 @ K ) r z K. Taking rim in (K, G,Si), i = 1,2, such that rG1uria = a(u) for all a in K. we have ( K , G , S i ) = XatGrioK (from Theorem 3.1.27) and rRr = L,rEC r(K 0 1)(1 0K M 1 , , 0rlr)r =

K’r(rl,, 0 rZr)r.

Now r(rln O r2r)r

((x o 1 - 1

= I’

o

P ( x ) ) / ( ~ - P ( x ) ) 1 ) (rln O r2r)r

f (’

= (rln@TZr)

n n

((o(.x)@ 1- 1 @ p s ( x ) ) / ( a ( x ) - p a ( . x ) ) O1)r

P?e

= (r1,@r2r)

((a(.u)-pdx))@ l / ( a ( x ) - ~ ( u ) ) @ I)r.

p f e

If cr # 7 then taking p = cr7-l gives a(x)-p7(x) = 0, so r ( r l n O r l r ) r= 0. When a = 7 then the big product is 1, so r ( r 1 0 @ r 2 0 ) = r (rln@r2,,)r. Likewise r ( r l a @ r Z s ) r= r ( r l a @ r l o ) .

93.1.]

Fundamental Results

163

Now we have rRr = x o s G K ’ r ( r l a @r2a)r, where each r ( r l a @ r 2 a ) r is invertible in rRr (whose multiplicative unit is r ) ; we conclude by observing that conjugation by the r ( r l n@ rZn)rinduces G on K ’ and, for p = UT, W I , ,

0 r2,,)4- l r ( r l n 0r2n)r(rlr 0r2rb @ ryt,lr2nr2r)r= r(81(o,7)0Bz(a,T))r = ( B l ( a , ~ ) @ 8 2 ( c ,= ~ )()Br1 ( a , ~ ) . B 2 ( o , l)r ~)@ = (B1. B 2 ( 0 ,z) 0 1 ) r , = r(r;,;rtnrir

the canonical image in K ‘ of p1. P2(a,5 ) . QED What we have really done is find a map $: W 2(G. K

-

( 0 ) )-, Br,(F)

= [ ( K ,G, /?)Iwhen K is a Galois extension of F with Galois group G. by $(/I) We are ready for a beautiful result.

Theorem 3.1.36. Suppose K is a Galois extension of F with Galois group G. There is an isomorphism $: . H 2 ( G ,K - ( 0 ) )-,Br,(F) given by $(b) = [ ( K , G, P)]. Proof We have proved so far that ) I is a 1 : l correspondence of sets (“onto” follows from Proposition 3.1.26) that preserves the multiplicative structure, i.e., is a group homomorphism, so $ is an isomorphism. QED

For further study of the Brauer group, the question arises whether or not every division PI-ring is a crossed product. If so, we could apply the theory directly to central simple algebras by looking of the cohomology groups .P at the underlying division rings (cf. Exercise 7, for example). However, the answer is “NO,” and the crowning achievement so far of PI-theory is that the only known way to obtain an example of a noncrossed product division ring is by means of the PI-theory [Q,(Y) is actually the example, for suitable PI.]We shall return to this question in $3.3.

Exponents of Central Simple Algebras

We shall now use the crossed product theory to obtain very important information concerning the Brauer group. Definition 3.1.37. The exponent of a simple PI-algebra R [written exp(R)] is the smallest m such that [R]“’= 1 [in Br(Z(R))].

The exponent clearly is well defined. To obtain more information we shall now use the regular representation, recalling that M , ( K ) 2 M,(K)”Pnaturally.

164

CENTRAL SIMPLE ALGEBRAS

Theorem 3.1.38.

[Ch. 3

For every simple PI-algebra, exp(R)lindex(R).

Proof. Let F = Z ( R ) . Clearly we may replace R by any other representative of [ R ] E Br(F), so we may assume R is a K-crossed product (K, G,B) for some Galois extension K of F. Let n = deg(R) = [ K : F ] , m = index(R), and t = n/m, so that R z M , ( D )for some division ring D. Let B be a minimal right ideal of R. Then tm2 = [ B : D ] [ D : F ] = [ B : K ] [K : F ] , implying. [ B : K ] = m. We use the regular representation to view R E End B , z M , ( K ) . In particular, taking r,, in R such that for all .K in K r , ‘ x r , = a(.u), and r,r, = r,,p(a, T ) for all a, T in G, we let (xi;)) denote the element of M , ( K ) corresponding to r,; then, for all b in B, we have

b(xlp”)B(o,T ) = br,,p(a, T ) = br,,r, = (b(.$)))r, = br,(T(x:T)))= b(x$’)(s(x$))).

Hence (xiT))p(a,T ) = (X$’)(T(X:;))). Taking determinants, letting x, = det(xI;)), yieldsx,,B(o,T)”’= x,r(x,).ByProposition 3.1.31 andTheorem 3.1.35,we have [ ( K , G, p)]” = 1, as desired. QED We can get even more precise information about the exponent. Theorem 3.1.39. Suppose R is central simple, and p is a prime number. Zfp)index(R) rhm pJexp(R). Proof.

Let K be as in Theorem 3.1.21. Obviously

exP(R @z(R)K)I~xP(R) and

plexp(R @Z(R)K). QED

Theorem 3.1.40. Zf D is a division PI-ring with center F and deg(D) primes p i , and for m i E Z f , then D 2 D , @ F . . . @ F D kfor suitable division riiigs Di, with deg(Di) = p?, and Z ( D i )= F . = n7=,pTi .for

Proof. Let it = deg(D). Since [D]“ = 1 we can find RiEBr(F) such that [ D ] = [ R I ] . . . [ R k ]and exp(Ri)lpri.But writing Ri = MJDi) for suitable division rings Di,we have, for some u

Mu(D) 2 . ‘ M , , ( D l ) @ p . . . OM t k ( D k )

2

Mr(D1 @ F . . ‘ @ F D k ) ,

wheret = t , t2.*.fk. By Remark 3.1.l(iv),Dl@ ... 0D, isadivisionringwhich must then be isomorphic to D ; by Theorem 3.1.39 deg(Di)is a power of pi, so we conclude by the unique factorization of n. Cyclic Algebras

Cyclic algebras are of considerable interest for the following three reasons: (i) If R IS a K-crossed product and n = deg(R) is prime, then obviously Gal(K/L(R)) is cyclic (i.e., R is K-cyclic); (ii) by the famous

43.1 .]

Fundamental Results

165

AIbert-Brauer-Hasse-Noether theorem (cf. Albert [6 IB]), each division PIring whose center is an algebraic number field is cyclic; (iii) the structure of cyclic algebras is very clear, as we now shall see.

Proposition 3.1.41. Suppose R is K-cyclic and cr is a generator ofthe Galois group of K over Z ( R ) . Then there is some element r in R such that r " E Z ( R ) and r - l x r = cr(x)for all x in K . Proof. By Skolem-Noether, there is some r in R with r - l s r = cr(x) for all .Y in K . Hence r" E C , ( K ) = K . But then r" = r-lr'r = a(r"),SO r" is in the fixed subfield of K under cr, which is Z(R). QED

Corollary 3.1.42. I f R is K-cjdic, then there is some r in R such that for n = deg(R), r" E Z(R) and r'Q. Z(R),forall i < 11.

Proposition 3.1.41 says that for K-cyclic algebras we can take the factor set 1:G x G + K as follows, where C = (0'10 < i d n- l}:P(cr',crj) = 1 for i+j < n, and P(d, d)= r" for i + j 3 I?, where r is in Proposition 3.1.41. Since r " E Z ( R ) , we can rewrite a typical K-cyclic algebra (with center F ) in the form (K, O,a ) where o is the generator of Gal(K/F) and a = r" E F . Conversely, given a Galois extension K over F with cyclic Galois group G = (0'10 d i d n-l}: and given a in F , form P : G x G + K by P(cr',aj) = 1 for i+j < n and P(d, a ' ) = a for i + j 3 17 ; the ensuing crossed product algebra ( K , G, b ) is obviously of the form ( K ,cr, a). Thus cyclic K-algebras are characterized by the triples (K, cr, a), where cr # e is an automorphism of K and c r ~ Kwith cr(a) = a. Write N , ( x ) for the norm x c r ( x ) ~ ~ ~ c r " - ' ( x ) of an element x in K , and write N , ( K ) = {N,(s)I.x E K } .

Proposition 3.1.43. Let K be a cyclic extension of F , with Galois group generated by 0, and let a l , a2E F . (i) [ ( K , o , a , ) ] [ ( K , a , a 2 ) ]= [ ( K , a , a l a 2 ) lin WF). (ii) ( K , 0,C L ~=) M,(F) E N,(K). Proof. (i) Obvious from Theorem 3.1.35.

(ii) First note, by (i), that [M,(F)] = [(K,cr,l")] = [(K,a, I)], so M , ( F ) z (K,cr,l). Thus there is some ro in M J F ) such that M J F ) = Cy=]rbK, with r: = 1 and r i ' x r o = a ( x ) for all x in K . Now refer to the discussion before Definition 3.1.30. If M,(F) = ( K ,cr, a,) then, taking r I such that rl = a, and rL1xr1 = cr(x)for all .Y in K , we have rI = roxu for some xu in K , so a1 = rl = (rox,)" = 1 . N"(.Y,,). Conversely, if N , ( x ) = a1 then write r , = r o x ; rl = N,(x) = a1 and we can write M,(F) = ri K .t ( K , cr, ul). QED

xy=

Corollary 3.1.44 (Wedderburn). I f a E F anda'4 N,(K),for all0 < i < n, then ( K ,cr, a ) is a division ring qfesponent n.

166

CENTRAL SIMPLE ALGEBRAS

Corollary 3.1 -45. ring ifla $ N , ( K ) .

Suppose n is prime and

E E F.

[Ch. 3

(K,a, a ) is a division

Clear from Proposition 3.1.43, because ( K , a , a ) is not a division ring iff ( K ,a,a ) 2 .M,(F). QED Proof.

These results still give some of the most effective ways to construct division rings, which we now exemplify. Proposition 3.1.46. For any Jield 4 and any n, there i s a cyclic dioisiorz algebru of degree and exponent n, whose center is a suhfieltl of‘ &A11 . . ., A,,, ).

,

Proof. Let C = b[A,,.. ., A n + J, the licommuting indeterminates, and let K = C$(A,,...,A~+~),the field of fractions of C. Then K has an a&) = A3,. . .,a(&) = 1,; letting F automorphism 0 such that a(Al)= i2, be the fixed subfield of K under a,we have G = Gal(K/F) is cyclic of order n, generated by a. Moreover, F 2 4(An+,). Let D = (K,o,E.,, +,). We shall conclude, using Corollary 3.1.44, by showing for all 0 c i < n that A t + , $ N , ( K ) . Indeed, suppose on the contrary c , , c , ~ C with A:+, = N,(c,c; 1. Then A;+ N,(c,) = N n ( c 2 ) .But, modulo n, the total degree of the left side (resp. right side) is i (resp. 0), which is absurd since i f 0 (modulo n). QED

’

Corollary 3.1.47. Suppose D i s an arbitrary division riny ojdeyree m. Then for any numher n relatively prime to m, there is an exterision D‘ of D that is a division riiig of degree mn. Proof. Let R = DIAl,...,An+,] and let D, be the ring of central quotients of R . Since R is a domain of PI-degree m, clearly D , is a division ring of degree nt; letting F = Z(D) and F , = F ( A , , . . . , I . ” . + , ) , we see D , :D 0,; F , . Now using Proposition 3.1.46, we build a cyclic division F-algebra D , of degree n and center a subfield K of F , . Then D OFK is a division ring, so D @ , D , z ( D O F K ) O K Dis2 a division ring of degree mn, by Remark 3.1.l(ic). QED

The Reduced Trace

We now look at a classical notion, the “reduced trace” of a central simple algebra. The idea is quite easy. Suppose R is simple of degree n, and F = Z ( R ) . Take a splitting field K of R . We define tr(r), the reduced trace of an element r in R, to be its trace in R O F K = M , ( K ) . There are two properties we want to obtain-that trfr) is independent of the choice of the . . , X , , 1, X,, . .. , X , , ) . field K , and tr(r)EF. Put h = C2n-1(Xy-1..

$3.1. ]

Fundamental Results

167

Remark 3.1.48. If h(r, X , , . . . , X , ) is not a GI of R, then [ F ( r ) : F ]= n, and the minimal polynomial of r (over F ) is the characteristic polynomial of r in M , ( K ) . [Indeed, let p ( A ) be the minimal polynomial of r (over K ) in M , ( K ) . Clearly h(r, X , , . . . ,X , ) is not a GI of M , ( K ) , implying deg(p(A)) 3 n, so p ( A ) is the characteristic polynomial of r. Obviously p ( A ) divides the minimal polynomial of r over F, since K [ I ] is a principal ideal domain; by Corollary 1.5.19 we conclude p ( I ) is the minimal polynomial of r over F.] Theorem 3.1.49. W i t h the notation as preceding Remark 3.1.48, tr(r) E F and is well dejined for euery r in R. Proof. If F is finite then R z M , ( F ) by Wedderburn's theorem, and we are done. Thus we assume F is infinite. Since deg(R) = n, obviously by Proposition 3.1.6, h is not an identity of R. Thus h(x, X,,. . .,X,) is not a GI of R for some x in R . By the Vandermonde argument, we conclude h(x ar, X,, . . . ,X,) is not a GI of R for all but a finite number of a in F. Hence, by Remark 3.1.48, for each of these a, tr(x+crr)E F and is well defined. But for a1 # a,, tr(r) = (a1-a,)-'(tr(x+a,r)-ttr(x+a,r)). QED Thereisa pretty,"generic" wayofviewingTheorem 3.1.49;cf.Corollary A.15 in Appendix A.

+

Involutions of Central Simple Algebras Write Unit(R) for the set of invertible elements of R, a multiplicative group. There is a group homomorphism [: Unit(R) + Group of inner automorphisms of R, where [ ( r ) is the inner automorphism x -+ r - ' x r . W e shall refer to this [frequently. Note that ker([) = Z(R) n Unit(R). Our point of view actually will be to consider the group of all automorphisms and antiautomorphisms of a ring. The involutions each have order 2 (whence the name). Write Invol(R) for {involutions of R of the first kind}. Remark 3.1.50. If (*) is an antiautomorphism of R and r e Unit(R), then (r-l.Kr)* = r*x*(r-')* = r*.Y*(r*)-'; thus i ( r ) * = * [ ( ( r * ) - ' ) . Definition 3.1.51. Suppose (R, * ) is a ring with involution; r is (*)nice if r* = fr (i.e., if r is (*)-symmetric or (*)-antisymmetric). Lemma 3.1.52. Suppose R is prime. I f Invol(R) and r in Unit(R), then r* = r' = f r .

* = J [ ( r ) fbr

(*), ( J ) in

Proof. J = J** = JJ[(r)* = [(r)* = * [ ( ( r * ) - ' ) = J c ( r ) [ ( ( r * ) - ' ) , implying r(r*)-' ~ k e r ( [ )E Z(R), so r* = :r for some Z E Z ( R ) . Hence r = r**

168

CENTRAL SIMPLE ALGEBRAS

[Ch. 3

= (:r)* = ?r, so ( z 2- I )r = 0, implying 0 = z2 - 1 = ( 2 - I ) ( = + 1). Since Z(R) is a domain, we conclude := 1. Thus r is (*)-nice, and obviously r* = r’. QED

We are ready for a major step in the characterization of the involutions of central simple algebras with involution of the first kind. Theorem 3.1.53 (Albert). Zf’ R is a simple PI-algebru, with (*), (J)EInvol(R), then * = J ( ( r ) for some (J)-nicer in Unit(R). Coiziiersrlj,, if J ElnvoljR) und r e UnitfR) is (J)-nice.then J [ ( r ) E Invol(R). Proof. J * is a Z(R)-algebra automorphism of R so, by SkolemNoether, J* = i ( r ) for some r in Unit (R). Applying J on the left yields * = J [ ( r ) , and r is (J)-nice by Lemma 3.1.52. The converse is obvious. QED (*), (J) in Invol(R) are equivalent if Definition 3.1.54. some (*)-symmetric r in Unit(R).

* = J [ ( r ) for

To examine Definition 3.1.54, we make a little computation. Remark 3.1.55. Suppose (*), ( J ) ~ I n v o l ( R and ) * = J<(rl). Also, suppose ~ , E Rand r;l = p i r i for p i = & 1, i = 1,2. Then ( r 2 r l ) *= r ~ r ~ ’ ~ r , = 011P2) (r2r1).

r.x

Remark 3.1.56. Suppose r, x E R with x regular, and r x E Z(R). Then = xr. [Indeed, x ( r x ) = rx‘, implying [x, r ] x = 0, so [x. r] = 0.1

Theorem 3.1.57. ( i ) Dejnition 3.1.54 is indeed un equioulence relation. (ii) I f R is simple PI then Invol(R) has at most two equivalence classes. (iii) Jf’(*), (J)EInvol(R)and * = J ( ( r ) f o r some (*)-antisymmetric r in Unit@), then (*) a d ( J ) are not equiualent. Proof. (i) Reflexivity is obvious, symmetry is obvious [if * = J i ( r ) then J = * [ ( r - ’ ) ] ,and transitivity is immediate from Remark 3.1.55. (ii) Suppose (.I1), (J,), and ( J 3 ) are in Invol(R). We claim that two of them are equivalent. This is obvious unless J , = J 2 [ ( r l ) and J , = J 3 ( ( r 2 ) for suitable (Ji)-antisymmetric ri in R . But then J , = J 3 [ ( r 2 r 1 ) , and (r2rl) is (J,)-symmetric by Remark 3.1.55, so ( J , ) and ( J 3 ) are equivalent. (iii) Suppose * = J [ ( r i ) for ri in Unit(R), i = 1,2, with rf = - I , and r z = r 2 . Then r 1 r y 1E ker([) c Z(R), and ( r l r y l ) * = - r I r y 1 (by Remark 3.1.55), which is impossible. Q E D

To gain further information we turn to M,(F), which we recall has the transpose involution (t), as well as the canonical symplectic involution (s) when n is even. Corollary 3.1.58.

When n is even and F has characreristic # 2, (t) und

43.1.]

Fundamental Results

169

(s) are inequivalent involutions of' M,(F). When n is odd, or when F has characteristic 2, Invol(M,(F)) has only one equivalence class.

Proof. When n is even, we are done by Theorem 3.1.57(iii). When char(F) = 2, all (*)-nice matrices are (t)-symmetric, so there is only one equivalence class. When n is odd, Unit(M,(F)) has no (t)-antisymmetric elements, so again, by Theorem 3.1.57, every involution of first kind is equivalent to (t). QED Definition 3.1.59. Let R = M,(F). The equivalence class of (t) is called the class of orthogonal-type involutions; the equivalence class of (s) (when n is even and F has characteristic # 2 ) is called the class of symplect ic-type involutions.

Thus, every involution of the first kind of M,(F) has either orthogonal type or symplectic type. We shall soon gain further information in terms of polynomial identities. The key fact needed is an easy result from linear algebra. which unfbrtunately is overlooked in many courses in linear algebra. We shall use the well-known decomposition of an F-vector space V into "characteristic subspaces" 6 with respect to a linear transformation (cf. Herstein [64B, corollary to theorem 6.N] ; i.e., if all the characteristic roots a l , . . . ,ak of a linear transformation r lie in F, we put = {,vE VI v(r-ai)"' = 0 for some m } , 1 d i 6 k , and note that V = V, @ . . . @ V,. Moreover, since [ 6 :F ] < co,we have for some mi,K(r - ai)""= 0. Proposition 3.1.60. Suppose ) E F. I f r E M,(F) is nonsingular, and if F contains square roots of all the characteristic roots a l , .. . ,ak of r, then for some p ( I ) in F[A] we have r = p(r)'.

v

Proof. Let V = F("). a right M,(F)-module. and let be the characteristic subspaces of V with respect to r, 1 < i ,< k . Since (A - a , )"'I, . . . , (A - ak)"'"are relatively prime, one sees easily by means of the Chinese remainder theorem (Lemma 1.7.15) that it suffices to find polynomials p i @ ) in F [ I ] such that v ( r - p i ( r ) ' ) = 0, 1 < i d k . In other words, restricting the action of r to &, we may replace V by 8, and assume that for some a in F and for some m, V ( r - a)"' = 0, so that (r-a)" = 0. Write y = r - a . Then the formal binomial expansion of (a+y)"2 is a l / z ++a- iizY - a - 312 y 2 /8 .... Since y"' = 0 this expression has only (m + I ) terms ; rewriting y = r - a and writing 1 in place of r gives us the desired p i @ ) in F[A]. QED Theorem 3.1.61. Suppose F is a$eld with 9, and (*)~Invol(M,(F)) has orthogonal (resp. symplectic) type. Then in some extension field K of F, extending {*) naturally to M J K ) = M,(F)@.K, we have

170

[Ch. 3

CENTRAL SIMPLE ALGEBRAS

( M , ( K ) ,*) .t (M,IK), t ) (resp. ( M , ( K ) ,*) z ( M , ( K ) , s ) ) .In particular, infinite, .Y(M,(F), *) = .Y(M,(F), t ) (resp. 9 ( M , ( F ) ,*) = f ( M , ( F ) , s ) ) .

if F is

Proof. Suppose (*) has orthogonal type. Then * = ti(r) for some r in Unit(R) with r' = r* = r. Let K be an extension field of F containing the square roots of all characteristic roots of r. By Proposition 3.1.60 there is some ro in M,( K I with rb = r t = ro and rg = r. We claim x -+ r; lxr0 gives the desired isomorphism ( M , ( K ) ,t) -+ ( M , ( K ) ,*). Indeed, .Y' Hr i ' x L r 0 = r i l r x * r - ' r o = ( r i l x r o ) * , as desired. The second assertion is immediate from Corollary 2.3.32. QED

This is very good, because in 92.5 we determined precise information about (M,(F), t ) and (M,(F), s). Unfortunately, this argument fails in characteristic 2, because there is no way even of taking the square root of the matrix e , , f u , , in M,(F). In fact, Theorem 3.1.61 itself fails, because clearly 4 ( M , ( F ) , t ) # .Y(M,(F),s) even in characteristic 2. In this case, we need a more sophisticated analysis of involutions, in terms of bilinear forms. Rather than digress so far to make this study now, we request the reader interested in characteristic 2 to turn to a more general theory (of primitive rings with involution) developed in $7.3, and note that Theorem 7.3.20 (with Example 7.3.14) implies the following result (independently of Theorem 3.1.61). Theorem 3.1.62. For any infinite field F , and any (*) E Invol(M,(F)), either j ( M , ( F ) , *) = Y ( M , ( F ) ,t) or .f(M,(F), *) = % f ( M , ( F ) , s ) . Corollary 3.1.63. If R is prime of PI-class n and (*) E Invol(R), then either R is u mutrrx algebra over a finite field, or .f(R,*) = .Y(M,(Z(R)),t), or $(R,*) = .Y(M,(Z(R)),s).

If 2 is finite, then we are done by Corollary 3.1.4; if Z then pass to Q,(R, *) and split it. QED Proof.

IS

infinite,

Corollary 3.1.64. Suppose R is an infinite, prime PI-ring. Then there are at most three possible distinct T-ideals 9 ( R , *) for the various possible involutions (*) of R .

Either (*) is of the second kind, and thus special, or (*) is of the first kind, and Corollary 3.1.63 applies. QED Proof.

Let us use these results to acquire a general result about (*)-PI rings Theorem 3.1.65 (Amitsur). Suppose R is a PI-ring, with involution (*), such that ( R ,* I satisjes a polynomial identity of degree d. T h e n (S,,)k is an identity of R for some k ; moreover, ifR is semiprime, we can take k = 1. Proof.

By Amitsur's method, we may assume (R,*) (and thus R ) is

43.1 .]

Fundamental Results

171

semiprime. We appeal to results of $2.2. By a subdirect decomposition, we may assume ( R , *) is prime. If R is not prime then every multilinear identity of ( R , * ) is special, so we are done by Amitsur-Levitzki (because S,[,,,, is even an identity of R ) . If R is prime then, noting that ( R , * ) is multiequivalent to (R[A],*), we may assume Z ( R ) is infinite. We conclude with Corollary 3.1.63 and Remark 2.5.14. QED (Note that without Theorem 7.3.20 we would be forced to require $ E R in the above results). There is a famous related question, due t o Herstein (actually in a more special form): "If ( R , * ) satisfies a polynomial identity, does R satisfy a PI?'So far in this book, we have not touched this question, answered affirmatively by Amitsur [69] after positive partial results were obtained by Herstein [67] and Martindale [69a] ; the proof is given in 47.4, as a consequence of a theorem o n generalized identities.

Involutions and Maximal Subfields Involutions can be studied in terms of crossed products; to do this, we relate them to maximal subfields. Proposition 3.1.66. Suppose D is a division PI-ring with ( * ) E Invol(D), and K is a Z(D)-subfield of D. I f K has an automorphism D over Z ( D ) of degree d 2, then the map D*:K + (cr(K))* can be lijed to an inner automorphism of D with respect to a (*)-nice element. Moreover, ifcr # 1 or if K is nonmaximal, then this element caii be taken at our whim to be (*)-symmetric (resp. (*)-antisymmetric).

Proof. Let K , = (a(K))*. Since K is commutative, cr*: K + K , is an isomorphism, which can thus be extended to some [(d). Then d*d-'x(d*d-')-' = d*o(x)*(d*)-' = (d-'olx)d)* = (o(cr(x))*)* = x for all .Y in K so, writing A = C , ( K ) , we have d*d-' E A , implying d* E Ad. Thus d* +d E Ad and d* - d E Ad. Since every nonzero element of Ad could be used in place of d , we have proved the first assertion, and are finished unless every element of Ad is antisymmetric (resp. symmetric); i.e., for fixed p in { - l , l ) , we would have (ad)* = pad for all a E A. Since 1 E A, we would have d* = pd, so pad = (ad)* = d*a* = pda*, implying &'ad = a* for all U E A. Hence, * : A+ A* would be an isomorphism, implying A were commutative. This is false if K is nonmaximal. Even if K is maximal, then x* = d-'xd = cr(x)*, for all x in K , implying cr = 1, so we have arrived at a contradiction when cr # 1. QED Proposition 3.1.67.

Suppose D is a dicision PI-ring with involution of the.first kind and K is a Z(D)-subfield of D having an automorphism D over

172

[Ch. 3

C E N T R A L SIMPLE A L G E B R A S

Z(D) of degree d 2. Then there is some (*) in Invol(D) whose restriction to K is cr. Moreorer. if cr # 1 or if'K is nonma.xima1, we can take ( * ) to be whicheuer tj'pe wc* choose.

Proof. Let J ~ I n v o l ( D )Then . by Proposition 3.1.66 there is a J-nice element d such that J - '.ud = .(.u)-' for all x in K . Define * = J ( ( d - ' 1 as in Theorem 3.1.53. For all x in K . x* = dx-'J-' = d(a(a(s)))-'d-' = ~ ( x ) , proving the first assertion. If cr # 1 or if K is nonmaximal then we can choose J to be J-symmetric or J-antisymmetric. Choosing properly makes (*) whichever type we choose. QED Theorem 3.1.68 (Albert). Suppose R = ( K , G , f i ) , F = Z(R) and K has an automorphism p of degree 2 that commutes with all automorphisms in G. Write K O (resp. Fo).for thefixed subfield of K (resp. F ) under p ; for all O,T in G, dejne P(o,T) = fi(o,T ) ~ ( / ?7( o) , )K~O . Then p e.urends to un inwlution ( * ) of R i f [ ( K n , G , fl)] = [I] in Br(F,).

Proof. Note that G restricts to a group of automorphisms of K , , over Fo. Take r, in R as in Theorem 3.1.29. ( 3 )By Remark 3.1.50, for each cr in G [(r,*)*= * L ( ( r , ) . - ' ) ; since (*) commutes with cr on K , we see that <(r,*)and [ ( ( r , , ) - * ) act the same on K , so r,*r,EC,(K) = K . Thus r,*r,EKo. Let y, = r;r,,. By Proposition 3.1.31 we only need show is associated to 1, which happens because

/ ? ( ~ , 7 ) 8 ( ~=,ror ~- )1rorrrrro(r~i)F1 * * * * = rG1r,ri(yrrr-')(yvri l ) r u r ~ i r ' r G ; ' r u T - l ( y r ) y , , lruTyill r~ = (~-'cr-lcr~(~,~))(cr-'cr~(y~))y~' = Y-T(Y,)Y,'.

=

=

(-=) By Proposition 3.1.31, there are y , in K O such that J?,T(J,,)Y;' fl(a, T ) for all u. T in G . We make r,* = y,ril by defining (*) by the rule

(XnGGr,x,)* = x,TEGp(xn)ynr;' for all x , in K . By Theorem 3.1.29 2 --I re = B k e ) = Y,Y, p(P(e,e ) ) F 1= yep@(e,e))F1, implying y,r, = p(8(e,e ) ) = p(rJ Thus the restriction of (*) to K is p, for (r,x)* = p(x)p(r,) = p ( r p x ) . Now (*) is an antiautomorphism because ((rmx,) (rrxl))*= (rnrP(cr17)7(x,)x,)* = B(a,T)*T(x,)*x,*.v,,~;' =

61(,'

T)4.61T(Xu)*X:(r,,p(a,7))- = y,T(y,)s(x,)*x:r; I ) (y,x,*r;') = (r,x,)*(r.x,)*.

= (v,x:r;

Hence ( r - ')*

=

( r * ) - ' for all r in Unit(R). so

'

(r,x,)** = (p(x,)y,r; )* = ((r,p(x,)- ' y ; =( ~ ; ' y ; ~ y ~ ; '= ) -r,x, ~

'I*)-

for all x, in K.Thus (*) has degree 2, so is an involution. QED

'r,

'

43.1.]

Fundamental Results

Theorem 3.1.69

R

=

173

(Albert). The,fdlowing assertions are equivalent 011

(K,G,p):

(i) (ii) identity (iii)

R has an involution of the,first kind; R has an involution of the first kind, whose restriction to K is the automorphism ; [R]' = 1.

Proof. ( i ) o (ii) by Proposition 3.1.67. (ii) e (iii) by Theorem 3.1.68, with p taken to be the identity automorphism. Q E D Characterization of Involutions of the First Kind

To utilize Theorem 3.1.69 we want to know that "having an involution" is an invariant of the Brauer group. Proposition 3.1.70. Suppose D is u division ring, and F = Z ( D ) has an automorphism p . For m , n arbitrury, p extends to an involution of M,(D) iffp extends to an involution of M,(D).

Proof. We may assume m = 1 (by passing from m t o 1 to n). If (*) is an involution of D then M,(D) has the involution ( x d i j e i j ) *= C d t e j i (for d i j E D). Conversely, let R = M,(D) = x D e i j . Suppose R has a n involution (*) and let T = x F e i j c R. There is a canonical homomorphism T + ZFeJ, given by eijt-.ez (for {e;ll < i,j < n ) is also a set of matrix units), so, by Skolem-Noether there is some r in Unit(R) with eJ = r e i i F 1 for all i,j. Then e z r = reij, so (r*)-'reij

=

(r*)-'e*.r JI

=

(e.ir-l)*r = (r-'e*.)*r = e i j ( r * ) - ' r ,

implying ( r * ) - ' r E C R ( T ) = D. Write d = (r*)-'r, so r = r*d. If d = 1, take y = r = r*; otherwise, take y = r-r* = r*(d- 1) E Unit(R). By Theorem 3.1.53 *c(y) is an involution (.of I) R, sending eij to y-'e:y = (yeijy-')* = (reijr-')* = e$* = e j i ; thus for any d in D, dJeij= (ejid)J = (deji)J= eijdJ, implying d J E C , ( T ) = D. Therefore ( J ) induces a n involution of D. Q E D Theorem 3.1.71 (Albert). Suppose R issimplePI. R hasan involution of theJirst kind i$exp(R) = 2.

Proof. In view of Proposition 3.1.70, we may replace R by any representative in Br(Z(R)). In particular, we may assume R is a crossed product, and are done by Theorem 3.1.69. QED There is a slicker proof of Theorem 3.1.71, which has been generalized to

174

CENTRAL SIMPLE ALGEBRAS

[Ch. 3

Azumaya algebras by Saltman [78c], but we used Albert's proof because the basic idea is very straightforward.

$3.2. Positive General Results about Maximal Subfields of Division Rings

In this section, we ask the following questions. If D is a division ring of degree n then is D a crossed product? If, moreover, D has an involution of first kind (so that necessarily I I is a power of 2, by Theorem 3.139) then is D a crossed product? Note that this question is asked without reference to Z(D). The object here is to prove the known positive results, which are quite scant. The answer is "Yes" for n = 2,3,4,6, and 12; in the (*)-case, the answer is "Yes" for n = 2,4,8. (In $3.3 we shall obtain the celebrated result of Amitsur [72a] that there exist noncrossed product division rings.) Many of our results will come from a very pretty method of Wedderburn [21] of studying arbitrary division rings; we also introduce the study of the generic division algebras liD(4, n). Remark 3.2.1. If deg(D) = 2 then, letting K be a maximal separable subfield of D, obviously K is Galois (since 2! = 2), so D is a K-crossed product.

Let us look more generally at the prime degree case

For every Z(D)-subfield K of a division ring D of Remark3.2.2. degree n, [ K :Z(DJ]Jn; in particular, if K # Z(D) and n is prime, then K is maximal. Theorem 3.2.3 (Albert). Suppose D is a division ring and n = deg(D) is prime. D is a crossed product i$for some d $Z(D) we haue d" E Z ( D ) . Proof. ( 3 )is Corollary 3.1.42. Conversely, let F = Z(D) and K = F(d), so that [K:F] = n. If K has a primitive nth root q of 1 then obviously K is a cyclic extension of F, and we are done. Otherwise, we shall assume the reader knows something about cyclotomic polynomials (cf. Herstein [64B, pp. 320-3211). Let F' be the field obtained by adjoining q formally to F , i.e., F' = F[l]/p(A)F[I], where p is the minimal polynomial ofq. Let D' = D O FF' and K' = K OFF'.Since [F':F] < n, D' is a division ring by Corollary 3.1.19, and K' is a subfield; since deg(D') = n is prime, clearly [K':F'] = n, and K' is a cyclic extension of F' with generating automorphism a : d t + q d . By Proposition 3.1 41 we have some r in D such that rdr- = pi and r " E F . Then d-'rd = q r ; letting L! = F'(r), we see that D' is an L'-crossed product,

93.2.1

Positive General Results

175

with Gal(L:/F') cyclic generated by an automorphism p given by conjugation by d . Now D'€Br(F'); we would like to conclude by showing D' 2 D " O F F ' for some cyclic D" with center F (since then, by Corollary 3.1.22, D z D"). T o d o this, we need to look a little more closely at roots of 1 ; the method chosen is quite computational. Let s be a n automorphism generating Gal(F'/F); T induces an automorphism 1 O r of DOFF'= D', which we also call T . Now s ( q ) = qi for some i. Then r i d - ' = qid = s(q)d = ~ ( q d=) s ( r d r - ') = s ( r ) d s ( r ) - ' , implying r - ' z ( r ) E K ' ; write t ( r ) = ri.Y for .Y in K ' . Let a = r"E F'. Then ~ ( a=) ( t ( r ) ) "= (r'x)" = aiN,(.u), where recall N , ( x ) = .u,.,(.u)...cr"-'(.u)~ F'. Proceeding inductively, we have, for every j , zj(a) = ai'N,(.yj) for some s j in K ' . Now let u = [F' : F ] and, taking i, uo such that uuo = ii, = 1 (modulo n), define uk = uoik, and i' = xi= i k U k E uu0 = 1 (modulo n ) . Let a' = ' ( ~ ~ ( a ) Then ) " ~ . 2' = l ( a i x ~ N , ( . u k )=" ~'N,,(n;=,.u;~) k = aN,(.u') for some x' in F' [since ii divides (i'- l), and any iith power of a is a norm]. Thus by Proposition 3.1.43 we may ussume r" = a'. The point of this maneuver is that writing FL = ( ~ " ~ z E F 'we ) , have s ( a ' ) ( a ' ) - ' ~ F ~ . [Indeed, t ( ~ ' ) ( a ' ) = -~ ( T ~ + ' ( C L ) ) " ~ ( ? ~ ( C I ) ) - "E' ~F:, since u,i = u k - 1 (modulo n).] Thus ~ ( a '= ) (a')'ay for some a1 in F'. Therefore we can define an automorphism r' on L' by

n;=

n:=

n;=,

-[:: )

t

pjrJ

n-- 1

=

C ~ ( [ j ~ ) a { r ~for j all

pi in F'.

j=o

Although we need not have t(L') G L', i and T have the same restriction to F'. Moreover, n-

I

n- 1

tp(j3,)p(ct',)p(r'') =

2 ~(@~)a{$jr'J

j= 0

so r' and p commute; since IT and LI are relatively prime, f and p generate a cyclic group. It follows that L: is a cyclic extension of F , so the fixed subfield of L: under r' is a cyclic extension of F . Extend ? to an automorphism of D', and let D" be the fixed subring. Obviously D' z D"O,F', as desired. Q E D The Generic Division Rings

Before continuing, we want to bring in an extremely important division ring. Actually, we have seen it many times before, but did not note it was a division ring, and for this reason we give it a new name. I n what follows, 4 is a commutative domain.

I76

CENTRAL SIMPLE ALGEBRAS

[Ch. 3

U D ( 4 ,n ) = 4,,(Y ) , the ring of central quotients of Definition 3.2.4. thegeneric matrix algebra. Note that we have not stipulated how many generic matrices we used. It is convenient to work with an infinite number, but in fact two are enough (cf. Exercise 2.4.2). Remark 3.2.5. Every element of U D ( 4 , n ) has the form f ( Y 1 ,..., x ) g ( Y,,..., where J’E~,{ Y } and g E Z ( & , ( Y ) ) ; that is, . f ( X , , . . . , X , ) is not an identity of M,,(4[9]), and g ( X , , . . . , X , ) is M,,(4[<])central.

x)-’,

Theorem 3.2.6

(Amitsur). U D ( 4 ,n ) is a dicision algehru.

Proof. We recall from $2.4 that U D ( 4 , n )is simple of degree n. If not a division algebra, then U D ( 4 , n ) would contain a matric unit e l z , i.e., for some .f(Yl,. . ., Y,) in 4,,{ Y] and g( Y,,. . ., y ) in Z(+,{ Y ) ) we would have (.fg-’)’ = 0, s0.f‘ = 0. Then for any infinite simple algebra R of degree n we would have some r , , . . . , r , in R such that r = . f ( r , , . . ., r,) # 0. but r 2 = 0. By Proposition 3.1.46 we could take R to be a division algebra, which is absurd, because then R has no nonzero nilpotent elements. QED

The proof of Theorem 3.2.6 is most instructive, because it exemplifies how we pass information from U D ( 4 ,n ) to arbitrary simple algebras of degree n ; we just used this to obtain “negative” information about C ’ D ( 4 , n ) (i.e., it has no nonzero nilpotent elements). Here is another result of the same sort. Lemma 3.2.7. Suppose A is an algebra with center Z , und in general write A’ = A and A‘ = A OzAk-’; i.e., Ak = A Q z ... B z A . Lnder this notation, fi C’D(4,n)’ has a set of t 2 matric units and A is simple oj’degrue ti, then Ak also has a set of t 2 matric units.

Proof. Let C = Z(+,,{ Y ) ) ,and let C‘ be the field of fractions of C . Since U D ( 4 ,n ) 2 $,{ Y } O r C’, we have U D ( 4 ,n)k +,,{ Y } k@(. C‘.Let R = 4,,{Y } k and, viewing C G Z ( R ) by c + 1 Q * * * @lc, let S = C-{O}; then we have R, = U D ( 4 ,n)k. Let (eij[l 6 i,.j s t } be a set of matric units for R,. Then we can write e 1.1. = rijs- 1 , 1 < i,j 6 t , for suitable rij in R , s in S . Let B = ker(v,). where v,: R + Rs is the canonical homomorphism r + rl - I . Then for all j # u,

((x:

rijruvE B, rijrju- riusE B, and = r i i )- s) E B. Thus, for some s , in S we = 0 and s 1 ( ( ~ ~ = , r i i ) = - -0. s) have for all I d i, j , u, ti < t , sl(rijruc-Sjuri,,s) Now ss, # 0 in S , so there is some homomorphism 1,9:4,,{ Y ] + R such that +(ssI) # 0. A is a C-algebra under the action g ( Y , , . .., Yu)a= g(+( Y,), ...,+( Yu))a, and induces a canonical homomorphism $: R -,Ak by

+

IL P . . - 0 +

R--

A @ , . . . @ , . A + A @ z . . . @ z A = A‘,

43.2.1

Positive General Results

177

the second map induced by I): C + Z . Obviously, $(ss,) = 10 ... 0l$(.ssl) # 0, so $(ssl)E Z . Thus { $ ( r ) $ ( s ) - ' 1 1 d i,j < t } is a set of matric units for Ak. QED Theorem 3.2.8.

U D ( 4 ,n ) has exponent n in the Brauer group.

By Lemma 3.2.7, if exp(LiD(4,n))= k, then for every simple algebra A of degree n, exp(A) ,< k . By Proposition 3.1.46 k 2 n, so k = n . QED Proof.

The reader should see that this technique could also be used to examine splitting fields. In a similar vein, LID(#, n ) has the most "general" elements one could want, namely the generic matrices Thus, when we want to make computations (for division rings) and be sure they do not degenerate, it is convenient to work in U D ( 4 , n ) . The following result justifies this procedure.

x.

Remark3.2.9. I f 0 # f i j ( Y l ,..., Y,)E4,,(YJ, 1 6 i < m , 1 G j 6 2 , t h e n for each division algebra D of degree n there are elements d , , . . . ,d , of D such thatf;,(d,, . . . ,d,)fi2(d,,. . . ,d , ) - ' # 0 for all i. [Indeed, let

.f = I T= l . f i l ( X l r , X,).fi2(Xl7 X,); ' ' '

' ' ' 3

then f is not anidentityofb,,{ Y},sois not an identity ofD ;weconclude by taking d , , . . .,d, with f ( d , , . . .,d,) # 0.1

In the following sense, U D ( 4 , n ) is the best place to look when trying to verify a question about maximal subfields for all division rings of a given degree. Proposition 3.2.10 (Amitsur). Suppose that F = Z ( U D ( # , n)), and U D ( 4 , n) has a (not necessarily maxirnal) subfeld K Galois over F with Galois group G. Then every division algebra D of degree n has a subfeld Galois over Z ( D )with Galois group isomorphic to G .

Proof. We just need to encode as much information as we can into elements of U D ( 4 , n ) . Write Z = Z(4,,( Y}). By the primitive root theorem, write K = F ( f h - ' ) for some f ( Yl, . .. , Y,) in q5n{ Y} and h in 2 c F. Then K = F[f]. Let p = xy'o h J i be the minimal polynomial of A where each h i € F , and h, # 0. Multiplying by a suitable element of Z , we may assume each h i E Z . Let f = f l g - ' , f i g ~ ' ....,,f,g-' be the roots o f p in K , where each . f i bn{ ~ X } and g E Z . Each a in G is determined by its action onJ i.e., a(,f) is somefi; let x i j =fi-.fi, 1 < i < j 6 in. Also take h, such that f h , = h,a( f ) ; multiplying by a suitable element of Z , we may assume h,E4,,{ Y } . Finally, for all a,z in G write h;'h,h, =.fi,,g;,' # 0 for suitable f,,, in 4,,{Y} and in Z . Obviously the,f,,, all are in K .

178

CENTRAL SIMPLE ALGEBRAS

[Ch. 3

By Remark 3.2.9 we can find a specialization of the to D, sending each for, gni (1 < i < j d tn, 0, T E G ) to respective nonzero in D. Let 2 = Z ( D ) and K = Z ( f ) elements f , h,, g. fi, xij, h,, for, = z[,fl. Then xy=ohif = 0, implying [ R : Z ] < m. But, writing 0(x) = hi'xh,, for x in K , we have 0(f;) is one of theJ, so 0 is an automorphism of K ;in view of the xij,each of these 6 is distinct and, in view of t h e f z and G, E = ot for all o, 7 in G. Thus K has a Galois group (over F') containing an isomorphic copy G of G . But Cy=ohiib' splits in K , so Gal(R/F) has order m,and hence must equal G. QED

6 h,, g, .A,_ xij, - h,.

Wedderburn's Method We shall now turn to a n extremely important method of studying D [ i ] for an arbitrary division ring D due to Wedderburn [31]. Let F = Z ( D ) throughout. The basic idea is quite easy. Since any nonzero g(A) in DC3.1 can for suitable k , with d, # 0, we can be written uniquely in the form C~=octiIi define deg(g) = k ; then deg(fg) = deg(f) +deg(g), implying instantly that D[A3 is a domain. Also, in this notation, for each d in D define g(d) = ,did'.

x;=

Remark 3.2.11 (Division algorithm). Given ,fig in Drill. we can "divide" f into g by writing g = 4f+ r, where q,r E D[A] and either r = 0 or deg(r) < d e g ( f ) ; moreover, r and q are uniquely determined in this procedure. I f f = i..-d, then r = g(d). (Proof is an easy modification of Herstein [64B, lemma 3.181 left to the reader.) Definition 3.1.12. f d i c i d e s g (in D [ A ] ) if g = qj'for some q in D[A]. For emphasis, if g E D[A], we sometimes write g ( A ) instead of g. Remark 3.2.1 3. ( A - d ) divides g ( I ) - g ( d ) . (Obvious from Remark 3.2.1 1.) In particular, (A-d) divides giff g f d ) = 0. Proposition 3.2.14. Suppose g. h~ D[A], and ( A - d o ) dii1idt.s gh and not h. Let d = h(d,). Then ( A -ddod-') dizlides g. Proof. By Remark 3.2.13 A-do divides h(A)-h(d,) = h-d, and d # 0. Thus I - d o divides gh-gd, so by hypothesis A - d o divides gd. Write gd = q(A)(A - d o ) for suitable q. Then g = q(A)d- (d(A- do)d- ) = q ( I ) d - . ( A - d d o d - ' ) , so i.-dd0d-' divides g. QED

Most of the further theory is based on applying Proposition 3.2.14. The minimal polynomial of an element d in D is the monic p in F[A] of lowest degree such that p ( d ) = 0. Viewing F[%]= Z ( D [ I ] ) , we see that P E %(D[A]).

$3.2.1

Positive General Results

179

Obviously, p is the minimal polynomial of every conjugate of d ; we look for a converse. I f g E D[A] and g(d) = 0.for all conjugates d of a given Lemma 3.2.15. element xo in D, then the minimal polynomial of xo divides g. Proof. Let p be the minimal polynomial of x., If the result were false, we could take a counterexample of minimal degree, g = C:=,diAi; multiplying on the left by d; we may assume d, = 1. Now for any conjugate d of x, and any y in D , obviously y d y - ' is also a conjugate of x, so 0 = Ef=,di(ydy-')' = E f = , d i y d i y - ' , implying C:=,(y-'d,y)d' = 0; also z : = , d i d i = 0, so 0 = ~ f = o ( ~ ~ - ' d i y - d i = ) d -i y - ' x [ y , d i ] d i , implying z [ y , d , ] d i = 0. Letting h, = [ y , g ] = ~ : = , [ y , d , ] A ' , we have h,(d) = 0 for every conjugate d of x,; moreover, deg(h,) < k - 1 since [y,d,] = [ y , 11 = 0, so by induction p divides h,. Write g = qp+r, with deg(r) < deg(p). Then, for all y in D , h, = [ y , q ] p + [ y , r ] , implying [ y , r] = 0 ; hence r E Z ( D [ A ] )= F[A]. But clearly r(.Yo) = g(u,)-q(xo)p(x0) = 0, since the coefficients of p are in F ; this contradicts p being the minimal polynomial of so.Thus the lemma is true. Q E D

',

=

Theorem 3.2.16. Suppose ~ E F [ A . ]is irreducible (in F [ A ] ) . If p ( d , ) 0 for some d , in D then, in D[A], p can be written as a product of linear

factors in D[A]. Proof. Write p = g(L)(A-d,)...(L-d,) with k as large as possible; by assumption k 3 1. Let h = (A-d,)... ( 2 - d ' ) . We claim that h(d) = 0 for every conjugate d of d , . Indeed, if h ( d ) # 0, then by Proposition 3.2.14, putting d,,, = h ( d ) d h ( d ) - ' , we have (A-d,+,) divides g ; writing g = g'(A.)(A-dk+l) and h' = (A-d,+ ,)h, we have p = g'h', contrary to the maximality of k . Thus, by Lemma 3.2.15 the minimal polynomial of d , divides h, and must then be p , since p is irreducible. Thus p = h. QED

As nice as this theorem is, for its application we are really more interested in its proof, viz., the following observations. Remark 3.2.17. In writing p = ( A - d , ) . . . ( A - d , ) we could make the following selections:

in Theorem 3.2.16,

(i) d, = [d, d,]d,[d, d l ] - ' , where ci is any element of D - C,(d); (ii) d 3 = [d,, d,]d,[d,, d l ] - ' if [d,, d , ] # 0. [Indeed, for (i), if [ d , d , ] # 0 then dd,d-' # d , , so, writing p we use Proposition 3.2.14 to show A - d 2 divides q, where dz

=

(dd,d-'-d,)(ddld-')(dd,d-'-dl)-'

=

[ d , d i ] d i [ d ,d 1 l - l .

= q(A)(A-d,),

= [d,d,]d-'ddld-'d[d,d,]-'

180

CENTRAL SIMPLE ALGEBRAS

[Ch. 3

To see (ii), write p = q(A)(A--d,)(;.-d,) = q ( A ) ( A 2 - ( d l + d z ) A + d 2 d l ) .By hypothesis, ( d , ) , - ( d , + d,)d2 d , d , = [d,, d , ] # 0. Hence, by Proposition 3.2.14 we can take d, = [dz,dl]d,[d,,d,]-'.]

+

Remark 3.2.18. If in Theorem 3.2.16 p = (A--dk)..-(A--dI) then for each i , p = ( n - d , ) . . . ( A - d , ) ( E . - d k ) . . . ( A - d i + , ) . [Indeed, put r = ( A - d i ) . - . (A-dL) and s = t A - d , ) . . . ( I - d , + , ) , and apply Remark 3.1.56.)

The Cases n= 3 and n= 4

We shall now show for n = 3 that every division ring of degree 3 is cyclic. Definition 3.2.19. but r"EZ(R).

An element r of a ring R is n-cenrral if r$Z(R)

Theorem 3.2.20. Suppose n is prime. Every ditlision algebra of' degree n is cyclic i$'UD(d, n) hus an n-central element. Proof. (=) I S obvious even when n is not necessarily prime by ~ Y ) and Corollary 3.1.42. Conversely, suppose j g - is n-central for , f ' #,( g E Z($,,{ Y } ) .Then j'is also n-central ; thus every division algebra of degree n has an n-central element, and is cyclic by Theorem 3.2.3. QED

Thus it is imperative to verify whether or not UD($,n) has n-central elements. (Equivalently, one may ask whether 4"{Y ) has n-central elements.) Note that [ Y,, Y,] is 2-central for 4,{ Y } . One can get some information concerning multilinear polynomials (cf. Exercises 12,14)and a very pretty result can be found in Saltman [78bP]. For ti = 3, there is a beautiful positive answer by Wedderburn [21]. Theorem 3.2.21. Every division ring of' degree 3 is cyclic:. I n fact, [ Y l , [Y,, Y,]Y,[ Yz, Y , ] - ' ] is a 3-central element of' UD(4,3).fbrevery 4. Proof. Using Theorem 3.2.20, it suffices to show for arbitrary 4 that D = U D ( ~ , Ihas I ) a 3-central element. Let F = Z(UD(4,n)); let d , = Y, and d = Y,. Obviously, [ d , d , ] # 0, so by Theorem 3.2.16, taking the minimal polynomial p(2 1 =r C;= a i l i of d , , for suitable cli in F we have p = (A - d , ) . ( l - d , ) ( A - d , ) ; by Remark 3.2.17 we can taked, = [ d , d , ] d , [ d , t l , ] - ' . Let .Y = [ d , , d , ] . Note that -a2 = d , + d 2 + d , ; commuting with d , yields [ d , , d , ] = [d,,d,] = .Y. Similarly, commuting a, with d , yields [ d 2 , d J = .Y. By Remark 3.2.18, for every cyclic permutation 7c of 1, 2, 3, we have p = (L-dn3)(A-dn2)(/ -dnl), and we saw [dnl,dnl]= fs.

$3.2.1

181

Positive General Results

We claim that s$ F . Indeed, the M , (+( <)), specializing Y l ~ r =l z ; = , t i e i i

and

are generic matrices; working in

Y 2 b r 2= ~ a e , 2 + ~ , e 1 3 + ~ s e z , + < , e 3 1 ,

and lettingr = [r2, rl]r1[r2, r , ] - I , it sufficestoshow [rl, 1.1$ F . But thisisclear because r has nonzero off-diagonal entries in the second column, proving the claim. Now, for any cyclic permutation TI in Sym(3), [dnl,dn2]# 0. [For otherwise p = di1'pd,, = ( A - d ~ , 1 d " 3 d " , ) ( ~ - d , 2 ) ( ~ - d * l )so di11dn3dn1 = d n 3 , implying [d,,,dn3] = 0 ; likewise [dn2,dn3]= 0, and we conclude [ d l , d z ] = 0, which is false.] Thus, by Remark 3.2.17(ii) we could take dn3 = [ d F 2d, , I]dn2[dn2,d, ,I- = xd, 21 ;sinced,, isdetermined by d,, and d, l , d, must equal this value. Taking subscripts modulo 3, we have shown d,, = ud,?c-' for each i. Thus x3 commutes with each di, so s3E C , ( F ( d , ) ) n C,(F(d,)) = F ( d , ) n F ( d 2 )= F . (We have used repeatedly Remark 3.2.2.) Therefore s is 3-central. QED ~

Say a field extension K of F is quadratic if [K:F] = 2. Turning to the case n = 4, we ask the natural question, "Does every division ring of degree 4 have a quadratic extension of the center?" For the sake of later results, we give a positive answer more generally, without assuming n = 4. Remark 3.2.22. Suppose D has an element d , of degree 4 over F = Z(D), with tr(d,) = 0, and let p = I a + + 2 L 2 + a , L + a o be the minimal polynomial of d , . In the notation of Theorem 3.2.16 we have

p

=

(A2+a'A+b')(A2+d+b)

for suitable elements a, b, a', b' of D, with b

= d2d1 and a =

-

(d, + d 2 ) .

Lemma 3.2.23. Assume the hypotheses and notation of Remark 3.2.22. Zf3l[D:F],and [F(a):F] d 4, then either a 2 € F or [F(a2):F]= 2. Proof. Matching coefficients in p yields 0 = a'+a, so a' as the equations a2 = a'a+b+b'= a1 = a'b go

Substituting b'

= a&'

= -a,

-a2+b+b',

+ b'a = - ab + b'a,

= b'b.

into the first two equations now yields

as well

182

[Ch. 3

CENTRAL SIMPLE ALGEBRAS

Case I. [a, h]

'

(a,b- - b)a, so - [(a,t~-' + b ) 2 - 4 a , ) a z = ((a 2 + u2)' -4aO~ o2 = 0. Then a1 =

=

(a2)3+2a2(a2)2+(a~-44610)a2.

Thus [ F ( a 2 ) : F ]51 3, so by hypothesis [ F ( a 2 ) : F ]= 2 or a 2 eF. Case 11. [a, b] # 0. By (I), 0 = [a,, b ] = - [ a 2 , b ] , implying a 4 F(u2),so [ F ( a 2 ) : F ]< 4. Thus [ F ( a 2 ) : F ]= 2 or a 2 e F . QED Proposition 3.2.24. UD(4,4) has a quadratic Galois extension of Z(UD(4,4)),grneratedby theelement ( [ Y 3 , [ Y 1Y, 2 ] ' ] [ Y 3 , [ Y l ,Y 2 ] ] - 1 ) 2 . Proof. Let d , = [ Y l , Y,], so that Remark 3.2.22 holds. By Remark 3.2.17wecan taked, = [ Y 3 . d , ] d l [ Y I , d l ] - 1 , s o a= =

-(d, +[Y3.dlldl[Y3.dll-1) = - Y3, d:] [y, 4 1 -

-(dl[Y3,dll+[Y3,dl]dl)[~3,dl]-1

9

Thus, in view of Lemma 3.2.23, it suffices to prove a4$Z(UD(4, 4 ) ) .This is seen by the following specialization to M4(4(()), taking subscripts modulo 4 : 4

KH

C

4

4

1 ei+l,i,

Siei.i+l,

~ 2 + + i= 1

i= 1

and

~

3

C

ei-+l,i; i=l

x:=

indeed, then o -+ ( < i - < i - 2 ) e i i ,whose fourth power is not scalar, implying a' is separable. QED Remark 3.2.25.

The reader should note we have also proved that

([ Yz, [ Y,, Y2]'][ I;, [ Y l , Y2]]-1)2generatesaGaloisextensionofdegree20ver Z(UD(h4)).

It is most useful to have these quadratic Galois extensions of the center, as we shall now see. Proposition 3.2.26. Suppose D is a diuision ring with center F, und d E D such that F ( d ) is a quadratic Galois extension of D . Suppose x E D and d , = [ x , d ] # 0 ; then d # d ; ' d d , e F ( d ) , [ d : , d ] = 0, and F ( d : ) c F ( d , ) . Proof. We have suitable a l , a 2 in F such that d 2 = a l d + a z . Then d d l + d , d = [ . u , d Z ] = c i l [ . ~ , d ] = a l d l , s o d d l =d,(a,-d).Clearlyd#a,-d [since otherwise 2d = a, E F, implying 2 = 0 and so a1 = 0, contrary to F ( d ) being Galois over F ] . Thus d # d ; ' d d , e F ( d ) . But dd: = dl(al - d ) d , = d l a l d , - d , ( d d , ) = &a, -d:(a, - d ) = d : d ; so [d:, d ] = 0 and d , $ F ( d : ) (since [ d , , d ] # 0). QED

Here is a technical observation that is useful. Remark 3.2.27.

Suppose D is a division ring with center F, and d E D

53.2.1

Positive General Results

183

such that F ( d ) is quadratic. If x E D such that d # x - l d x ~F ( d ) and if x ' ~ F ( d )then F , x , and d generate a division ring D, of degree 2, with Z ( D , ) = F. [Indeed, obviously 1, d, x , and xd are an F-basis of D,, and so [Dl:F] = 4; also D , is closed under multiplication. Thus D , is a division ring of degree < 2 ; since D , is not commutative, we have deg(Dl) = 2, so Z ( D , ) = F.] Now we can give an explicit version, due to Rowen [78a], of an old theorem of Albert. Write Z,for Z72.Z. Theorem 3.2.28. Every dioision ring of degree 4 has a Galois extension K of' the center F having Galois group Z, x Z,; moreover, 11 D = UD(4,4), we can take K = F(.Y,)F(.Y,),where .Y, = ([Y,, [ Yl, YJ']. [Y,,[Y,, Y,]]-')z and x, = [Y,, Y,]'. Proof. By Proposition 3.2.10 we need only prove the second assertion. By Proposition 3.2.24 F ( x , ) is a quadratic Galois extension of F . Clearly [Y4,x1] # 0 since x , $ F ; letting d , = [ Y4,xl] (and d = x , ) in Proposition 3.2.26, we have x, # d ; ' x , d , E F ( x , ) , Id:, x,] = 0, and F ( d : ) c F ( d , ) . Since x2 = d:, we have [F(x,): F] is a proper divisor of 4, so is 1 or 2, and K = F(.y1)F(.y2)is a field. Moreover, .Y;$ F. (This is clear from the specialization used to prove Proposition 3.2.24; details left to the reader.) Thus F ( x , ) is a Galois extension of F , and by standard Galois theory we can conclude by showing F ( x , ) n F ( x 2 ) = F , which we now claim. Otherwise x , E F ( x , ) , implying by Remark 3.2.27 that x 1 and x , generate a division subring D , of D = UD(4,4) of degree 2 and center F ; then D z D , O F C D ( D 1 implying ), exp(D) = 2, contrary to Theorem 3.2.8. Thus the claim is established. QED

We can now summarize the known positive results about when all division PI-rings of a certain degree are crossed products. Theorem 3.2.29 (Wedderburn-Albert). Suppose D is a dioision ring ofdegree 1, 2, 3,4,6, or 12. Then D is an abelian crossed product. Proof. We have seen the result to hold for 1 < n < 4. The cases n and n = 12 follow immediately by Theorem 3.1.40. QED

=

6

Algebra of Generic Matrices with Involution

Comparing the proof of Theorem 3.2.28 to the proof in Albert [61B], we see the similarity that the key step is obtaining a quadratic extension of the center (which we did with Wedderburn's method). Having done this, one

I84

CENTRAL SIMPLE ALGEBRAS

[Ch. 3

wants to build it into the desired maximal subfield. The use of generic matrices is most opportune here, because it enables one easily t o dispose of a degenerate case which occupies over half of Albert’s proof. Thus, in our study of division rings with involution, we first look for a “generic” one. There is an obvious candidate. Definition 3.2.30. Let 4,,{Y , Y‘) denote the subalgebra of M,,(4[5]) generated by all generic matrices Y,, Y,, ... and their involutes Y;, Y;, . . . under the canonical sympletic involution (s) of M,,(4[5]). We define 4,,{ Y, Y ’ }analogously,using thetranspose(t)inplaceof(s).(Ofcourse wetaken even when considering (s).) Using the results of Chapter 2, the reader should easily prove the following results. Remark 3.2.31. The involution (s) of M,,(4[5]) restricts to an involution (s) on 4,,{ Y , Y’}, and (4,,{Y , Ys},s) is relatively free in the class of algebras with involution < (M,,(4[(]), s). If 4 is a domain, then the algebra of central quotients +,(Y, Y’) of 4,,{Y, Y’) is simple of degree n, and ( s ) induces an involution of the first kind on 4,,(Y , Y s ) .Likewise, the algebra of central quotients 4,,(Y, Y ‘ ) is simple, and (t) induces an involution of the first kind on 4,JY , Y‘). Thus +,,( Y , Y s )and (b,,(Y,Y‘) have exponent 2. For simplicity we shall assume from now on that 4 is a field with 3. [Even in characteristic 2 one should get the same kind of results by considering polynomial identities of rings with bilinear forms (cf. Theorem 7.3.20), but the procedure is considerably more complicated : the characteristic 2 case is in fact open.] In this case, every division algebra of exponent 2 has involutions both of orthogonal and symplectic type by Theorem 3.1.69 and Theorem 3. I .57, so with the identical argument used in Theorem 3.2.6 one has Proposition 3.2.32. Suppose 3 ~ 4 I.f there is a division algebra of exponent 2 and degree n, then 4,,( Y , Y s )and 4,,(Y,Y ‘ )are division algebras.

So we now have an involutory analog of the generic division rings, although there is no chance for 4,,( Y , Y s ) and $,,(Y, Y ‘ ) to be division rings unless n is a power of 2. (It is easy to see that they are division rings when n is a power of 2, using Proposition 3.2.32; cf. Exercise 3.3.5.) Proposition 3.2.33. I f 4,,(Y,Y s ) has a subfield Galois ouer Z($,,(Y , Y s ) )with Galois group G, then every division algebra D of exponent 2 and degree n has a subfield Galois over Z ( D ) with Galois group isomorphic to G. Proof. Identical to proof of Proposition 3.2.10. QED

$3..2 .]

185

Positive General Results Simple PI-Rings with Involution of First Kind

Why d o we concentrate on (s) instead of the more familiar (t)? Proposition 3.2.34. I f R is u siiiiple ring ofdegree ti with symplectic iiirolutiori ( s ) ,aiid r = rs E R, theii r is ulyebraic qfdegree < 4 2 oiler Z(R). Proof. Splitting ( R , s ) to (M,(F),s) for some field F, we see (in M,(F)) that ii is even and r has degree dr1/2 over F by Theorem 2.5.10. Hence, by Corollary 1.5.19 1, r , . . . , F 2 are Z(R)-dependent. OED

Let us now t r y to determine the structure of simple PI-rings with involution of the first kind. The first nontrivial case is for degree 2. This case is so important that every simple ring of degree 2 and center F is called a quaternion F-algebra. Remark 3.2.35. Iff E 4 and R is a simple PI-algebra, then every quadratic extension of Z(R) contains a 2-central element. [Indeed, if r2 + a , r + a , = 0, then ( r ~ ( , / 2E) Z(R).] ~

+

I f * E F arid R is a quaternion F-algebra, then Proposition 3.2.36. there are elements r l , r2 in R such that r: E F, r: E F , rl r2 = - r 2 r l ,and 1, r l , r,, r 1 r 2are an F-basis OfR. Proof. IfR t M,(F),thentaker, = e I 2 + e , , a n d r 2=el,-e,,.Thuswe may assume R is a division ring. By Remark 3.2.35,in any maximal subfield K we have a 2-central element r , . Then a , + a , r , + a 1 - a 2 r l defines an automorphism of K ; by Skolem-Noether, r2rlr; = - r , for some r 2 in R, implying r2rl = - r l r z . Now r: E C , ( K ) = K , so F(r:) K . Hence [F(r:):F] < [ K :F] = 2, implying r:E F. It remains to show 1, r,, r2, and r l r z are linearly independent. Suppose a , +a,r, +a3r, + ~ 4 r l r z= 0. Then O = [0, r , ] - [a3r2,r1]+[a4r,r2,r1] = 2rlrz(-a3+ct4rl), implying a3 = ~ 4 ~ so 1 , u3 - a4 = 0. Hence a1 = - a z r , so aI = a, = 0. QED

1

Remark 3.2.37. exponent 2.

Any tensor product of quaternion F-algebras has

Proposition 3.1.67 gives an easy proof of a theorem of Albert. Theorem 3.2.38. Suppose 3.4. Every division algebra D of degree 4 and exponent 2 is a tensor product of quaternion Z(D)-subalgebras. In.fact, if ( * ) ~ I n v o l ( Dis ) symplectic, theii D is a tensor product of quaternion Z(D)subalgebras which are invariant under (*).

Proof. We shall prove trivially the second assertion, which clearly implies the first assertion. Let F = Z(D) and take .Y* = x $ F . Then [ F ( x ) : F ] = 2, so there is a nontrivial automorphism of F(x), which by

186

CENTRAL SIMPLE ALGEBRAS

[Ch. 3

Proposition 3.1.67 is given by a symmetric element d of D. Thus [ F ( d ) :F ] = 2, and F(d2)c F ( d ) , implying d 2 E F . By Remark 3.2.27 (with x and d reversed) x and d generate a quaternion F-subalgebra A of D, and clearly A is (*)-invariant. Then D z A O FC,(A). Q E D We proceed to degree 8. Lemma 3.2.39. Let D = # 8 ( Y , Y s ) ,F = ZP), d , = [Yl d , = [ Y3 - Y;, d ,Id,[ Y3- Y;, d,] - I , and 1~ = d1 + d 2 =

+ Y:, Yz - Y,],

[Y3- Y;,d:][Y3- Y,",d,]-'.

Assume D is a dicision ring. Then [F(a2):F ] = 2

Proof. Since ( M 4 ( 6 [ 5 ] )0 M4(#[5])OP, 0 ) E ( M 8 ( m ) , s), we see by Proposition 3.2.24 that a2 $F. O n the other hand, tr(dl) = 0, and d; = d, (so that [ F ( d , ) : F ] < 4); by Lemma 3.2.23 we will be done if we can show [ F ( a ) : F ]= 4. Let w = [Y3- Y;",,]. Clearly ws = w, so J = s{(w-') is a symplectic involution of D and aJ = waSw-' = w([Y3- Yj',d:]w-')"w-' = [ Y , - Y;, d f 1 w - I = a. Hence a is J-symmetric proving [ F ( a ) :F] =4. QED Theorem 3.2.40 (Rowen). Assume $ E 6.Every division algebra D of exponent 2 and degree 8 is a K-crossed product ,for some Galois extension K of Z(D) having Galois group Z,x Z,x Z,. Proof. By Proposition 3.2.33 we may assume D = #,( Y, Y b ) . Let F = Z(D). We are done if D contains a quaternion F-subaigebra A, because then by Theorem 3.2.38 C , ( A ) will be a tensor product of quaternion F-subalgebras, so D will be a tensor product of quaternion F-subalgebras. Thus we may assume that D does nor have a quaternion F-subalgebra. (This is just a device to bypass specialization arguments.) Let K, be a quadratic extension of F (given by Lemma 3.2.39). By Remark 3.2.35 K = F ( x ) for some 2-central x in D. Then x H - x induces the automorphism of K , over F; by Proposition 3.1.67 there is some symplectic (*)~Invol(D) with x* = -x. By Proposition 3.1.66 with 0 = 1 we have some (*)-symmetric y such that y - ' x y = -x. Let K, = F ( y 2 ) . Since [ F ( y ) : F ]< 4 and y 2 x = xy2, we get [ K , : F ] d 2. If y 2 c K 1 ,then y 2 and x generate a quaternion F-subalgebra of D,contrary to assumption. Thus K , n K , = F and [K, : F ] = 2. Now let T be the nontrivial automorphism of K , over F , and let 0 be the automorphism of K , K , , such that ~ ( x=) - x and ~ ( x '=) ~ ( x ' for ) all x' in K , . By Proposition 3.1.66 we have some (*)-symmetric d with d-'xd =a(.)* = x and d-'x'd = z(x') for all x' in K,. Let K , = F(d2). If K 3 L K , K 2 then d 2 = a1+a,x+a,y2+a4xy2 for suitable a, in F, so also d2 = (dz)* = a l - a,x+a3y2-a4xy2,irnplying2d2 = 2a, +2a3y2,sod2E K , ;

43.3.1

The Generic Division Rings

187

then y 2 and d would generate a quaternion F-subalgebra of D, contrary to assumption. Therefore K , $ K , K , and K = K , K , K 3 is our field. QED The structure of division PI-rings of degree 8 and exponent 2 is now known in the following sense: There is an example which is not a tensor product of quaternion subalgebras (due to Amitsur-Rowen-Tignol [78P]), but Tignol [78] proved for every such division ring D that M , ( D ) is a crossed product that is a tensor product of quaternion subalgebras (cf. Exercises 1-7). Here are two recent theorems which are beyond the scope of this book: Theorem (Rosset [77P]). Every member ofBr(F) of prime index has a representative that is an abelian crossed product. Theorem (Snider [78P]). When F contains every member of Br(F) of index 4 has a representative that is a tensor product of cyclic subalgebras.

$3.3. The Generic Division Rings

In $3.2 we used U D ( 6 , n ) to prove some positive results about division rings of small degree. In this section we shall obtain the famous result of Amitsur, that UD(Q,n) is not a crossed product for n divisible by 8 or the square of an odd prime; this result has been extended for characteristic p by several people. Later, we examine the generic affine division rings, focusing on the center. Before presenting Amitsur’s theorem, we would like to describe it briefly. We have seen how to build crossed products from factor sets; if the factor sets are particularly nice (such as in the cyclic case), there often are good criteria to check that these crossed products are division rings. However, it is exceedingly difficult to build division PI-rings without some construction which automatically makes them crossed products. The great mathematician Albert spent years trying to build a noncrossed product of degree 5. Until Amitsur, all attention on the crossed product question was focused on the “simplest” open case, namely prime degree. Amitsur’s idea was to look instead at the opposite extreme, where the degree n is highly composite. Suppose we can construct two (crossed product) division algebras D , and D , both of degree n, such that a given group G cannot appear as the Galois group both of a subfield of D , Galois over Z ( D , ) , and of a subfield of D , Galois over Z ( D , ) . Then, by Proposition 3.2.10 G cannot be the Galois group of a subfield of U D ( $ , n ) Galois over Z(UD(4,n)).If we can do this for every G of order n then UD(+,n) fails to be a crossed product. In practice, D , will be cyclic and D , will be an abelian

188

CENTRAL SIMPLE ALGEBRAS

[Ch. 3

crossed product with respect to a group which is a direct sum of cyclic groups. Incidentally, this idea is useless for prime degree, because then there is only one possible Galois group, so we cannot then play two crossed products against each other.

Skew Polynomial Rings and Their Rings of Central Quotients

In view of the above discussion, we need to find examples of division rings. The first example is a generalization of the construction of a cyclic crossed product. Definition 3.3.1. Suppose R is a ring with an injection c : R .+ R (not necessarily an automorphism). The skew polynomial ring R [ p ; e] is defined to have the additive structure of the polynomials ~ ~ = , r i pwhere i. ri E R, k arbitrary 3 0, with the multiplicative structure induced by the rule pr = e ( r ) p . Remark 3.3.2. R[p;e] is indeed a ring. Every automorphism T of R commuting with u extends to an automorphism of R [ p , r ~ ]by the rule z ( x r i p i ) = xt(r,)pi. Remark 3.3.3. There is a degree function, deg(xf=,ri$) = k (where rk # 0), defined on all nonzero elements of R [ p ; 171.If R is a domain, then for every nonzero pl, p z in R[p;a], deg(p,p,) = deg(p,)+deg(p,), implying R [ p ; c] is also a domain.

Exercise 1 gives us one connection between this construction and the cyclic construction, but we occupy ourselves now with an easy iterative procedure. Namely, suppose el,. . . ,e,are commuting automorphisms of a domain R . We define R , = R, R , = R[pl ;el]; inductively, having obtained Rkr we extend c k + to Rk by the rule c k + (pi)= pi for all i 6 k , and define R k + , = Rk[pk+ ;o k + ,I. We shall hold this notation for the next few results. Remark 3.3.4. Iterating Remarks 3.3.3and 3.3.2, we see R , is a domain, and ol,. . . ,a,extend naturally tocommutingautomorphismsof R,, with oj(,ui) = p i for all j # i.

We have the following situation in mind. Suppose F is a field whose characteristic is relatively prime to n, and assume F contains a primitive nth root v] of 1. (Then F contains n distinct nth roots of 1). Let G be an abelian group of order n. so we can write G = G I x ...x G, for suitable t , where each Gj is cyclic, with generator oj having order m j dividing n . Let m be the least common multiple of W I ~..,.,m,,the “exponent” of G . Let R = F[.u,, . . . ,s,]. where each xi is a commuting indeterminate over F , and let

43.3.1

The Generic Division Rings

189

each aj act on R by aj(cr) = cr for all CI in F , aj(.xi) = -xi for all i # j, and crj(.xj) = q j x j , where q j = is a primitive mjth root of 1. Now form R, as described above. . . A monomial of R , means an element .Y? . . .xi'& I . . .pi2', . where 0 < i,, . . . , izt< co.Clearly the monomials of R, are a basis of R, as F-vector ,_._, i2,,aix;. . . xf'p;' I . . .pi2' we write h to be the nonzero space. If h = 1 . . . pthat ~ 2 1(i,, ..., i,,) is largest according to the term ~ i . x ~ 1 . . . . x ~ p ~ *such lexicographic ordering. This notion is very useful because of the following easy observation. +

xi=(il

Remark 3.3.5. For any h = h , h , in R, we have h = h,h,. Also, for any h,,h, in R,, h,h, = qkh2hI for suitable k E Z + , and ( h l ) ' " ~ Z ( R ,(Just ). check this for monomials.) Furthermore, if g E Z ( R , ) then @ EZ(R,). (Indeed, for any monomial h, gh = hg, implying gh = hg, yielding the desired result.) Theorem 3.3.6. W i t h the notation as above, R , has PI-degree n ; writing D for its ring of central quotients, we have D is a division ring of degree n with center F(xY', . . . ,.xy',pTI, . . .,117') (isomorphic to the Jield of ,fractions of a polynomial ring) and maximal subjield

K

=

F ( s , , . . . , x,, p y ' , . . . ,@),

and D is a K-crossed product with Galois group G. Proof. Since every x i commutes with each p j " ~ we , see that the set B = { p ~ - . . p ~ ' dl Oij d m j for eachj} is a basis of R , over the commutative R [ p y ' , . ..,p?], so R, is a PI-ring, and B is also a basis of D over K . On the other hand, iffg-' €CD(K),where,fe R, and g E Z ( R , ) , thenfEC,(K). Thus [ x i , f l = 0 for all i, implying at once that ~ E KHence . f - f ^ € C , ( K ) ; by induction on the number of monomials ofJ; we concludefEK. This proves K = C,(K), so K is maximal and deg(D),= [D:K] = m , ...m, = n. Let

F,

=

F(x;"', . . . ,xyt, py1,. . . , p T ) .

Clearly F , c Z(D) and K is a Galois extension of F with Galois group G. Thus [ K : F,] = n. But [ K :Z(D)] = n, so F , = Z(D). QED (Although Theorem 3.3.6 is a special case of Amitsur-Saltman [78], the reader should apply these methods to that paper. To utilize this construction properly, we bring in some easy Galois theory. Proposition 3.3.7. Suppose F is any Jield of characteristic relatively prime to n, containing a primitive nth root of 1, and L is a cyclic Galois extension of F with Galois group generated by CT.Then L = F ( a ) ,for some a E L such that o ( a ) = qa and u" E F .

190

CENTRAL SIMPLE ALGEBRAS

[Ch. 3

= q" = 1. By Corollary Proof. Obviously a(?) = q, so qa(q)...o"-'(q) 3.1.34 q = &)a . for some U E L. Thus o(a) = ?a, and @'(a)= via for each i. Hence [ F ( a ) : F ] 2 n, so F ( a ) = L. Also o(a") = (a(a))" = q"an = a", implying ~ " E F QED .

Theorem 3.3.8. With the assumption and notation us in Theorem 3.3.6, suppose L is an arbitrary Z(D)-subfield of D with arbitrary automorphisms a,z,fixing Z ( D ) . Then (T" = 1, T" = 1 , and (TT = TO. Proof. First suppose om# 1. By Skolem-Noether, CJ can be extended to an inner automorphism of L, i.e., for some hE D we have h-'ah = a(a) for all U E L .Multiplying h by a suitable element of Z(R,), we may assume hER,. Let u be the order of 0 , let q' = f'", i.e., q' is a primitive uth root of 1, and let L , be the fixed subfield of L under o. By Proposition 3.3.7 we have L = L , ( a ) for some a in L such that o(a) = q'a and a" E L , ;we may assume a E R,. Now ah = ?'ha and so for some i not divisible by n, ah" = qih"a. Thus ah = q'ha and ah"' = qihma,contrary to hmE Z ( D ) by Remark 3.3.5. This proves CT" = 1. Since o was arbitrary, we have T" = 1 for any automorphism T of L. Continuing. take p = ( T T C J - ~ T - ' . If p # 1 then for some U E Land some power q' # 1 of q we have p ( a ) = q'u, arguing as above. Then using Skolem-Noether and multiplying through by some element of Z ( R , ) , we may assume U E R,, and have h,, h, in R, such that (r is given by conjugation by h,, and T is given by conjugation by h,; i.e., q'a = p(a) = ~ T C T - ' T - ' ( U ) = h2h,h~'h;'ah,h,h;'h~'. Write h; =As; ', where g i E Z ( R , ) and f i E R,, i = 1.2. Then we have gi = J h i and- q*'ug-ig-i ~ 2 h ~ f ~ f ~ a h l h , fTherefore lf,. ii =.Liti, i = l , Z , and q'i%jf$i = h 2 h , , f , f ~ ~ h , h , f , , f , ;since G i € Z ( R , )and h;' =fi:i;', we get q'i -j , j, h-lh;ihh h h - 'h;'. But &;-'&hi = qih for some power qi of q by 2

1

2

Remark 3.3.5. Thus q'& = i i ; that p = I . Q E D

' q 2 q,& = h, contrary to q' # 1. We conclude

Corollary 3.3.9. Every Z(D)-subjield of'D has an abelian Galois group Of'e.upoiie~irdividing m, where D is as in Theorem 3.3.6.

Suppose G is u direct sum of cyclic groups ofprime Corollary 3.3.10. order. Then every Z(D)-subfield of D has u Galois group which is a direct sum of cyclic groups of prime order. Proof. Apply Corollary 3.3.9 to the structure of finite abelian groups. QED

Write Z, for the cyclic group Z/pZ. Proposition 3.3.11. With the notation as i n Theorem 3.3.6, suppose G is cyclic (i.e.. t = 1). Then for every prime p , D does not have a subfield L Galois over Z ( D ) having Galois group 7 , x Z, x Z,.

43.3.1

The Generic Division Rings

191

Proof. Otherwise, letting F , = Z ( D ) and appealing to Proposition 3.3.7, we have L = L l L 2 L 3 , where Li = F,(y,), ~ P FE , , and elements hi, 1 6 i < 3, such that hiyi 5 qyihi (where q is a primitive pth root of 1 ) and hiyj = jSjh, for all i # j . Multiplying by a suitable element of the center, we may assume the yi and hi are in R, = R,. Now, using Remark 3.3.5, we have . ? P E Z ( R ~ )h,i i i = q i ~ ~ and h ~ , hipj = ?,.hi and p i j j = ijjifor all i # j . It follows that { j ; ~ j $ ~ ~ t ~<; ~u ,~, u2? l O u 3 < p ) is a set of F,-independent, commuting p-central elements (including one element in F , itself) and that there are p 3 elements in this set. We claim this is too many. . . Indeed, each of these supposed elements has the form cr.u’,pJ,for suitable a in F ;dividing out by the proper element of F , , we may assume cr = I and 0 6 i , j < n. So we want to show that there are not p3 commuting elements each of the form . y i p { , with 0 6 i,j < n and ( . Y I ~ { ) ~ EFor F , . ease of , ) (xp’)” ~ notation write .Y for x i , p for p , and cr = CJ{. Then ( ~ l p c ’ = = sa(.u)..-oP-’(.u)piP~F. Hence j p = n or 0, so j = n / p or j = 0; also . u c r ( . ~ ) ~ ~ ~ cFr,~implying - ~ ( x ) ~i = n / p or i = 0. But this gives only p z pairs ( i ,j ) , so, as we claimed, p3 was too many. Q E D Noncrossed Product Theorems Having done virtually nothing so far, we are ready nevertheless to prove most of Amitsur’s noncrossed product theorems. (Amitsur mostly). Suppose q5 is ajield with characTheorem 3.3.12 teristic relatiiiely prime to 11. (i) If’ U D ( 4 , n ) has a subjield Galois over the center with Galois group H , then H is a direct sum of cyclic groups ofprime order. (ii) l f UD(q5,n ) has a subfield Galois oiler the center of’degree k, theii k cannot haue a cubic,factor ( > 1 ). I n particular, if I I i s diuisible bj1 a cube (> I ) , rhen CJD(q5.17)is not a crossed product. Proof. (i) follows from Proposition 3.2.10 and Corollary 3.3.10, where F = q5(q), a primitive nth root of I . (ii) follows from (i), Proposition 32.10, and Proposition 3.3.1 1. Q E D Division Algebras over Fields Having a Complete, Discrete Valuation Actually, Theorem 3.3.12 can be improved a bit, using a considerable deeper analysis of field theory, which we shall go into because of the importance of the non-crossed product theorems. We require the theory of complete, discrete valuations. Definition 3.3.13. A discrete iduatioii of a field E is a group homomorphism 11 from the multiplicative (abelian) group E - (0; to the

192

CENTRAL SIMPLE ALGEBRAS

[Ch. 3

additive group 2, such that u(a+p) ?' max(u(a), up))for all u # - p in E . The valuation ririg of u is { 0} u { a E Elo(a) < 01, and is denoted E,. We let c, denoteanelement ofE,,suchthatu(c, )ismaximal negative. WriteE,,forthefield E,/Jac(E,). Remark 3.3.14. For all a in E u(cE)lu(a).Thus, for any natural number k we may replace u by the valuation u' given by u'(a) = - ku(a)/u(c,),thereby having u(cE)= - k . Remark 3.3.15. E,, is a valuation ring with unique maximal ideal {0)u { a E El c(a) < 0). (Proof is left to the reader.) Moreover, every element of E , can be written in the form uck for suitable u E Unit(E,,) and i 2 0. (Proof left to reader.)

There is a very nice converse to Remark 3.3.15 (cf. Exercise 3); consequently, a n y valuation ring of a discrete valuation will be called a discrete ualuatiori ring. It is easy to see that the completion (cf. $1.10) of a discrete valuation ring is a discrete valuation ring; we shall call a discrete valuation complcte if its valuation ring is complete. Example 3.3.16. Let C be a (commutative) principal ideal domain and let E be the field of fractions of C. Fix a prime element p of C .Then every element ofE-(0) hastheformp'c,c;',whereiE ~andc,,c,arenotdivisiblebyp.We define u(p'c,c; ') = - i, a discrete valuation. The completion of the valuation ring of E will be called the p-udization of C . Example 3.3.17. In Example 3.3.16, let C = Z and p be a prime number. The p-adization of Z is called the ring qfp-adic integers, and is quite well known; the field of fractions is called the ring of p-adic numbers. Example 3.3.18. In Example 3.3.16, let C = K [ A ] , where K is a field, and let p = 1.The p-adization of C is denoted K[[A]]. Example 3.3.19. Let us examine Example 3.3.18 in more detail. and slightly greater generality. Given a division ring D, define the ring of.forrna1 power series D' to be the set of formal expressions x:=odiAi, rli in D, subject odill'+~:i"0,,di2;1i = )3?,(di, +di2)Ai and multiplication to addition ( ~ : i " = , d i , l i ) ( Z : = , , d j 2= ~j) (~i+j=kdildj2)Ak.Clearly D'isaring,and has a submonoid S = f d i 2 E D'ld, # 01.

xi".:

xrzo

In the above notation, we can identify D[L] G D' by sending X : f = , d i l i when D is a field, we have D' 3 D[[A]:] E Db. We claim that D' = DI,,which will yield equality at each stage. In fact, the proof is applicable ,fbr any division ring D. In order to d o this, we need only show each element s of S is invertible in D'. Write

-,ZzodiA',where di= 0 for all i > t . Consequently,

$3.3.1

The Generic Division Rings

193

s = Z ~ o d , 2and ‘ s1 = 1 - d ; ’ s = ~ ~ o d , Awhere 1 , do = 0 and d, = - d ; ’ d , for all i 2 1. One can carry out formally the infinite sum 1 +s, +sf + . - . = s2 for some s2 in S, because the “coefficient” of each A‘ is a sum of a finite number ofterms. Thenone hass,d;’s = (l+s,+s:+ ...)( I - s , ) = 1. (This last assertion is clear, but not obvious!) Likewise s has a left inverse, and we have proved the claim. This justifies writing the ring of formal power series of D as D[[A]]. Let S’ = {A’li b 0) in D[[2]]. We call D[[L]],. the ring of Laurent series ouer D, written D ( ( 2 ) ) . Remark 3.3.20. For any division ring D, D[[A]] is a domain and D ( ( 2 ) )is a division ring. Consequently, D ( ( A ) )is the ring of central quotients of D[[2]]. If F is a field then F ( ( I ) ) is a field with a complete, discrete valuation. (All straightforward, from Examples 3.3.18 and 3.3.19.)

One of the division ring examples used in Amitsur’s noncrossed product theorem was a cyclic division ring whose center is the field of p-adic numbers. (This treatment was used by Jacobson [75B], for example.) In order to be more characteristic free, we shall work instead with a field of Laurent series, but the point of course is that we have a complete, discrete valuation ring for the center. Risman [77b] has obtained precise information by examining the ramifications of the situation, and we shall present Risman’s results, which (in certain situations) actually give the best noncrossed product results now available. We start by looking a bit closer at discrete valuations. The theory of discrete valuations appears in many books. Its exposition here would be a considerable diversion, so, rather than develop the necessary theory, I shall refer to one of the standard texts (Weiss [63B, Chapters 2 and 31). The reader may also consult Jacobson [75B, pp. 104-1091, for a skilful treatment of many of the needed results, modulo some of the theory (such as Hensel’s lemma). Definition 3.3.21. Suppose u is a discrete valuation of a field E . Let GE = f u ( a ) ( aE E ) , a cyclic subgroup of .Z generated by u(cE).

We shall carry the following conventions through this discussion of discrete valuations: F has a complete, discrete valuation u such that u(c,) = - n, and E is a field extension of F with [ E :F ] = n. We use repeatedly the following result. [63B, 2-2-10 and 2-1-21, There is Q unique, discrete ualuatioti Weiss of E whose restriction to F is u, and t h i s ualuatiori is complete. Now clearly G, is a subgroup of G E .Moreover, F , E E , and by Remark 3.3.15 Jac(F,) E Jac(E,), so we can view F , E J!?,. Define the ramification index e ( E / F )to be the index of G, in G,, and the residue class d e g r e e f ( E / F ) to be [E,:F,]. Note that e = u(cF)/u(cE) = -n/u(cE).

194

CENTRAL SIMPLE ALGEBRAS

Weiss

[63B, 2-3-21. n

=

[Ch. 3

e(E/F),f(E/F).

We say E/F is unramijed if e(E/F) = 1 and E, is a separable extension of F , ; in this case (Weiss [63B, 3-2-71) E, is a separable extension of F,. Now we consider more than one extension at a time. If L , and L , are subfields of a field L, we write L , L , for the subfield of L generated by L , and L,, called the compositum o f L , and L , in L . Remark 3.3.22. [f F , f ( E I F )= f ( E / E , ) f ( E , / F ) .

c El

E E then e ( E / F )= e(E/El)e(E,/F) and

Weiss [63R, 3-2-10]. There is a unique maximal subjield El of E such that E,/F is unramijied. W e call E l the inertiafield OfE (ooer F ) . Remark 3.3.23. If K is an F-field contained in the inertial field of E, then K / F is unramified. (Indeed, K,,/F,. is separable, and by Remark 3.3.22 e ( K / F )divides 1 , and is thus 1.) Corollary 3.3.24. I f L , and L , are unramiJied extensions of F and heir compositum L , L, exists in some F$eld L, then L , L,/F is unramijied. Proof.

so L , L ,

Let K be the inertia field of L over F . Then L , E K and L , E K , thus L,L, is unramified. QED

E K , and

We can use these results decisively in a certain situation. Proposition 3.3.25. Suppose L , is a Jinite-dimensional Galois e.xtension of F , and L , is an arbitrary extension of F. Let K be a subjield of L,, maximal with respect to K isomorphic to a subjeld of L,. Then there is a,field L 2 L , containing an isomorphic copy L', of L,, such that L,Z, 2 L, OKEl.

Proof. Clearly L , is Galois over K . Write L , = K(ol), and let p ( 1 ) (in K [ A ] )be the minimal (irreducible) polynomial of a. We view p in L2[A]by identifying K with a subfield of L,. Then for an irreducible factor p1 of p in L2[;I] we see that L,[I]/L,[A]p, is a field that contains an isomorphic copy of L,, so the coeficients of p , are contained in an extension of K (in I,,) isomorphic to a subfield of L,. By assumption on K , all the coefficients of p 1 must be in K , implying p , = p. Now take L to be the field L2[A]/L2[A]P

L2

OK ( K [ L ] / K [ A ] p = ) L 2 OK L l ,

and we are done. QED Proposition 3.3.26. Suppose E is a cyclic (Galois) extension O J F whose Galois group is generated by 0,and e(E/F) = 1. Then (E,a,c,) is a division ring of exponent n.

93.3.1

The Generic Division Rings

195

Proof. In view of Corollary 3.1.44 it suffices to prove that if CLE N,(E) then nli. But if c: = N J x ) for x in E , then u(x)n = u ( N , ( x ) )= v(c3) = -in, so v ( x ) = -i. But e ( E / F ) = 1, so n l u ( x ) . Hence n l i . QED

Theorem 3.3.27. Suppose E is an unramified, Galois extension of F, whose Galois group is generated by rs, let A = ( E , o , c , ) , and let q be the characteristic of F , . (i) Any F-subjeld of A which is unramified over F, is isomorphic to a subfield of E. (ii) Suppose q / n and F does not contain nth roots of 1 other than k l . Then every F-subfield L' (of A ) abelian Galois over F has a subfield L, such that [E:L] < 2 and L is isomorphic to a subfield of E . Proof. (i) Let L be an F-subfield of A which is unramified over F ; using Proposition 3.3.25, for a maximal subfield K of E isomorphic to a subfield of L we can write the compositum E' of L and E as L OKE . Now let t = [ K :F] and let A , = C,,,(K).By Theorem 3.1.8, index(A,) = n/t. Moreover, E is a Galois extension of K whose Galois group is (d),and we can write A , 2 ( E , d , c F ) . Now writing z for 1 @a', we have ( E ' , z, c F )= L O K( E , a', c F )= L O KA , , which, for i = [L:K], has index dividing n/t,. On the other hand, E' is an unramified extension of K , SO by Proposition 3.3.26 [E':L] divides n/ti. But E E E', SO n / t = [E:K] < [E':K] d n/t, implying E = E'; therefore L is isomorphic to a subfield of E' = E .

(ii) Let L be the inertial subfield of L!. By part (i), L is isomorphic to a subfield of E, so it suffices to prove [L':L]< 2. Since the characteristic of F is relatively prime to n, we have by Weiss [63B, 3-4-31 L' = L(x) for some x in L: such that xk E L, where k = e(L' :L). If k < 2 then [L' :L] 6 2 and we are done, so assume k > 2. We shall then arrive at a contradiction to the assumption F has no primitive kth roots of 1. L' is Galois over L, so clearly L: has a primitive kth root q of 1. Then __ v ( q ) k = u(1) = 0, so ~ ( q=) 0, and i j ~ F ( q ) , .Applying Hensel's lemma (cf. Weiss [63B, 2-2-11) to the cyclotomic polynomial (which is separable because ,Ik- 1 is separable over we see F(q) is an unramified extension of F, implying q EL. Now let a be the automorphism of L' over L such that a(x) = qx, and let z be an automorphism of L' over F such that t ( q ) = q i # q for suitable i. Then u ( z ( x ) )= u(x), implying t ( x ) = xu for some u E L'with u(u) = 0; now m ( x ) = z(qx) = $xu and ~ ( x=)@(xu)= qxa(u); since Gal(C/F) is abelian, we see qixu = qxa(u), implying cr(u)u-l = qi-'. Now a restricts to an automorphism of E , over F,, which induces naturally an automorphisrn (3 of over with a(C)ii-' = qi-' # 1. On the other hand, by Weiss [63B, 3-53] 5 = 1, contrary to 5 ( C ) # U. Thus k < 2 after all. QED

E),

E,

196

CENTRAL SIMPLE ALGEBRAS

[Ch. 3

Now let 4 be a field of characteristic 4, with q f n , such that 4 contains no roots of 1 other than f 1. Let E , = 4(xl,..., x,,), where the .xi are commuting indeterminants over 4, and let F , be the fixed subfield under the automorphism r~ of E given by a(x,) = x2, a(xz) = x3,. . . ,a(x,,) = xi. Put E = E,[[d]] and F = F,[[1]]. Clearly the hypotheses of Theorem 3.3.27 and Proposition 3.3.26 are satisfied, so we have proved Theorem 3.3.28. Let 4 be a fteld of ‘characteristic 4, with q l n , such that 4 contains no roots of 1 other than f 1. Then there is a division algebra of degree n such that any subfield which is abelian Galois over the center has Galois group which is either cyclic or has a cyclic subgroup of index 2.

More Noncrossed Product Theorems Theorem 3.3.29. For 4 as in Theorem 3.3.28, U D ( 4 ,n ) does not have a subfteld Galois over the center ofdimension divisible by an odd square > 1. Proof. The Galois group is a direct product of cyclic groups of prime order by Theorem 3.3.12, so we conclude with Proposition 3.2.10 and Theorem 3.3.28. QED

For emphasis, we put together Theorems 3.3.29 and 3.3.12 in the case

4 = Q. (Amitsur). U D ( Q , n) is nor a crossed product $ 8 l n Theorem 3.3.30. or ifm’ln for m odd > 1. We have not treated the case where n is divisible by the characteristic of

4, which has been solved in very pretty papers of Amitsur-Saltman [78] and Saltman [78b], in the sense that the conclusion of Theorem 3.3.12 holds, regardless of characteristic. Saltman [78a] also extended these results when the exponent of the division ring is divisible by a cube (or by an odd square in certain cases). Incidentally, the case n = 5 is still wide open. The Center of the Ring of Generic Matrices One of the most important open questions in PI-theory is whether the center of the affine generic matrix algebra F,{ Yl, ..., yk} is purely transcendental over its center. If this were known to be true in general, then a host of new theorems about division algebras would emerge by the use of Bloch [74]. Unfortunately this is only known for n = 2, 3: and 4, the latter case having only been solved very recently by Formanek. Part of Fonnanek’s method works for all n ; we shall present that theory here

$3.3.1

The Generic Division Rings

197

(which is based on work by Procesi) and leave the special cases for the exercises. We turn to the paragraph preceding Remark 1.10.26 and use that notation, working inside hfn(&). is defined in Theorem 1.10.28. ,~ over F by Theorem 3.3.31. (i) K k , nis the subfield o ~ F ,generated all
+

all of which are in K ; thus Kk,,,E K . It remains to show that the generators Kk,,,,so we see immediately of K are all in Kk,,,.Each (!,!)E Kk,nand t\:)tif)~ K~,~. that for all i , j 2 2, ~ \ : ) < ! Y ) ~ $ ) EThus t i( lUi I2) ~ (1213 UF).

. . ti:;!

=

(t\:lt!tl)"'(tl'i!tf,:')-'(t\'!t!3!t1~1)EKk,n,

as desired, proving K = Kk,n. (ii) Clearly n takes cycles to cycles, so K is invariant under each n in Sym(n). Working inside Mn(Fk,,,), identify F k , , , with the set of scalar matrices, as usual, and let Sym(n) act on a matrix via its action on the ,ajje..) = z(tlij)eij.Any element tl of Fk,,, can be entries, i.e., Z(Z;+~= '2 viewed as f i(Y;, Y2,...,&)f,( Y;, Y,, ...,&)- I, where f,,f' are Fn{Y;,Y,, ..., &)-central polynomials. Now look at the entries of o! as rational functions p i j in the entries of Y;, Y,,. . ., &; thus p l l = p , , = ... = p,,,, and p i j = 0 for i # j. One sees immediately that u must be in K , and, = x ; = , p i i e i i= o!, so moreover, for ncSym(n), n(a) = Cy=lpni,Rieii tl E KSym("), the fixed field of K under Sym(n). This proves Fk,,,E KSym(,,). Now we let K , = Fk,,,(<\'?,.. .,t:!,)) and consider R , = R K , E M , ( K ) . First note that n y L f ( Y ; ' - t i f ' ) = n ~ ~ ~ ( < ! ! , ) - t $ ) ) e , ,implying ,,, en,,€R , ; likewise each eiiE R , . Hence each eiiY;'ejjER , . Also R , is a prime finite dimensional K ,-algebra, implying each regular element is invertible, from which it is a simple matter to show that each eijE R , ; hence K E R , , so R , = M,,(K),implying K = K , .

x;,j=

198

[Ch. 3

C E N T R A L SIMPLE ALGEBRAS

(This also proves that K , splits R , and thus, by conjugation by see that K , splits F,{ Y;, Y2, . . . , yk}.) Moreover, since all <:!' are roots of the characteristic polynomial of Y, [viewed in M,(Fk,,J], whose coefficients lie in F,,,, we see that K is algebraic over F,,, (and in fact must be Galois), and [ K :FkJ < n!. But - K. and [ K :KSy"'(")]= n !. Thus Fk3,#= KSYm("). Fk., K S ? " ' ( n ) c (iii) Since K is algebraic over F,,,, we may apply Theorem 1.10.28 to get trdeg(F,.,/F) = trcleg(K/F) = (k- 1)n2+ 1. (iv) By checking transcendence degrees, it suffices to find ( k - 2 ) ) ~ ~ elements generating Fk,, (as a field) over F2,,. Well, let t = n', and suppose x,,. ..,x, are an F,,,-basis of F,(Y,, Y2).Note that x,,. . .,xt are also an Fk,nbasis of F,( Y , ,. . . , yk), viewing F,( Y,, Y2) c F,( Yl, . . . , yk). Let F be the F2,ngenerated by [tr(Y,xi)13d i < k, 1 < j d t ) . It is enough to subfield of Fkbn show F = Fk.,,,in biew, of part (iii). Let D = F,(Y,, Y2)F[G F,,(Y1.. .., Yk)], a division ring with center F. For any i, we have = ,ctijxi for suitable Now each aij in F,,,. Hence for each u, tr(xx,) = ~>=,ctijtr(xj.xu). tr( yix,) E F and tr(.ujx,) E F2., E F. So we have n2 equations (one for each . also know that the value of u) for the n2 "unknowns" c t i I , . . . , a i r We equations are consistent (because they have a solution in F,,,,) and so, by Cramer's rule, the solution is in F2.,,. Thus each c l l j E F2,,,,which means X E D . Hence D = F,(Y,,. .., q),so F = Z(D) = Z(F,(Y,...., Y,)) = FL.,, as desired. QED a

= e l , +xr=2<\:'eii,we

x:=

In view of part fiv) above, the entire transcendence question rests on the case k = 2. Example 3.3.32. For 17 = 2, k = 2,

and Sym(2) acts by permuting

(y:

with ($\, i

=

1,2; hence, as is easily seen,

which is purely transcendental over F. Thus the case n Example 3.3.33. K 2 , 3= F(t',',I C(1) 9

-229

=2

is solved.

(2) (2) ( 2 ) (2) ( 2 ) v(2) ( 2 ) ( 2 ) ( 2 ) 2 ( 2 ) 22,533~412521,51353115235~2,51~(23~31).

e ( 2 ) 5'2) 9

11,

Formanek computed the fixed subfield, and we shall describe his method in Exercise 4. For n 2 4, one presently needs to use other "generic" kinds of division algebras, in order to apply Bloch's theorem cited above (cf. Snider [78P]).

Exercises

Ch. 31

199

EXERCl SES

43.1 1. IfR = M,(D)foradivisionringD,and A,B aresimplefinitedimensionalZ(R)-subalgebras of R, then every isomorphism from A to B can be extended t o an inner automorphism of R. ( H i n t : Modify Theorem 3.1.2.) 2. (Jacobson) I f D is a division ring algebraic over a finite field F, then D is commutative. [Hint:Ifd E D - F , thenF(d)isGaloisover F ; Exercise 1 providesafinitenoncommutativedivision ring, which is a contradiction.] Now prove that in any ring R, iffor each r in R there is suitable n(r) > 1 in H + such that r""' = r, then R is commutative. [Hirit: Show Jac(R) = 0, and all primitive images are fields.] This result has led t o many theorems about rings with conditions resembling polynomial identities; fortherootsofthissubject,seeHerstein[68B,Chapter 3 with its references.] 3. For any factor set 8 :G x G + K -{O} there is a n associated factor set 8':G x G + K - {0}such that P(u,e ) = P(e, u) = I for all u in G. 4. If K is a field extension of F then - O F K yields a homomorphism Br(F) + Br(K) with kernel Br,(F). 5. Suppose R is central simple of index p'q, where p is a prime not dividing q. Then there is a chain of fields K Oc K , c ... c K j , each K i Galois over K i + l with cyclic Galois group of order p, such that p[[K,:Z(R)] and index(R@,,,,Ki) = In fact, K j can be taken t o be a separable splitting subfield of R @ Z , H ) K O[Extensioe . hint: Take K as in Theorem 3.1.21; replacing R by RaZ,,)K, we may assume deg(R) = p j . Now let F = Z(R), R = M,(D), and take a maximal separable subfield L of D, contained in a Galois extension L'. Let K O be the fixed subfield of L' under a p-Sylow subgroup. D a F K , is a division ring, so K j = L B F K ois a field. Finish using easy properties of p-groups.] 6. Prove Theorem 3.1.39, using only Exercise 5 and properties of cyclic algebras. 7. Suppose IGI < co. For all n > 0, IGl.f"(G,H) = 0. Interpret this for the Brauer group. 8. For any field F, M J F ) has a matrix with n distinct characteristic values. (Hint: Take the companion matrix of an irreducible polynomial.) 9. (Amitsur) Here is another method of obtaining splitting subfields. Let t = n - n and suppose ,f(X,, ..., X , ) is t-normal. If .s E M , ( F ) with characteristic values pl,...,fin,write D(r) = r l i , j ( p i - p j ) 2 . Then for all .xl ,...,. Y, in M,(F).f([.u,.u,], ..., [.Y,.Y,],.Y,+~ ,..., -ud) = D ( x ) / ' ( x , ,. _ ,xd), . ( H i n t : Assume Y is diagonal.) For every simple ring R of degree n gn([X,+,, Xl], . . , , [ X d + , ,X,], X,+ , . ,X,) is R-central. [ H i n t : Using Corollary 3.1.5, reduce to thecaseR = M,(F).Then R hasanelement xwith D(.u) # 0,soxgeneratesamaximalcommutative subring C of R, which is separable when C is a field.] 10. For every Azumaya algebra R of rank nz. the polynomial g of Exercise 9 is R-central, and IEg(R)+. 11. If ( R, * ) is simple and (*) is of the second kind, then either (R,*) is special or R is a matrix algebra over a finite field with automorphism of degree 2. 12. (Rowen [75a]) Determine the complete set of inclusions between . f ( M , ( F ) , t ) and .F(M*m(F),S). 13. (Scharlau [75]) Suppose D is a division PI-ring with antiautomorphism ( J ) fixing Z ( D ) . D has an involution of the first kind iffJ2 = [ ( x ) for some x in D with x.9 = I.

,,,

$3.2 Assume through Exercise 7 that F is a field with f. 1. Given a, b in F - {0}, let the symbol (a, b ; F ) denote the quaternion F-algebra F+F.u + F y + F . s y , where .uy = - y ~ .Y* , = a, and y2 = b. Show (u, b ; F ) is a quaternion F-algebra

200

CENTRAL SIMPLE ALGEBRAS

[Ch. 3

determined up to isomorphism, and (a. b :F ) 2 (b,a ;F). Also, for c E F, (a, b ; F ) O F(c,b ; F ) 2 M 2 ( F ) @ F (ac,b ; F). 2. If ( a , , b , ; F ) : ( a 2 , b 2 ; Ff) M , ( F ) , then, for suitable a e F - { O } , ( a , , b , ; F )o ( a , b , ; F ) 4c ( a , b , : F ) r ( a , , b , ; F ) . ( H i n t : Writing D = F + F x , + F v l +F.u,y, = F + F x , t Fy, +Fx,y, with xr = ai and y' = b , replace x i by [yl. y,].) 3. Suppose R is simple of degree 4, exponent 2, F = Z(R),and F ( x , , x , ) is a maximal subfield ofR witha, = .xZ~F.Then,forsuitablebi in F-{0}, R P ( P , . ~ , ; F ) O ~ ( ~ , , ~ ~ ; F ) . 4. (Albert) Suppose A is a quaternion C-algebra, and C is a quadratic extension of F. A has an involution (*)with F = Z(A,*),iffA z Q@,C for some quaternion F-algebra Q; in this case, we can take Q invariant under (*). 5. (Modified from Tignol [78]) Suppose A is simple of degree 4,exponent 2, and has an involution (*) of the second kind. Then, for K = Z(A) and F = Z(A,*), there are quaternion Falgebras Q1,Q2, and Q:,, such that M,(A) P Q cOFQ2@FQj@FK. [Extensive hint: Assume A is a division ring. Then, by Lemma 3.2.23 and Exercise 3, A = (a,,b, ; K ) m K ( a , , b , : K )for suitable ai # 0 in F. Let fii = bib: E F . By Theorem 3.1.68, (a,,& ; F ) 2 (a,,/&; FYp P (a,,B,;F).ByExercise?.forsomea # Oin F , M , ( F ) 2 (ala,/3,; F ) ( a 2 a . p 2 ; F )o (a,,?,/&:F). Now M , ( A ) : ( a , a , b , : K ) @ ( a , a , b , ; K ) O ( a , b , b , ; K ) , each of which has an involution of second kind, so apply Exercise 4.1 6. (Tignol[78]) If A is simple of degree 8, exponent 2, then M,(A) is a tensor product of quaternion subalgebras (Hint: Take the centralizer of a 2-central element and apply Exercise 5.) 7. Suppose A is as in Exercise 6. If A is not a tensor product of quaternion subalgebras. then M , ( A ) is a crossed product with Galois group Z,x Z,x Z, x L,. 8. (Converse to Proposition 3.1.67) Suppose D is a division ring with symplectic involution (*), K is a maximal subfield Galois over Z ( D ) , and (*) induces an automorphism u of K. Then D # I . 9. IfDisadivision ring,eachelement algebraic over Z(D)ofdegree n, then D is PI ofdegree n. Conclude Frobenius' theorem ; each algebraic noncommutative Rdivision algebra is (-1, -1;R). 10. Assume i e 4 and D is a division algebra. A set S = { d ] , .. .,d,} of 2-central elements is "good" if for each i # j didj = f d j d i and S n C,(di) # S n C,(d,). For deg(D) = 2" D is a tensor product of quaternion subalgebras iff D has a good set of 4"'- 1 elements. #.( Y, Y') is a tensor product of quaternion subalgebras iff every division algebra of degree n and exponent 2 is a tensor product of quaternion subalgebras. 11. (Rowen [74c]) Let R = Z,{Y) and C = Z(R). R O c R is not prime and not torsionfree over C. (Hint: Letting $ : R + %,(Y) be the canonical injection, and for

show0 # r e k e r ( & O $ ) Indeed,ifr = Othenbyanargumentonhomogeneity[Y,, YL] @ [ Y , ' , Y,] = 0, which is absurd; on the other hand, [ Y:, Y,] = (tr Y,)[Y,, Y,].) 12. If #"{ Y) has an n-central element, then it has a completely homogeneous n-central element. 13. Call a ring R ordered if R has a total order ( > )such that for any rl > 0, r2 > 0, and r3 < 0, we have rlr2 > 0, r , + r 2 > 0, and - r3 > 0. Then R is a domain of characteristic 0, and - 1 < 0. Prove Albert's theorem, that every ordered division ring is a field. [Hint: Otherwise take d # 0 of trace 0. By Wedderburn's method, 0 = tr(d) is a sum of conjugates of d, contrary to d > 0 or d i0.1 Hence every ordered PI-ring is a field. 14. (Leron) If F ik a field without nth roots of 1 other than 1, then L'D(F,n) has no multilinear n-central elements. 15. Write out the maximal subfield of &(Y. Y") given in Theorem 3.2.40.

Ch. 31

Exercises

20 1

93.3 I. If u is an automorphism of order

17

of a field K and u(y) = y e K , then

( K , ~ ,: YK[P;u]/(P"-Y-'). ) What is the analogue of Theorem 3.3.6 for R prime PI? 3. If C is a valuation ring of a field F and there is some c in C such that every element of C has the form uc' for suitable t( E Unit(C) and i 2 0, then F has a discrete valuation given by u ( ( ~ ~ c ~ ) ( u ~=cj-i. j-~) 4. (Formanek [78P]) Using Example 3.3.33 and the theory of elementary symmetric functions (cf. Lang [65B, pp. 132-1341, reduce the question of transcendence of F3,kover F to the following assertion: Let tl, t2,t3,)tbe commuting indeterminates over F and let L be the fixed subfield of F(t1,t2,t3,y) under the action of Sym(3) acting on the subscripts of ti,also with n ( y ) = y for A = (123) and n ( y ) =t,t2[3)i-1for A = (12); then L is purely transcendental over F. To prove this assertion, let a = 5 1 + 5 2 + 5 3 r b = 5 1 5 ~ + 5 2 5 3 + 5 ~ 5 1 , c = (1t253, d =y+cfl, A = ( ~ l - 5 2 ) ( 5 2 - 5 ~ ) ( 5 J - t 1 ) land e = ( ~ - t , t ~ t ; ~ ~ - ~Then )A. e2 = (d2 -4c)A'; using this equality, solve for a in terms of b , = b i d , c1 = c/a3, d , = d/a, and el = e/a4, which are thus a transcendence base for L over F . 5. With notation as in Theorem 3.3.6, if G has exponent 2 then D has an involution of the first kind (cf. Exercise 3.2.10). 2.

CHAPTER 4

EXTENSIONS OF PI-RINGS One of the famous problems in ring theory is Kurosch’s problem, If R is an affine algebraic algebra over a field F , then is [ R :F ] finite? This is false in general, by a theorem of Golod [64]-Shafarevich; modern PI-theory came into existence when Kaplansky [50] solved Kurosch’s problem in the affirmative (using reductions due to Jacobson) in case R is PI. Shortly thereafter Shirshov [57a, b, c] proved the following generalization, through combinatorial analysis: If R is a PI-ring which is finitely generated and integral over Z ( R ) ,then [ R : Z ( R ) ]< cx). Incredibly, Shirshov’s result was completely unknown in the West for about 15 years, because of its obscure title (as well as its being in Russian), but has now become the key to the very pretty theory of integral PI-ring extensions. We shall give this theory in greater generality, for “integral extensions” of PI-rings. This extra complication takes some effort to digest, but is well worthwhile, because it leads to a very clear method of analyzing finitely generated algebras, in particular affine algebras. Our approach follows a very rich (but condensed) paper of Schelter [78], with some simplifications due to the use of the Capelli polynomial.

$4.1. integral and Algebraic Extensions of PI-Rings

The object of this section is to develop a little theory of “integral” and “algebraic” elements, generalizing results of $1.9, for use in the study of PIring extensions. Throughout this section, assume W is a ring and R is a W ring. Our first step is to define “integral” and “algebraic” elements of R (over W ) . The obvious way is to replace polynomials by generalized polynomials (with coefficients in W ) . The ensuing definition, and most of the results of this section, are due to Schelter [76a, 781. Definition 4.1.1. An element r of R is W-algebraic if there is a generalized polynomial f ( X , ) = ~ ~ = o f i ( Xfor, )suitable t such that eachf; is a generalized monomial,f;(r) # 0, and f ( r ) = 0. Ifl; = X i we say r is W integral (the smallest such t is the degree of r). R is W-inteyral (resp. W -

202

r $4.1.I

Integral and Algebraic Extensions

203

algebraic) if every element of R is W-integral (resp. W-algebraic). If every element of R is W-integral of degree < t , we say R is integral of bounded degree. If r E C,( W) is W-algebraic (resp. W-integral) then clearly r satisfies an equation C:=owiri = 0 such that w,r' # 0 (resp. w,= 1). Definition 4.1.2. Given elements rl ,...., r, of R , let W{rl ,..., r,} denote the subring of R generated by r l , . . . ,r,. Call R a finitely generated extension of W if R = W{rl,. . . ,r,) with ri E C,( W) for each i. R is a finitely spanned extension of W if R = Wri for suitable u, with each ri E C,( W).

xy=

Throughout this chapter, the reader may prefer to treat the special case W E Z ( R ) ,which is still of great interest.

Integral PI-Extensions We now improve some results of $1.9 (with independent proof) in the following situation: Definition 4.1.3. PI-extension of W. Theorem 4.1.4,

A PI-ring which is an extension of W is called a

GU kolds,for every W-integral PI-extension R of W.

Proof. Let P I L P , be prime ideals of W, and let Q1 be a prime ideal of R with Q 1 n W = P , . Then take Q2 2 Q1, maximal with respect to Q2 n W E P,. Let R = R/Q2 and W = W/(Q, n W) 2 ( W +Q2)/Q2, a subring of R . Thus R is an integral PI-extension of W.Moreover, R is prime. We claim that Q, n W = P , or, equivalently, P , = 0, where denotes the canonical homomorphic image. Otherwise there is a nonzero element Z in P , n Z ( w ) , since W is a prime PI-ring. Let A = Q, + z R . Obviously A d R and A n W d W . Suppose a E A n W. Then a = Z = W for suitable r in R and W in W. Now r satisfies some equation r ' + x : ; A , h ( f ) = 0 for suitable generalized i. Thus 0 = ( Z F ~ + ~ ~ Z & F ~ ( Zimplying F), monomials ,fi(X,) having degree a'= (F)'= - ~ ~ ; ~ z ' - % ( w ) so ~ Pu'EP,. ,, Thus ( ( A n W)+P,)/P, is a nil ideal of the prime PI-ring W/P,, implying A n W c P,, that is, (Q, + z R ) n W E P , . By hypothesis on Q,, we get z E Q,, implying Z = 0, contrary to the choice of z. Hence P , = 0, so Q z n W = P,, as desired. QED Corollary 4.1.5.

Proof.

LO koldsjor every W-integral PI-extension of W.

Immediate from Corollary 1.9.7. QED

204

[Ch. 4

EXTENSIONS OF PI-RINGS

Lemma 4.1.6. Suppose R is prime, W G R, and ~ E Z ( R ) - { O )is Walgebraic. Then N' n zR # 0. Proof. Write C:=,,wizi = 0, with t minimal such that w,zf # 0. Then w,, # 0 because otherwise 0 = z x : = lwizi-' = z R ~ ~ Z ~ w , +since , z ' ; R is prime

and t is minimal, we would get z = 0. But then w,,

= - Z;:=,

wiz' E W n zR. QED

Proposition 4.1.7. Suppose R is a prime PI-ring and Z ( R ) is algebraic over W z R. I f W nJac(R) = 0, then Jac(R) = 0. Proof. By Lemma 4.1.6, we cannot have z # 0 in Z(R) n Jac(R), so Z(R)n Jac(R) = 0, implying Jac(R) = 0. QED

Theorem 4.1.8.

!fR is an integral PI-extension of' W, then INC holds.

Proof. Suppose PI c P2 are prime ideals of R lying over the same prime ideal P of W . Then, passing to R = RIP;, we see that has a nonzero element r of Z(R).By Lemma 4.1.6,O # W n G W n Suppose 0 # W E for w i n W. Then w E P2n W = P , contrary to W # 0. QED

p; E.

p;

So integral PI-extensions satisfy the consequences of Proposition 1.9.14. Bergman [74aP] has proved, without any PI-assumption, that finitely spanned extensions satisfy GU, cf. Exercises 2-6. One can also describe the Jacobson radical of PI-extensions. Theorem 4.1.9. Suppose R is an algebraic PI-extension of W satisfying GU. (i) Jac(W) = W n Jac(R). (ii) If R is prime and Jac(W) = 0, then Jac(R) = 0. Proof. (i) Since all primitive ideals of a PI-ring are maximal, the assertion follows immediately from Corollary 1.9.9. (ii) W nJac(R) = 0 by (i), so we are done by Proposition 4.1.7. QED

54.2. Formal Words and Shirshov's Solution to the Kurosch Problem

This section will deal with "words" (cf. Definition 1.1.1). The motivating idea is that combinatorial results about words may be translated to properties of free algebras, to yield PI-theorems. This will be the central theme of Chapter 6 ; for the present, we shall obtain some straightforward properties about words which will yield an affirmative solution to Kurosch's problem for PI-rings. Definition 4.2.1.

A word w is l-initial if w has the form (1")w' for

$4.2.1

Formal Words and Shirshov's Solution

205

some u B 0, where 1 does not occur in w'. A partition of w is a way of writing w = w1w2...w,, for suitable m, such that each wj is 1-initial; the partition of w is minimal if rn is as small as possible. Remark 4.2.2. Every word has a partition and a unique minimal partition. For example, the (10,5)-word (5421132134) has a partition (54) (2) (1) (132) (13) (4), and the minimal partition is (542) (1 132) (134).

We order our words partially, according to Definition 1.1.3. Definition 4.2.3. A word w is d-decomposible with d-decomposition w1 ... wd if w n l ... wnd < w for each # 1 in Sym(d). Remark 4.2.4. If w = wlw' and w' has (d - 1)-decomposition w2".wd with each w j < wl, then w has d-decomposition w1w2..'wd. In the presence of a polynomial identity of degree d , d-decomposition is very important, yielding a method of framing an inductive argument on the order given on words. To utilize this idea, we must now prove that there are an abundant number of decomposible words.

' x Z + --t Z ' such that There is a junction p : Z f x Z Lemma 4.2.5. for every k 2 p(t, u, d), each ( k , ?)-word has either (i) a subword of the form w; or (ii) a d-decomposible subword. Proof. Simultaneous induction on d and t. Since every word is 1decomposible, we may take P(t, u, 1 ) = 1 ; likewise 1" is a subword of every ( k , 1)-word for k 2 u, so we may take p( 1, u, d ) = u. Suppose inductively that we have p ( t ' , u , d - l ) for all t' in L', and also that we have p(t-l,u,d). Now set t' = ut'J(f-'*"+d) and (inductively)

p(t,u,d) = (p(t- l,u,d)+u- l)p(t',u,d- 1). Suppose w is a (k,t)-word not satisfying (i) or (ii); we shall prove that k < P(t, 4 4. Let w1 ... w, be the minimal partition of w, and write wi = luiGi,where 1 does not occur in G,.Then hi is a word in { 2,. . .,t } so we are done if any Gi has length 2/3(t- l,u,d); also, (i) holds if any ui 2 u. Thus we may assume for all i that 8, has length < / 3 ( t - l,u,d), and that u , < u. Call a 1-initial word admissible if it satisfies these two properties. Incidentally, each wi has length <(B(t-l,u,d)+u-l), so k d (p(t-l,u,d)+u-l)m. Clearly there are fewer than t' distinct admissible 1-initial words; ordering them lexicographically, assign to each 1-initial word u a corresponding number t l ( u ) , yielding an order-preserving bijection of {admissible 1-initial words} with the set { 1,. . . ,c'}. Now consider w' = a(w2)... a(w,), an (m - 1, t')-word. Suppose rn 3 p(t', u, d - 1). Then either (i) holds and we are done, or w'

206

[Ch. 4

EXTENSIONS OF PI-RINGS

has a ( d - l)-deconiposible subword a(w,)....(wj) for suitablej > i 2 2. In the latter case w i . - . w j has a decomposition w;’...wi, where each w: is a ...,w,} and thus starts with 1. But C3i-l does product of words from {w2, not start with 1, so w i - l > w$’ for allj; by Remark 4.2.4, h - l w i . . . wi is a ddecomposible subword of w. So either (i) or ( 1 1 ) hold unless m < P(t‘, u, d - l), implying

k < ( P ( t - - l , ~ , d ) + u - l ) P ( t ’ . u , d - l ) = B ( t , u , d ) . QED Condition (i) of Lemma 4.2.5 can be further improved. Lemma 4.2.6. l f w has length 3 d und w is not o f f h e f o r m (w’)jfor any j > 1, then wZdhas 4 d-decomposible subward. Proof. Write M ’ = (ili2...id)w‘,where w’ is a suitable subword (possibly blank). Define w1 = w and w, = (iuiu+l... id)w’(il. . ~ i . - ~ ) , for 2 < u 6 d. Now, for each u 6 d , w, # w ; otherwise we would see easily, by matching terms of w and w,,, that w has the form (i1i2...i,,- l ) j for suitable j , contrary to hypothesis. It follows immediately that w l , . . .,wd are all distinct (for if w, = wq with u < q, then w = wq+ -,, which we just saw was impossible). Thus, for some R in Sym(d), w n l > wn2- * * > W , ~ .But obviously each wi is a subword of w2. Thus, writing w2 = wIwiwi’, we have W2d =

wJk 1 (wn1 w L l

W h 2 ) ‘ ’ ’ ( w n ( d - 1 )w;(d- 1

(wndwid)*

,,, 5

Setting ui = w , ~ w ~ , w ~ , ~1 + i < d, and vd = WndWLdr we see immediately that v1 > v 2 > ..- > vd, proving u1 ... vd is a d-decomposible subword of w2‘. QED (Shirshov’s theorem). There is u juncrion P, where u 2 2d and all k 2 p ( t ,u, d ) , every (k, t)-word has either (i) a subword of the ,form wU,, with w,, of length < d , or (ii) a d-decomposible subword. Theorem 4.2.7

p : L + x Z+x Z+-+ Z+,such that for all

Proof.

Apply Lemma 4.2.6 to Lemma 4.2.5. QED

Suppose A c R are algebras and R is a finitely generated extension A{xl, ..., xf) of A, where each x i € C R ( A ) .(We are not assuming that A is commutative.) Although the x i are not indeterminates, we shall speak of a “monomial” in the xi to denote a term x i , ... xir in R, and will say that this term has length k and corresponds to the (k,t)-word ( i l . . . i k ). Note that there are tk monomials in the xi of length k, although some of them may be equal (in R ) by some fluke. We are ready for Shirshov’s solution to Kurosch’s problem.

R

Theorem 4.2.8 (Levitzki-Kaplansky-Shirshov-Schelter). Suppose = A { x l ,. . . ,x,} satisjies a polynomial identity f ( X l , .. . ,X d ) ,with each xi in

54.2.1

Formal Words and Shirshov's Solution

207

C , ( A ) , and every monomial in the x i of length < d is integral over A . Then R is a finite-dimensional leji (resp. right) A-module; in fact, letting fi be as in Theorem 4.2.7, and putting u = max{2d, degree of integrality of each x i , . * * xik of length < d } , we have R spanned by all monomials in the x i of length G B(t, u, 4. Proof. Let B = {monomials in the x i of length < f i ( t , u , d ) } . We shall prove the theorem by showing that any monomial in the x i is spanned (over A ) by elements of B. Well, consider a typical r = x i , ...xiL,corresponding to the (k, t)-word w = (il ... ik). To prove that r is spanned by B, we may assume inductively that every monomial in the x i of smaller length is spanned by B, and that every monomial in the x i of length k , corresponding to a smaller (k, t)-word, is spanned by B . Now if k < P(t, u,d), then X E B and we are done. Thus, assume k 3 P(c, u,d). Then w satisfies conclusion (i) or (ii) of Theorem 4.2.7. If (i) holds then w has some subword w:, where wo has length < d . But writing wo = (jl...J,) and ro = x j , . . . x j q , we have by hypothesis that rg = uirb for suitable ui in A . Now r = r'r:r'' for suitable r', r" that are monomials in the x i . Moreover, r'rir" = air'rbr"; each r'rbr'' has length < k and so is spanned by elements of B , by induction on k . Hence r is also spanned by elements of B. If (ii) holds, then w has some &decomposition w 1 ... wd. Letting r j be the monomial in the x i corresponding to w j , 1 < J ,< d, we write r = r'rl . . - r d r " for r', r" that are monomials in the x i . Now, we may assume f ( X , , . . .,x d ) = X I ... Xd+x,n~~ym(d)-((l)]ClnXnl . . . x n dThen . r = -xClnr'rnl *..Fndr".But each r'rnl ...rndr" is a monomial in the x i , corresponding a smaller ( k , t ) word and thus, by induction, is spanned by B. Hence r is spanned by B. QED

xY , d:

Corollary 4.2.9. I f R is a finitely generated, integral, PI-extension of C S Z ( R ) ,then R is ajnite-dimensional C-module.

Theorem 4.2.8 can also be used to give a positive answer to a special case of Burnside's problem for groups (cf. Exercise 1). The nil analog of Kurosch's problem was already done, very easily, in Proposition 1.6.34 (cf. Exercise 1.6.6), and Shirshov's extension has an easy, noncombinatorial analog given in Exercise 1.11.20. This can be used to obtain an alternate, noncombinatorial proof of Theorem 4.2.8, given by Procesi [73B, p. 1521, which is in fact shorter than the proof given here. Each proof has its own elegance, and Shirshov's proof serves as a good introduction to the method of applying combinatorics to PI-theory, typical of some of the best work of the Russian School (cf. Chapter 6).

208

EXTENSIONS OF PI-RINGS

[Ch. 4

The converse to Theorem 4.2.8 due to Park-Schelter [78], is given in Exercise 4.1.9. Every finitely spanned PI-extension of W is integral (and thus satisfies GU, LO, and INC). The hypotheses of part of Theorem 4.2.8 can be weakened, in the sense that if R is spanned over a central subring C by integral elements, then R is integral over C. (This is due to Amitsur-Small [78P]; the case C is a field is in Procesi [73B, p. 1331).

$4.3. The Characteristic Closure of a Prime PI-Ring

Before developing the theory further, we need an extremely important construction, found by Razmyslov [74b] and Schelter [78]. We shall more or less follow Schelter; similar work was also done by Amitsur-Small. Everything follows from the following observation. Theorem 4.3.1. Suppose R c M,(F), and suppose g,(R) # 0. Then there is an F-basis r , , . . .,rn2of M , ( F ) , with all the ri E R, whereby we maji view Reg(R) C M,,(F). Let C be the Z(R)-subalgebra of F generated by the characteristic value? of all elements qf Reg(R), and let R' = RC G M,(F). Then g,(R)+ a C, so Rg,(R)a R'. Proof. Taking ri such that gn(rl,r2,.. .) # 0, we see that rl,. .. ,rnZare a basis of M,(F). The rest follows immediately from Theorem 1.4.12(iii), where we view Reg(R) as matrices by their action on the n2-dimensional Fri. QED vector space

x::

With the notation and hypotheses as in Theorem Remark 4.3.2. Then 4.3.1, suppose S is a submonoid of F - { O ) with S ngn(R)# 0. R' c R, [viewed as a subring of M,(F)]. [Indeed, there exist rl,. . . ,rd in R, with gn(rl, ..., rd) = 1; thus, for any rER viewed in Reg(R), Theorem 1.4.12(iii)yields (for any characteristic value ak of r), ak = akgn(r,,..., r d ) = x i , + . . . + i , = k g n ( r,..., l r i l r,r'r,r,+l,... . r d ) ~ R s ,

where t

=

n'.]

Definition 4.3.3. With the notation as in Theorem 4.3.1, R' is called the characteristic closure of R.

Why did we view Reg(R) c M,,(F) instead of more simply in M,(F)? Because that is the easiest way of applying Theorem 1.4.12. [Amitsur-Small [78P] developed a method of working in M , ( K ) , where K is a maximal subfield of Q,(R) generated by one element.] The point of passing to R' = RC is that by the Cayley-Hamilton theorem each element of R is integral of degree 6-n' over C ; this leads us to

84.3.1 Proposition 4.3.4. finitely spanned over C.

The Characteristic Closure

209

With the notation as in Theorem 4.3.1, R' is

xf=

Proof. Suppose x E R'. We can write x = I xici for suitable . Y ~ ER , c ~ E Cin, which case x ~ C [ x...,.x ~ , k ] , which is finitely spanned over C by Theorem 4.2.8. QED

This result is quite useful, enabling us to reduce some assertions about prime PI-rings to prime PI-rings which are integral extensions of the center. However, even more striking results can be obtained under the further assumption that R is a finitely generated extension of a subring W . Recall the function fi of Theorem 4.2.8. Theorem 4.3.5. Suppose R = W { x ,,...,.Y,} is an extension of W , g,(R) # 0, and R G M,(F). Let C , be the Z(R)-subalgebra Of'F generated by the characteristic values of all monomials in x , , . . . ,x , of length
a

a

Proof. Obviously g,(R)+ C, and R g , ( R ) a Rb, in view of Theorem 4.3.1 ; the other assertion is Theorem 4.2.8. QED Remark 4.3.6. In Theorem 4.3.5, C , is a finitely generated (commutative) extension of Z ( R ) .

Raising Prime Ideals

R' and Rb are useful because they have this ideal Rg,(R), which is also an ideal of R . We shall exploit this fact to determine some information about Spec(R') and Spec(Rb). The situation is very clear when we pick out just what is needed to examine GU, INC, and LO. Theorem 4.3.7. Suppose R c R' is an extension of rings of PI-class n, such that R g , ( R ) a R'. Then.for any P E Spec,(R) there is exactly one member of Spec(R') lying over P that contains every R' such that A' n R G P. I n particular, .for any P,,P,ESpec,(R) we have LO(P,), GU(P,,Pzj, and INC(P in the extension R c R'.

A'a

Proof. First note that (R n PR')Rg,(R) G PR'Rg,(R) G PRg,(R) G P. Thus R n PR' c P [since g,(R) $ PI. Take P' to be (PR',P)-maximal. Now if R' and A ' n R E P then A'g,(R)R G A'ng,(R)R E P c P', so A' c P'. The rest of the theorem follows immediately. QED

A'a

210

[Ch. 4

EXTENSIONS OF PI-RINGS

Theorem 4.3.8. Suppose R c R' is an extension of rings of PI-class n, with Rg.(R)d R'. If PI,P,eSpec,(R) and P, is Rg,(R)-minimal, then GU(P,, P,) holds and LO(P,) holds.

Proof. By Proposition 1.9.6 it suffices to show that every (0,P,)maximal ideal P; of R' lies over P,. Let A' = P,+ Rg,(R)aR'. If r E A' n R, then r = x 1 +x, for suitable x 1 in P2 and x, in Rg,(R) s P,. But x, = r - . ~ , E Pz r ,R c P 2 . Hence A' n R 5 P, so, by assumption on P i , A' = P,, implying Rg,(R) c_ P i . Now Rg,(R) c P, n R 5 P,, implying P, n R = P, [since P, was Rg,(R)-minimal]. QED

Corollary 4.3.9. I f R E R' is an extension of rings of PI-class n with Rg,(R)d R', then LO(P) holdsfor every minimal prime ideal P of R. Proof.

Either P ~Spec,(R),or P is Rg,(R)-minimal.

QED

Here is a related result about the Jacobson radical. Proposition 4.3.10. if R E R' is an extension of rings of PI-class n with Rg,(R)d R', rhen g,,(R)Jac(R)E Jac(R') and g,(R)Jac(R') E Jac(R); in particular,,for R' prime, Jac(R) = 0 ifJac(R') = 0. Proof. g,(R)Jac(R) is a quasi-invertible ideal of R but is also a left ideal of R', proving g,(R)Jac(R) c Jac(R'). The other assertion is analogous; cf. Lemma 1.5.26. QED

94.4. Finitely Generated PI-Extensions

In this section we shall apply the earlier results of this chapter, in particular Theorem 4.3.5 to finitely generated extensions. The basic idea is as follows: Suppose R is a prime, finitely generated PI-extension of W, and let Rb and Wd be as in Theorem 4.3.5. Now Rb is a finitely spanned extension of Wd, which is a finitely generated, central extension of W. In turn, Rb and R are intimately related. We already know something about finitely spanned PI-extensions, and we shall soon find out facts about finitely generated central extensions. Then we shall pass information from W to Wd to Rb to R, and have theorems.

Hilbert's Nullstellensatz Generalized, and Related Results

Proposition 4.4.1. I f a commutative domain R is jinitelji generated over a semiprimitive domain W, then R is semiprimitive. Proof.

Writing R = W [ z , ,. . . .zk] and proceeding inductively on k, we

54.4.1

Finitely Generated PI-Extensions

21 1

may assume R = W [ z ] .Jac( W [ A ] )= 0, so we are done unless z is algebraic L 0 for suitable c, in W with c, # 0. Write c = c,. over W . Hence ~ : = o c , z= Let .“ = {maximal ideals of W not containing c } . Let S = {c‘li 2 1). If P E // then P,~spec(W,). Since R , = W,[i] is W,-integral, some f?,~Spec(R,) lies over P,, so some P‘€Spec(R) lies over P . Now let denote the canonical image In RIP’. R t W[,F],a field, so Jac(R) = 0. Hence, by Proposition 1.6.44 W n Jac(R) G n{PE rid} = 0, so by Proposition 4.1.7 Jac(R) = 0. QED A ring is Jacobson if every prime ideal is the Definition 4.4.2. intersection of primitive ideals. Remark 4.4.3. R is Jacobson iff every semiprime homomorphic image is semiprimitive.

Jacobson rings arose in connection with the Hilbert Nullstellensatz, which we recall is the statement, “Suppose F is an algebraically closed field and f;,fi,. . .,f, are in the commutative polynomial ring F [ A , ,..., A,]. I f f vanishes at all the common zeroes of,f,, . . .,,A, then some power of,f lies in the ideal generated by .1;,...,\,.”In view of the proof of the Hilbert Nullstellensatz (cf. Exercise l), we refer to the following result as a “substitute” Nullstellensatz. Remark 4.4.4. Any commutative, finitely generated extension of a Jacobson ring is Jacobson. (Immediate from Proposition 4.4.1.) In the next result, we shall use the easy fact that every finitely spanned extension of a commutative ring is integral (cf. Proposition 1.3.24). However, more general results can be obtained using the same basic techniques (cf. the exercises).

(Procesi). I f ’ R is a prime, Jinitely generated PITheorem 4.4.5 extension of a commutative, seiniprimitioe ring W , then R is semiprimitive.

Proof. Form Wd and Rb as in Theorem 4.3.5. Wd is a commutative, finitely generated extension of W, so Jac(W6) = 0, by Proposition 4.4.1. Rb is finitely spanned over Wg, so Jac(Rb) = 0 by Theorem 4.1.9(ii). Hence Jac(R) = 0 by Proposition 4.3.10. QED Corollary 4.4.6 (Noncommutative Nullstellensatz: Amitsur-Procesi). Aiiy PI-ring R which is a Jiriitely generated e.utensiori OJ a commutative

Jacobsoii ring is Jacobson. I n particular, Jac(R) is locally riilpotent. Proof.

Apply Theorem 4.4.5 to the prime images of R. QED

212

[Ch. 4

EXTENSIONS OF PI-RINGS

ACC(ldeals) and Related Notions

This last result inspires the question, “Is Jac(R) nilpotent under the hypotheses of Corollary 4.4.6?” This question is still open, but there is a satisfying answer due to Razmyslov [74b]-Schelter [78]. To obtain this result in its most comprehensive form (known so far), we look briefly at rings with ACC(idea1s) (cf. Definition 2.4.18 ff). Remark 4.4.7. If R satisfies ACC(ideals, resp. left ideals, resp. right ideals), then so does every homomorphic image of R . Theorem 4.4.8 (Hilbert basis theorem). If R satisfies ACClidrals, resp. left ideals, resp. righr ideals), thert so does R [ i ] . Proof. We do ACC(idea1s); the other assertions are analogous. E Gj). Suppose Ad R[A]. and let Aj = {leading coefficients of ~ ( A ) Aldeg(p) Then each A j a R , and A , E A , G A 3 E ..., so for some rn A j = A,+1 for all j 2 m. By Proposition 2.4.22 we have A j = RaijR for suitable u j ~ Z + , a i Aj ej ; for each j < m,each i G u j , take p i j € A of degree Gj, having leading coefficient a i j .Then one sees easily that

xz,

m

A =

Y.

2 R[ll]pijRII].

QED

j= 1i= 1

Corollary 4.4.9. I f R satisfies ACC(ideals, resp. leji ideuls, resp. right ideals), then so does every finitely generated central extension of’ R . Proof. Every finitely generated central extension of R is a homomorphic image of some R[A,, . . . ,A k ] , so apply Remark 4.4.7 and induction to the Hilbert basis theorem. QED Proposition 4.4.10. Suppose R i s a j n i t e l y spanned extension o j W . l f W satisfies ACC‘(ideals, rrsp. leji ideals, resp. right ideals), then so does R.

Proof. Write R = Cy=,W r , , with each r i E C , ( W ) . We do ACC(idea1s). Suppose A , G ’4, E ..., with each A k a R . For each k < 03, each j < u. let A k j = A , n ( E { = l ~ f ‘ r iand ) B,,= f w ~ ~ ~ a - - w r for ~ ~some ~ , a. in ~ A - ~, ~ , ) . Then B k j aW, so equality eventually holds in B l j c B Z jc .... Now B k , r , = A k l . By induction on j , equality holds eventually in A l j G A,, E . . . for each j ; we are done since A,, = A,. QED Proposition 4.4.11. [ f a PI-ring R satisfies ACC(ideals), then Nil(R) i s nilpotent. Proof. Obviously by hypothesis L,(R), the sum of all nilpotent ideals of R , is nilpotent. But by Theorem 1.6.36 Nil(R)” c L,(R) for some n. Thus Nil(R) is nilpotent QED

34.4.1

Finitely Generated PI-Extensions

213

[Incidentally, the PI-hypothesis is necessary, although Levitzki proved that for any ring satisfying ACC(leff ideals), the nilradical is nilpotent.] Before continuing with the theory, we pause for a very Small example which is the source of virtually all counterexamples in this chapter.

,

+

Example 4.4.12. Let R be the subring (Ze, + Qe,, Qe,,) of M,(Q). R satisfies ACC(right ideals) but not ACC(left ideals). [Indeed, one sees easily that the only right ideals of R are 0, Qe, and all (Jell +Qe,,), ( I e , , + Q e , , + Q e , , ) for all 1 4 7 ; since Z satisfies ACC(ideals), R must satisfy ACC(right ideals). On the other hand, putting Li = { i 1 / 2 ' I n E Z}, we see L , e , , c L , e , , c ... is a chain of left ideals of R.]

Digression. The point of Example 4.4.12 was that if A , , A, are rings and M is an A , - A , bimodule [i.e., left A,-module and right A,-module such that (a,y)a, = a,(ya,) for all aiEAi, Y E M I ,then (Alel, + M e , , + A , e , , ) is a ring whose left (resp. right) ideal structure reflects the A,-module (resp. A,-module) structure of M . Thus one should expect the left-ideal structure and right-ideal structure of a PI-ring to be as distinct as one wishes. We also want to look at information about ACC(semiprime ideals), which is somewhat weaker than ACC(idea1s). Proposition 4.4.13. Suppose R scltis$es ACC(semiprime ideals). For any A d R, there are only a finite number of A-minimal primes.

Proof. Let .d= {BaRlthere are an infinite number of B-minimal primes}. Write A' for n{A-minimal primes of R}, a semiprime ideal. Obviously if A E . then ~ A ' E . ~ ,so .ci/ has semiprime ideals; choose a maximal semiprime ideal B E .d.Then clearly B is maximal among all ideals in .d. Obviously B is not prime, so B 2 B , B , for suitable ideals Bi3 B. But then B , $ . d , i = 1,2. Since any B-minimal prime contains either B, or B,, we see that there are only a finite number of these, contrary to B E d.The contradiction came from assuming .dnonempty. QED Remark 4.4.14. If a semiprime ring R has a finite number of minimal primes P , ,..., P,, then P , . . . P k ~ P , n . . . n P , ~ n { P ~ S p e c ( R )so J=0, each expression is 0.

Remark 4.4.14 will be used to reduce several proofs to the prime case. Theorem 4.4.15 (Schelter). Suppose R is a prime, finitely generated PI-extension of W . If W satisfies ACC(ideals, resp. left ideals, resp. right ideals), then so does Rb (with the notation as in Theorem 4.3.5). Proof. With the notation as in Theorem 4.3.5, Rb is a finitely spanned extension of Wg, which is a finitely generated central extension of W . QED

214

EXTENSIONS OF PI-RINGS

[Ch. 4

Corollary 4.4.16 (Schelter). Suppose R is a Jinitely generated PIextension of a ring satigvving ACC(ideals). Then R satisjies ACC(prime ideals). Proof. Suppose P , c P , c ... is a chain in Spec(R). Take i such that RIPi has minimal PI-class (say n); replacing R by RIP,, and P , c P , c *.. by ( P i + , / P i c ) (Pi+2/Pi)c - * . , we may assume R is prime, and each PjeSpec,(R). But this is impossible by Theorem 4.4.15and Theorem 4.3.7. QED

Other results of this type will be obtained in the exercises to 85.2; Theorem 4.4.16itself will be improved to Theorem 4.5.7. Nilpotence of the Jacobson Radical

We are now ready to show that the Jacobson radicals of many finitely generated PI-extensions are nilpotent. Theorem 4.4.17 (Schelter). Suppose R is a prime, Jinitely generated PI-extension ofa ring W satisfying ACC(ideafs). !f A 4 R, then Nil(R/A) is a jinite intersection ofprime ideals (ofRIA), and is nilpotent. Proof. Otherwise, let 9 = (PESpec(R)(R/Pis a counterexample to this replacing theorem}. By Corollary 4.4.16,some P ESpec(R) is maximal in .Y; R by RIP, we may assume the theorem holds in every proper prime image of R. Let Rb be as in Theorem 4.3.5;then Rb satisfies ACC(idea1s) by Theorem 4.4.15.Now g,(R)Aa Ro, so there are a only finite set of g,(R)Aminimal primes of Rb, which we call I",,...,P,,and, for some rn, ,Pi)" E g,(R)A. [Since Rb/g,(R)A satisfies ACC(ideals).] Let Pi =PinR#0,1 s i < u . Now R/(A + Pi) is a homomorphic image of RIP,. Hence, by assumption, R / ( A + P i ) has a finite set of prime ideals whose intersection is nilpotent; thus, for every i there exist Qij€Spec(R), 1 d j < ki (for some k i ) each containing A +Pi, such that each ( n i r , Qij)' c A +Pi for suitable t . Thus (n:L ,Qij)')"' E A ( P , ... P,)" E A , implying immediately i7j"r1Qij,which is nilpotent. QED Nil(R/A) =

(n;=

(ny=

,

+

Corollary 4.4.18. Suppose C is a commutatice Jacobson iloniuin satisfying ACC(ideals), and R is a Jinitely generated C-algebra. l f R d M , ( C [ S ] ) , then Jac(R) is nilpotent. Proof. If R = C{r,, ..., rk} then we can view R as a homomorphic image of C { X , , . . . ,X,} (by sending each X i-+ ri), and thus of C , ; Y,, . . . , Y,), a prime ring. Hence Theorem 4.4.17 is applicable, to show Nil(R) is nilpotent; thus, by Corollary 4.4.6Jac(R) is nilpotent. QED

$4.4.1

Finitely Generated PI-Extensions

215

(Razmyslov). !f'R is an qfine algebra satisfjing all Corollary 4.4.19 identities oj'n x 11 matrices, [hen Jac(R) is idpotent. A Small Counterexample

We have just seen that, for any affine algebra R satisfying the identities of 11 matrices, Jac(R) is nilpotent. However, R need not be admissible (cf. $1.6), as shown by the example of L. Small below. Any subset of R of the form Ann,(S) [resp. AnnX(S)] for suitable S c R will be called a left (resp. right) annihilator of R. (Compare with Definition 1.7.23.)

11

x

Remark 4.4.20. If R satisfies ACC(1eft annihilators, resp. right annihilators), then so does every subring of R. [Indeed, if for some subring T of R, Ann,.S, c Ann,.S, c ..., then Ann,S, c Ann,S, c ..'.I

If R is Uffit~eand udmissible, then R satisfies ACC(1efi Lemma 4.4.21. annihilators) and ACC(right annihilators). Proof. By hypothesis, R = F[rl,..., rk} 5 M , ( C ) for some 11, some commutative ring C. Thus we can write each ru as a matrix (c);'),1 d u d k ; renumbering the cir) as c , , c 2,..., cmr where in = n2k, we have R s M,(F[c,, . . . ,cm]), which satisfies ACC(left ideals) and ACC(right ideals). Thus, by Remark 4.4.20 R satisfies ACC(1eft annihilators) and ACC(right annihilators). Q E D

Example 4.4.22 (Small). Anaffinealgebra R over anarbitrary field F , such that R is a homomorphic image of a subring of a matrix ring (over a commutative ring), but R is not admissible. Let S = {A'(i2 0) c F[A], and let R , be the subring (F[A],e, + F [ i , ] , e , , +F[A]e,,) of M,(F[i],). For each i E Z , let Bi = A'F[A]e,,, a right ideal of R,. Then B, 3 B, 3 .... Now, partitioning M,(F[A],) as M,(M,(F[A],)), let R , be the subring of M,(R,) given by all { ( z ) l c t ~ F , . u , R,). y ~ Let A = {(;;)l.u~F[A]e,,), an ideal of R,, and let R = R,/A. Write - for the canonical image in RJA. Obviously R, and R, are affine, so R is affine. However, we claim that R does not satisfy ACC(right annihilators). In fact, for BI = c R , , we claim A n n ' ( F ) c A n n ' ( E ) c . . . . Indeed, for r = (g;) E R,, we have r E Ann'% iff Bir G A iff B,y G F [ I ] e , , iffy ~ i , . - ~ F [ A l e , ,so , the claim is now clear (since A-'F[A] c A-2F[I] c ...). Therefore R is not admissible by Lemma 4.4.2 I . Also note that Example 4.4.22 does not satisfy ACC (ideals); indeed each BIdR, soAnn'B;aR,.

,

Lewin has proved some very interesting results about affine PI-algebras, which we summarize briefly. First, Lewin [73] has shown that there are an

216

[Ch. 4

EXTENSIONS OF PI-RINGS

uncountable class of affine F-algebras d M , ( F ) which are not admissible. (This is a considerable improvement of Small’s counterexample given here.) Moreover, Lewin [74] proved that any affine PI-algebra R having nilpotent Jacobson radical is <M,(F) for some n. In view of Razmyslov’s theorem, we see that the following statements are equivalent for an affine PI-algebra R: (i) Jac(R) is nilpotent; (ii) R d M , ( F ) for some n. Razmyslov [74b] has shown that for any affine algebra R qf’characreristic 0 satisfying a Capelli identity, Jac(R) is nilpotent; thus, by Lewin’s theorem, R d M , ( F ) for some n.

Going Down

We now turn towards Schelter’s analog of Krull’s “going down” theorem. Lemma 4.4.23. Suppose R is an integral PI-extension of’ C c Z(R), and A 4 C . For vach element r of AR there is a suitable commutative polynomial f (A) wirh coeficients in A, such that for some integer q > deg(f) rq = , f ( r ) .

Proof. Write r = C:= ,auruand let T = C { r l , . . , r l ) c R. By Theorem 4.2.8 we can write T = If=Cx, for suitable x i € T . Write ruxi = ,cijUxj, 1 ,< i, j d d, 1 ,< u < t . Then Y.Y, = (xi= auciju)xj;letting a i j = xi= aucijuE .4, we see det(iSijr- a i j )= 0, yielding the assertion. QED

xy=

,

,

x&

(Going down: Schelter). Suppose R i s u prime PITheorem 4.4.24 ring, integral over a normal subring C C Z(R). If P, E P, ~ S p e c ( C )and P‘, ~ S p e c ( Rlies ) orer P,, then there i s some Pb ~ S p e c ( Rlying ) over Po, with Po E P; { r E R I O # r+P;EZ(R/Pl)} and S = C - P o . Let S1 is a submonoid of R. If we can obtain 0 be done by taking Po containing RPo, maximal such that Po n S , = fzl. [Indeed, clearly PoE Spec(R). If Po $ P, then 0 # (P,+P;);P‘,a (RIP‘,),a prime PI-ring, so Po contains an element of A , , which is impossible; thus Po E P,.Similarly, Po n C = Po.] So we need only show S , n R P , = 0. Suppose not. Then X S E R P , for some X E A , , S E S .Let p ( A ) be the minimal monic polynomial for xs (over C ) . By Gauss’ lemma (Proposition 1.9.41) p is irreducible. On the other hand, by Lemma 4.4.23 we have xs satisfying some monic polynomial q ( A ) = Ak +ql(A), where q1 is a polynomial of degree < k , whose coefficients are all in Po. Let F be the field of fractions of C. Since F[A] is a principal ideal domain, plq, so by Gauss’ lemma p divides q in C[A]. By passing to C[A], where C = C/Po, a domain, one has = Ak, so p = Ad for d = deg(p); thus Proof.

Let .4,

=

= { x s ) x ~ A , , s ~ SClearly f. S, S, n RP, = then we shall

94.4.1

Finitely Generated PI -Extensions

217

p = Ad+Ef:dcilli for suitable c i € Po, so (.us)d = - I ~ : ; C ~ (Thus X Sxd) ~ =. - x f : d c i ~ i - d ~(in i Rs). Now if the minimal polynomial of x in R, (over C,) had degree < d , then the minimal polynomial of xs in R, would have degree < d , and Gauss’ lemma would yield a contradiction to the assumed minimality of p . Thus I d+ c ~ s ~ is- the ~ Iminimal ~ polynomial of x; but .Y is integral over C so, again by Gauss’ lemma, each C. Now ( c ~ s ~ - ~ )=s tie ~ - Po, ~ so in fact each c i . F dE Po (since s $ Po). Hence xd = -Ef:,’ E P,R E P i , implying (s+ = 0 in RIP,, contrary to the hypothesis 0 # .Y + Pi E Z(R/P’,). Thus we have the desired contradiction to the supposition S, n RP, # @. Q E D

xfzd

Affine PI-Rings are Catenary

We are almost ready for Schelter’s theorem that all affine PI-rings are “catenary,” i.e., every prime of R is contained in a chain of Spec(R) whose length is rank(R). The only additional element we need is the following standard result quoted from the commutative theory: (Noether-Bourbaki [72B, Theorem V.3.11). Normalization lemma Suppose C is a commutative qjine algebra (over F ) with ideals A , c . .. c A,. Then there exists u transcendence basis x , , . . .,x, of C and n ( i ) E Z + , 1 < i < k, such that C is integral ouer F [ x , , ..., x,], and A in F [ x , , . . .,x,] is the ideal of’ F [ x , , . . .,x,] generated by x l r . . , x , ( ~for , each i < k. Although the proof is somewhat lengthy, the ideas are clear enough and are given in Exercise 12. Schelter’s key lemma now follows. Lemma 4.4.25. Suppose R is prime of PI-class n, and is integral over un a@ne F-algebra C L Z(R). For ar7p nonzero PESpec(R) we can find Po E Spec,(R) such that Po c P and rank(R/P,) = rank(R/P) 1.

+

,

Proof. Let A , = P n C and A = A, n g,(R)+. Clearly A , , A , are ideals of C, and are nonzero because P n Z ( R ) # 0 (and thus P n g,(R)+ # 0) and Z ( R ) is a domain integral over C . Now, by the normalization lemma, we have some s,,. . .,.urnin C and n( l ) , n ( 2 )E Z + such that for C , = F [ s , , . . . ,.urn] C is integral over C, and A i n C , is the ideal of C, generated by s , , . .., x,#(~,, i = 1, 2. R is integral over C,, which is a unique factorization domain, and thus normal. Let I = xyLzJxiC,. C,/I z F [ ~ , , x , ( , ) +,...,. , x,], so I ~ S p e c ( c , ) ; hence by “going down” there is some Po E Spec(R) lying over I with Po c P. Also Po E Spec,(R) since x, 4 Po. Now using Theorem 1.10.23 plus the fact

218

EXTENSIONS OF PI-RINGS

[Ch. 4

that integral PI-extensions preserve rank, we have rank(R/P,)

=

rank(C,/I) = m + 1 - 4 2 ) = 1 +rank(C,/(P n C1) ) = 1 rank(R/P).

+

QED Lemma 4.4.26 (Schelter). Suppose R' is any finitely generated PIextension of R, PESpec(R), and P' [ i n Spec(R')] is (PR', P)-maximal. Viewing RIP G R'IP, we have trdeg(Z(R'/P')/Z(R/P)) = 0. Proof. Passing to R'IP', we may assume P' = P = 0 and every nonzero ideal of R' intersects R nontrivially. Moreover, localizing at Z(R)- (0}, we get Z(R) is a field, so that R is simple. But then R' is simple. Clearly R' is finitely generated over Z(R), so we are done by Lemma 1.10.21. QED

Now we can show that all affine PI-rings are catenary Theorem 4.4.27 (Schelter). Suppose R I S a prime c@ne PI-rblg, and P E Spec(R). Let t = rank(P). Then rank(R) = t + rank(R/P). Proof. Induction on r. First assume t = 1. Then take Rb as in Theorem 4.3.5 and take P' E Spec(Rb) that is (PRb, P)-maximal (Corollary 4.3.9). By Lemma 4.4.26 and Theorem 1.10.23 rank(Rb/P') = rank(R/P). By Lemma 4.4.25 there exists Poc P' in Spec,,(Rd) such that rank(Rb(Po) = rank(Rb/P') + I . But, in view of Theorem 4.3.7, we have R n Pb # P . Since rank(P) = 1. we must have R P, Po= 0. But 0 is the unique ideal of Rb lying over 0 by Theorem 4.3.7. Thus Po= 0, and by Lemma 4.4.26 and Theorem 1.10.23 again

rank(R) = rank(Rb) = rank(Rb/P') + 1 = rank(R/P) + I , proving the theorem for t = 1. In general, there is a rank 1 prime P , c P such that rank(P/P,) = 1 - 1 in R/P ; by induction, rank(R/P,) = rank(R/P) + ( t - 1). But wejust proved rank(R) = rank(R/P,)+ 1 = rank(R/P)+ ( t - I ) + 1 = rank(R/P)+t. QED

$4.5. Generalizing the Razmyslov-Schelter Construction

A good rule of thumb for PI-theory is that virtually every theorem about central simple algebras can be generalized to A,-rings. Accordingly, the Razmyslov-Schelter construction of 34.3 could be further generalized, and would enable us to improve several theorems of 94.4. We have refrained so far from adding this bit of generality, to induce the reader to read the beautiful theorems of $4.4 before skipping to Chapter 5. Here is the basic idea. Suppose R is proper maximally central of rank n2,

$4.5.1

Razmylov-Schelter Construction

219

with 2 = Z(R); then [R:Z] = n2 so by Proposition 1.3.24 R is integral over Z of bounded degree n2. Although the coefficients are linked to the Cayley-Hamilton theorem, they do not seem to be uniquely obtained, so we start all over again to get them in a precise manner. By Theorem 1.8.50 g(rl,. . .,r,, rt+ l , j , . . .,rd,j)= 1 we know R is an A,-ring and, for t = n2, for suitable k E Z + and suitable elements of R. Applying Theorem 1.4.12 to any T in End,R yields the coefficients ak (for T ) in a well-defined formula! We call these clk the characteristic values of T . Of course this definition depends on the choice of r l , . ..,rt, rt + l $ b . ..,rdJr 1 6 j ,< k , but that clearly is no problem; we just fix these elements at the start.

x!=

Definition 4.5.1. Suppose R E A and A is a proper maximally central algebra. T h e characteristic value subring C of A is the subring of A generated by all characteristic values of Reg(R) [viewed in End,,,,A]. We call RC the characteristic closure qf R in A. Theorem 4.5.2. Suppose R c A and A is proper maximally central of rank n2. Let C be the characteristic ualue subring of A and R' = RC G A. Then g,(R)+ a C, Rg,(R) 4 R', and R' isJinitely spanned over C.

Proof.

Straightforward.

QED

Of course, we are mostly interested in finitely generated extensions Theorem 4.5.3. Suppose R = W { x l , ..., x,} is an extension of W , and R is contained in a proper maximally central algebra A of rank n2. Let Co be the subring of A generated by all characteristic values of all monomials in xl,. . . ,x, of length
Use Theorem 4.5.2 and Theorem 4.2.8.

QED

Before proceeding, we need examples of rings contained in proper maximally central algebras. Remark 4.5.4. Suppose C2,t1+,is an identity of R, g,, is R-central, and g,(R) contains a regular element c ; then for S = {c'li 2 0}, R,, is proper maximally central of rank n2, and R c R,,. Remark 4.5.5.

If n{PESpec,(R)} = 0, then R can be injected into is clearly proper maximally central of rank n2, by Theorem 1.8.50.

n{Q,(R/P)IPE Spec,(R)}, which

Now we are set for applications. We shall concentrate on pushing forward results from g4.4, omitting details of proof when they are the same as for corresponding theorems in $4.4.

220

EXTENSIONS OF

PI-RINGS

[Ch. 4

Theorem 4.5.6. Suppose R is a Jinitely generated PI-extension of W, contained in a proper maximally central algebra ojrank n2. If W satisfies ACC(ideals, resp. left ideals, resp. right ideals), then so does Rb, notation as in Theorem 4.5.3. Proof. Rb is a finitely spanned extension of a central extension of W. QED

By examining Spec,, closely, we can obtain a simultaneous generalization of Corollary 4.4.16 and a result of Procesi. Theorem 4.5.7. If R is a Jinitely generated PI-extension of a ring safisfying ACC(ideals), then R satisfies ACC(semiprime ideals).

Proof. Suppose B1 c B , c ... is a chain of semiprime ideals of R. Taking i such that RIB, has minimal PI-class (say n) and passing to RIB,, we may assume that R is semiprime of PI-class n, and PI-class (RIBj) = n for all j. For each j, write .Pj= {l‘~Spec(R)lB~ c P } . Obviously -9, =I B2 =I .... B j = n{P € . P j } . Let .’Pjn = YjnSpec,,(R), Bj, = n { P EY~,} and BJ = n { P E Yj-;:/Pj,,}.Now B;/Bl E B;/B; E ... in the semiprime PI-ring RIB‘, of PI-class 6 n - 1 ; by induction of n, we have, for some m, BJ = BJ+! for all j 3 m. Hence B,, c B,+ c B m + l , nc ... . Passing to RIB,,,,, we may assume the following properties: R is semiprime, n{PESpec,(R)) = 0, and B , c B , c ... with B j = Bj, for allj. But now, by Remark 4.5.5 we can inject R into a proper maximally central algebra A of rank n2. Now, in the notation of Theorem 4.5.3, KO satisfies ACC(idea1s) by Theorem 4.5.6. But Rg,,(R)B,R c Rg,,(R)B,R c ... are ideals of KO,so, for some rn’ we have Rg,,(R)B,R = Rg,,(R)B,+,R for all j b m’. In particular, gn(R)Bj+,E P for each P e p j , ; since g,,(R) qL P we conclude B j + I c P , so B j + G n(PE.Yjn) = B j , a contradiction to B j c B j + ] . QED One consequence of Theorem 4.5.7 is an improvement of Theorem 4.4.17. Theorem 4.5.8. I f R isfinitelq generated, semiprime PI-extension of a ring satisfying ACC(idea1s)and Id R, then Nil(R/I) is nilpotent. Proof. In view of Theorem 4.5.7 and Proposition 4.4.13, there are only a finite number of minimal primes of R, so, by Remark 4.4.14 the assertion passes to the prime case, which was Theorem 4.4.17. QED

EXERCISES $4.1 1. (Bass) An inicgral, central extension of R is Azurnaya of rank n2 iff R is Azumaya of rank n2.

Ch. 41

Exercises

22 1

Bergman [74aP, 74bPl has developed a lovely theory of extensions without any PIassumptions, using the ring R{X) introduced in $2.1. His method is tied in with GI-theory, and we present it in Exercises 2-6, as well as in Exercises 7.6.10 and 7.6.1 1. Write a = R{Xl, ..., X k i / I , where I is the ideal generated by all [ X i , . ] , 1 < f < k , and r E R . Any element f of R can be written xr,,h, where /I E . / / ( X ~ , ..., X k ) , the free monoid in the X ,= X i + l ; for PeSpec(R), write P-supp(,/) = {hlr,$ P i . Also write 4 a R d if A is an R-bimodule of f? [i.e., A is an R-submodule with A R s A , ] For AQ,R, say A is P-dcwsc, if for every k i n . / / ( X l , ..., R,)thereissuitabIerin R - P s u c h that r h ~ . 4 . 2. If A Q , d is P-dense, then for any h , ,..., h, in . K ( X , ,..., X k ) , u arbitrary, there is suitable r in R- P with rhiE A , I < i < u. 3. Suppose R is prime, A d R d ,and M E . / / ( X , . . . . . X , ) , such that A + R M is P-dense. If ,JE A and h~ P-supp(f) E M , then A + R ( M - ( h ] ) is also P-dense. [Extensire hint: Write x for the coefficient of h in 1: We claim we may assume 0-supp(f) c M . Indeed, write f = f ‘ + f ” . where ,f‘ E P a and f ” E RM. Take r E R - P such that r(O-supp(f’))G A + RM, and take r l in R - P such that rrix $ P. Now r r l f ’ G ( A + RM)P, so write rrl f ’ = 1; +fi,where fl E AP and ,/,ERPM = P M . The claim is established when we replacer by (rrLf-fl) = ( r r l f ” + f Z ) .But now xhE A + R ( M - { h ) ) d , R and the assertion follows forthwith.] 4. Suppose A Q d , P-supp(J’) = [ I ) for some f in A , and, for some finite set M c . / / ( X l , . . . , X k )A, + R M isP-dense.Then A i s P-denseand A n ( R - P ) # @.(Hint:Take M minimal; then M = 0 by reverse induction, since P-supp(fh) = { h } for any he M . ) 5. (Bergman) If R’ is a finitely spanned extension of R, then LO holds. [ H i n t : Let P€Spec(R).R’ is a homomorphic image RIA ofR, for suitable A , with A n R = 0. A + R M is Pdense for some finite M ; conclude ( A P R ) n R E P with the contrapositive of Exercise 4.1 6. Suppose R s R‘ and LO holds in every prime homomorphic image R ’ / P (from R/(R n P‘)). Then G U holds from R to R’. In particular, GU holds for all finitely spanned extensions. 7. (Schelter) Suppose R = W z , with each zieZ(R). Then R is W-integral of bounded degree. [ H i n t ; We have (r[Sij-w i j ) z j= 0. By reverse induction on k , show < t such that } for each k < u that for t = 2 ” - k there exists , ~ ; ~ ’ E W {ofX ~degree I (r‘cSij-,hyl(r))zj= 0. Indeed, immediate for k = u ; inductively, putting w$) = , h y ) ( r ) ,we have, for each i,

+

xy=, x:=,

x!=

k

O=

1 (+aij- w!;’)zj(r’ wL\))+ -

j= 1

k

wl:)

1 (fakj-

W$~)Z~,

j= 1

so you can take Jik-

’)

=

x;j;”aij+,f;y’x:-,h;*’j;;) -,Af)x; dkj+,fip@,

Now, for k = 1, ( r f - f j ; ’ ( r ) ) z l= 0.) 8. If R = I: Wri= with each, ri E C,( W ) ,then R is a homomorphic image of a subring of M J W ) . Thus every finitely spanned extension of a PI-ring is PI; cf. Exercise 1.8.2. 9. (Part-Schelter [78]) This generalizes Exercise 7. Every finitely spanned extension R of W is W-integral of bounded degree. [ H i n t : By Exercise 8, we may assume R = M , ( W ) . Prove that there exists some degree d ( n ) such that every monomial in r l , . . . r r d , n(with ) coefficients in W ) of degree > d ( n ) is a sum of monomials of lower degree. Let e = 1 -en”. For any r’, Y in R, and for each I(’ in W, letting e,,,re,, = w,e,,r’ewre,, = wlenn, we have r‘w(r- w o ) - w 1 E er‘ewre,, +Re, from which we conclude for any generalized polynomial ,/(XI,. .. ,X , ) and any r l r. . .,rt+ in R, that there is a suitable generalized polynomial y of lower degree with f ( r l , .. . ,r,)r,+ - g ( r l , . . . r, + ) E,f(er,e,.. . ,er,e)er,+lenn+ Re. Proceeding induc. we tively on n, using the erie, we can modify g to get , f ( r l , . .,r,)r,+ - g ( r l , . . ., r , +l ) ~ R eThus may assume each riERe. Then for all wi in W, rlwl...r,w, = e r l ( w l e r 2 ~ . . ~ ; _ 1 e r , w , ) +

.

,

222

EXTENSIONS OF PI-RINGS

[Ch. 4

ennrl(wler2 ... w,- ler,w,).By induction on n, we can eliminate the first term, leaving something in e,,Re, and (e,,Re)(e,,Ke) = 0.1

$4.2 1. (Herstein-Prwesi) If a multiplicative subgroup G of a PI-ring R is torsion (i.e., every element has finite order), then every finitely generated subgroup of G is finite. [Extensive hint: Replace G by an arbitrary finitely generated subgroup, generated by g,, . . . ,g k ;then letting C = Z. 1 s R, replace R by C{g,, .. . ,gk}.Now R = Cs,for suitable x, E G. and satisfies ACC(1eft ideals). IfGwereinfinitewecould take AaRmaximalwithrespect to(G + A ) / A infinite.ReplaceR by RIA. { r E R I ttr = 0 for some n in Z } Q R. and so is finitely generated as left ideal. and hence is 0. Now i f B aR withB' = Othen by Exercise 1.5.3, {b E BI 1 - b E G } isanadditivetorsionsubgroupof Rand is thus0: thus (C + B ) / B is infinite,so B = 0. Hence R is semiprime; show R is prime. Let T = Q,(R)withcenterF,dndletn = [T:F].Gcontainsabasis{x,, .... x-,}ofT.Giveng= six,, we can solve ai from the equation tr(gxj) = 1, at tr(u,xj), 1 < j < n. The traces of elements of G are sums of roots of 1 algebraic of degree < n ; there are a finite number of these, yielding only a finite number of possible a,. Hence G is finite.] 2. If T is an R-integral PI-extension of R, and if R is a W-integral extension of W ,then T is W-integral. (Hinr: Apply Exercise 4.1.9 to elements of T.)

,

54.4 1. Prove Hilbert's Nullstellensatz (cf. Kaplansky [70B, theorem 331, as follows, for an algebraically closed field F. Step I : Every polynomial ring F [ I , , . . . ,I,] is Jacobson. Step 2: If M is a maximal ideal of F[A,, . . . .A"], then M is generated by elements I, - a l , . . . , & - u M for suitable a, in F. Step 3 : Let A be the ideal of F I I l , . . . ,A"] generated by,f,,. . . .A. By Step Z,/is in every maximal ideal containing A , sofis in every prime ideal containing A ; thusfdE .4 for some d. 2. In the notation of $4.3, Rb is an extension of Wd satisfying GU, LO, and INC. (Hint: Use exercises from & I .)I. We now aim to modify the proof of Theorem 4.4.5 to account for the possibility W is noncommutative. 3. Suppose R is a prime PI-extension of W . If zeZ(R) is W-algebraic then z is Z(W)algebraic. (Extensite hnt: Replacing R by W {z}, we may assume R is a central extension of W . Suppose x w , z i = 0, and apply gn,where n is the PI-class of R, to make the coefficients central.) 4. If R is a prime PI-extension of W , and ZEZ(R),then either W { z }z W[A] or z is algebraic over Z ( W ) . 5. If R is a prime PI-extension of a simple subring W and if ZEZ(R). then either W { z }== W[A]or Wjz; is simple. 6. If R is a prime, finitely generated, central PI-extension of a semiprimitive subring W, then R is semiprimitive. (Hint: Mimic the proof of Proposition 4.4.1, using Exercises 4, 5. and Exercises 4.1.5-4.1.7.) 7. (Procesi) If R is a prime, finitely generated PI-extension of a semiprunitive ring, then R is semiprimitive. (Hint: Use Exercises 2, 3, 6.) 8. (Procesi) Any finitely generated PI-extension of a Jacobson ring is Jacobson. 9. (Procesi) If R is a simple, finitely generated PI-extension of an arbitrary ring W, then W is simple and R is finitedimensional over Z ( W ) . (Hint: In the notation of Theorem 4.3.5, R = Rb. Hence some finitely generated central extension of W is simple.) 10. (Procesi) If R is a prime, finitely generated PI-extension of W and if P is a maximal ideal of R, then P n W is a maximal ideal of W by Exercise 9. Conclude that Jac( W ) c_ Jac(R). 11. In Example 4.4 22, show Jac(R)' = 0. 12. Prove the normalization lemma. [Hint: We may assume C = F [ I , ,.... I,] for some m.

Ch. 41

Exercises

223

First assume k = 1 and A , = Cx, for some x1 = p(A,, . ..,A,,,). For 2 < i < m, formally put xi = A,-,I$, with d, to be chosen soon. Then Ai = xi+Af, so p(l,,x,+1:', ..., x,+A:..')-x, = 0, yielding, for large enough di an integral equation for A , over F[x,,.. .,xm]. Thus C is integral over F[x,,. ..J,,,], so x,,. ..,x, are algebraically independent. Since F[x,,. ..,xm] is normal, conclude A , n F [ x ,,..., xm] = x,F[x,, . ..,.xm]. Next, by induction on m, take k = 1 and A , arbitrary. Finish by induction on k . ] The special case ofTheorem 4.5.7 due to Procesi is obtained directly in the next two exercises, cf. Amitsur [70]. Exercise 13 is also needed in Saltman [78bP]. 13. Let T = M,(4[5]),R = Y ) . and I d R. I = R n T I T iff R/I G M J C ) for some commutative algebra C. 14. If 4 satisfies ACC(idea1s) then every finitely generated PI-algebra R satisfies ACC(semiprime ideals). [ H i n t : First take R = 4";Y , , . . ., Y,J .]

CHAPTER 5

NOETHERIAN PI-RINGS This chapter concerns PI-rings satisfying ACC(ideals), ACC(1eft ideals), and/or ACC(right ideals), and gives the various attempts to generalize the theory of commutative Noetherian rings. Although several very nice theorems are akailable, notably by Jategaonkar, progress has stalled somewhat after 1Y74,because the affine PI-theory (of Chapter 4) has turned out to be considerably more rewarding. The reader will note that modules enter into the theory much more forcefully than in preceding chapters, giving the theorems a new flavor.

95.1. Sufficient Conditions for a PI-Ring to Be Noetherian Definition 5.1 .l. A module M is left Noetherian if there is no infinite chain M , c M 2 ,:I... of submodules of M . (Right Noetherian is defined analogously for right modules.) R is Noetherian iff R is left and right Noetherian as an R-module.

Obviously R is Noetherian iff R satisfies ACC(1eft ideals) and ACC(right ideals). Various elementary facts concerning Noetherian rings were given in 44.4. Here is another. Remark 5.1.2. If M is a submodule of a finite direct sum of left Noet herian modules, then M is left Noetherian. (Proofis straightforward,and is left to the reader.) Hence any submodule of a finite dimensional module over a ring satisfying ACC(1eft ideals) is itself left Noetherian and thus finite dimensional. Formanek's Theorem

Now we present a variant of a theorem of Formanek [74] ; its proof is modified by Amitsur [75]. Theorem 5.1.3. Suppose CZn2 +, is an identity of' R , gn i s R-central, and gn(R)contoin\ a regular element of R . Then R is a Z(R)-submodule of a

724

r v . 1 .I

Sufficient Conditions for a PI-Ring

225

,finite-dimensional Z(R)-module. [ f R has a Noetherian subring R, 2 Z(R), then R is a jinite-dimensional R,-module (resp. right R,-module), and R is Noetherian. Proof. Suppose cEg,(R) is regular. By Theorem 1.4.21 Rc is a submodule of a finite-dimensional Z(R)-module, which is certainly a finitedimensional R,-module (resp. right R ,-module). But r H re gives us a module isomorphism, so we are done by Remark 5.1.2. QED

(Formanek). fj' R is prime and Z ( R ) is Noetherian, Corollary 5.1.4 then R is Noetherian and jinite-diniensional over Z(R). R has PI-class n, and all nonzero elements of Z ( R ) are regular. QED Proof.

Example 5.1.5. For n 2 4,F,,[ Y,, . . . , y k } does not satisfy ACC(ideals), in view of Example 4.4.22; hence its center is not Noetherian. Elaborating the argument of Theorem 5.1.3 gives a slight improvement of the main theorem of Formanek [74].

if R is a semiprime PI-ring and has a Noetherian Z(R)-subalgebra R,, then R is a finite-dimensional R,-module and is Noetherian. Theorem 5.1.6.

Proof. [Note that R , , and thus Z(R), satisfies ACC(annihi1ator ideals); cf. Proposition 1.7.31.1 By Theorem 1.7.34(iv) Q,(R) 2 Q1 @ ... @ Qk for ,tic-' for suitable simple PI-rings Qi, and by Theorem 1.7.20 1 = suitable ceZ(R), c i €Z(R) n Qi. Let e, = tic-'. Now R c C:= Rei and each Rei is a prime PI-ring containing R,ei, which is Noetherian (being a homomorphic image of Rl). Now, by Theorem 1.4.21 Re, is isomorphic to a submodule of a finite Z(Rei)-module, and clearly Z(Rei) is isomorphic to a . each Re, is Z(Re,)-submodule of Z(R) [by z h z c for z ~ Z ( R e , ) l Hence isomorphic to an R,e,-submodule of a finite dimensional R,e,-module; thus [Re,:R1] < a.ByRemark5.1.2,[R:R,] < coandRsatisfiesACC(1eftideals). By an analogous argument, R satisfies ACC(right ideals). QED

xr=

Cauchon's Theorem

We recall from Example 4.4.12 that a PI-ring may satisfy ACC(right ideals) without being Noetherian. However, for semiprime PI-rings the situation is much nicer, as first discovered by Cauchon [76] ; cf. Amitsur

PI. Theorem 5.1.7 (Cauchon's theorem). IfR is a prime PI-ring satis.fying ACC(ideaZs),then R is Noetherian.

226

[Ch. 5

NOETHERIAN PI-RINGS

Proof. Suppose R has PI-class n, and let r = n2. Take nonzero c = g,,(rl,.. .,rd)E gn(R).Let L be an arbitrary left ideal of R. For every s E L we have by Lemma 1.4.20 CS z zj(x)rjfor suitable zj(.u)Eg,(R) c Z ( R ) . Now let M = P. viewed as R-module. Letting z(x) = (I‘, (x), . . ,,z,(x))E Z(*’, one can easily verify that XxELz(.x)R= If= l z ( ~ i ) for R suitable xi E L. (Same idea as in the proof of Proposition 4.4.10.) Thus, given X E L , we can find a,, . . .,ak in R such that z(x) = z(xi)ai, so for all j zj(x) = lzj(xi)ai = Cf= laizj(.ui): then

,

xf=

f

k

k

/

f

cf=

\

k

k

x!=,

implying x = C:= ,aixi. We conclude that L = Rxi. Thus R satisfies ACC(1eft ideals J;symmetrically, R satisfies ACC(right ideals). QED Corollary 5.1.8 (Cauchon). If’ R is ACC(ideals), then R is Noetherian.

CI

semiprime ring saristving

Proof. By Proposition 4.4.13 and Remark 4.4.14, we can view R as a subdirect product o f Ri = R/Pi, i = 1, . . .,t, for suitable 1, suitable Pi E Spec(R), and each Ri is Noetherian. Thus each Ri is a left Noetherian R-module by the action r(x + P i ) = r.u + P I ,and so R 0. ‘ . 0R, is a left Noet herian R-module containing R as a submodule, so R is left Noetherian; likewise R is right Noetherian. QED

Here is another way of finding rings with ACC(idea1s) Remark 5.1.9. Suppose R satisfies ACC(idea1s)and S is a submonoid of Z(R). Then R I - ’ = R/Ann,S also satisfies ACC(ideals), and it follows easily that R, satisfies ACC(idea1s). The same argument holds for ACC(1eft ideals, resp. right ideals). The Eakin-Forrnanek Theorem

The question arises as to whether, given a Noetherian ring R, finite dimensional over a central subring C, we can conclude C is Noetherian. This was proved in the commutative case by Eakin, and a beautiful, general result was discobered by Formanek [73], whose proof was simplified greatly by Formanek’s trick of “forgetting” R was a ring! Namely, we consider modules over a commutative ring. Definition 5.1 .lo. If M is a C-module, an extended submodule is a submodule of the form A M for A a C . M satisfies ACC(extended submodules) if there is no infinite chain M I c M , . c ... of extended submodules.

$5.1.1

Sufficient Conditions for a PI-Ring

227

xf=,

Cyi is a C-module for y i E M. Remark 5.1.I1. Suppose M = For any ~ E CcM , = 0 iff cyi = Ofor all i, 1 < i < k . (Formanek). Suppose a commutatioe ring C has a Theorem 5.1 .I 2 faithful, finite-dimensional module M satisfying ACC(extended submodules). Then M is a lefi Noetherian C-module and C is a Noetheriun ring.

xt=

,Cyi for suitable y i € M. First we claim that it is Proof. Write M = enough to prove M is left Noetherian. Indeed, note by Remark 5.1.1 1 that C is a subdirect product of Ci = C/Ann,yi, 1 d i d k ; we have a module isomorphism C i2 C y i G M, so each C i would be a left Noetherian Cmodule, implying C is a left Noetherian C-module. Thus C would be Noetherian (because it is commutative), proving the claim. We shall thus prove the theorem by arriving at a contradiction from the assumption M is not left Noetherian. Let AM be an extended submodule of M, maximal with respect to MIAM not left Noetherian. MIAM is a faithful C/Ann,(M/AM)-module in the natural way; replacing M by MIAM, and C by C/Ann,.(M/AM), we may assume that for every 0 # B a C , M/BM is left Noetherian. Now, in view of Remark 5.1.1 1 and Zorn’s lemma there is a submodule M, of M, maximal with respect to M/M, being a faithful C-module. Take any submodule M’ of M with M , c M’. By assumption, MIM‘ is not faithful, so cM G M’ for some cEC. But cM = (cC)M is an extended submodule of M, implying M/cM is left Noetherian. Thus MIM’ is left Noetherian for all M‘ 3 M,, obviously implying MIM, is left Noetherian. But M/M, is also faithful, so by the claim of the first paragraph C is Noetherian; M being a finite-dimensional module over C, is thus left Noetherian, the desired contradiction. QED Corollary 5.1.13. Suppose R satisfies ACC(right ideals) and has a commutative subring C (not nrcessurily central!) over which R is a finitedimensional module. Then C is Noetherian.

Note that here we mixed ACC(right ideals) and module (which means left module by our convention), Formanek-Jategaonkar [74] proved that if R has ACC(1eft ideals) instead, then C is Noetherian anyway. However, for the important case of prime PI-rings it is easier to appeal to Cauchon’s theorem, yielding instantly Theorem 5.1.I4. Suppose R is u prime PI-ring satisfying ACC(ideals), aiid is a finite-dimensional module oi’er a commutative subring C. Then C is Noetherian.

NOWwe get the theorem of Formanek-Jategaonkar as a consequence in Exercise 7.

228

NOETHERIAN PI-RINGS

[Ch. 5

Example 5.1.15. We get in trouble when C is not commutative. Let W be Example 4.4.12, and R = M,(Q). R is a Noetherian ring and a finite dimensional W-module, but W is not Noetherian. Noetherian Counterexamples

Several questions arise from the theorems above. Most notably, one wonders whether the center of a Noetherian PI-ring R need be Noetherian. The answer is, “No!”, even if R is integral over Z ( R ) (cf. Exercise 2 ) . We shall now give ii class of examples (due to Cauchon [76]-Rowen [77a]) in which many pathological properties are to be found, including R not integral over Z ( R I [which, in oirw of Corollary 5.1.4. implies Z ( R ) is not Noetherian]. Recdl that a deriration D of a ring R is an additive map satisfying (.uy)D = (.uD)y+s(yD). Example 5.1.16. Let C be a commutative Noetherian domain with derivation D. We work in M 2 ( C ) . For any c‘ in C define i. = (cell+ c e , , + (cD)e12)EM,(C),and let H = (tic E C),acommutativesubring of M , ( C ) , isomorphic to C (via the map c -+ ?). Choose co E C and let R be the subring H+M,(C)co of M , ( C ) . R is spanned as H-module (and also as right H-module) by the elements {coeijI1 < i..j < 2) u { 1 ) . [Indeed, one need only show that every element of the form ccfleijis spanned by the coeijover H. For i = 1, cc0elj= i‘c,~.,~;for i = 2, c‘cOezj= k o e 2 j - E , c f l e , j , where c1 = cD.1 Thus R is Noetherian.

S

Remark 5.1.17. With the notation as in Example 5.1.16, let Then Rs = M 2 ( C S )proving , R is prime. Moreover, H n Z ( R )

= .(cLli3 1). = {c~cD = 0;.

Example 5.1.18 (Cauchon). A prime, Noetherian PI-ring not integral over its center. Let A = Q[&,%,] and P = L,A~spec(A).Let C = A, and and D the “ordinary” partial derivative apply Example 5.1.16 with co = i1 with respect to 2,. It is easy to see that Z ( R ) is the set of scalar matrices such that specializing I , - 0 gives an element of Q. Hence the element (A2ell+ A 2 e , , + e , z ) is not integral over Z ( R ) .

Cauchon’s example is interesting because it satisfies several “nice” properties( cf. Exercise 3). Example 5.1.1 9 (Rowen). A prime,affine, Noetherian PI-ring R such that Z ( R ) has a proper ideal A with AR = R . Let K be the field generated over Q by the commuting indeterminates .Y, y,, .vz, z , , z,, and let C be the Let Q-subalgebra of K generated by x, y , , y,, z , , z,, and ( I -J~,Z,)Z;~. C, = Q [ . Y , Z ~ . Z , , Z ~ ‘1 and extend the 0 derivation on C, to a derivation on

$5.2.1

The Theory of Noel herian P1-Rings

229

C,[y,,y2] by the conditions y,D = y 2 z 2 and y,D = y:; by restriction, D is also a derivation on C. We build R as in Example 5.1.16, with co = x. Obviously C is Noetherian, so R is prime Noetherian, and also clearly R is affine. Let Z = Z ( R ) and A = ( z , Z + z , Z ) a Z . Clearly 1 = z l y , + z 2 r , where r = ( 1 -y,z,)z;’, so AR = R . It remains to show A # Z . To d o this, we first show C n t,= { g E C l g D = 0). Indeed, (c) is trivial. Conversely, suppose gD = 0 and g = 1;=,,f;(y2)y\ for suitable f ) ~ C , [ y ~ ] , chosen with k minimal. The coefficient of y: in g D is ( h ( y 2 ) ) Dwhich , is thus 0; it follows that &(y2)is some nonzero element p in C,. If k > 0 then the coefficient of y:- in g D is 0 = (,f, - ( y 2 ) ) D+ kpy2z2,contrary to 1 # 0; thus k = Oand g = ~ E C , . Now if A = Z then zld, + z 2 d 2 = 1 for suitable d , € Z ; taking parts of degree 0 in x, we may assume die H , so we can write di= ?ifor suitable ci e C. By Remark 5.1.17 c,D = 0, so by the last paragraph C , E Cn C,. Thus tie Q [ ~ ~ , z ~ , z ; ~ ] , a n d z+, zc,,c 2 = I.Taking homogeneous part sin termsof z 2 , we may assume c1 = h,(z,) and c2 = h 2 ( z , ) z ; ’ for suitable h , ~ Q [ z , ] . But c2 E C, so c2 = ( ( 1 -y,z,)z; ‘)c for some C E C. Hence the degree of c2 in y, is at least 1, contrary to c2 E C , . This proves A # C, as desired. B. Mueller [77] has examined Example 5.1.19 in great depth, and has further generalized the class of Example 5.1.16. On the other hand, Mueller [76a] has techniques to analyze the prime spectrum of Noetherian PI-rings.

,

$5.2. The Theory of Noetherian PI-Rings

In this section we develop Noetherian PI-theory, with an eye toward developing the “classical” commutative results, such as the finite ranks of prime ideals. All the proofs require more module theory than required anywhere else in this book, and a close reading will show that polynomial identities enter only incidentally into many of the major results. The Principal Ideal Theorem

We start with perhaps the most basic theorem in Noetherian PI-rings, the “principal ideal theorem,” found by Jategaonkar [75a] ; later Jategaonkar [74b] showed that the PI-assumption is superfluous! Recently Chatters, Goldie, and Hajarnavis have found an easy proof of Jategaonkar’s results, based on an appropriate modification of the commutative proof in Kaplansky [70B]; we present their proof for PI-rings. First we need some easy facts about modules. Definition 5.2.1.

A composition series for a module M is a chain of

230

[Ch. 5

NOETHERIAN PI-RINGS

submodules M = M , 3 M , 2 ... 3 M , = 0 such that each M i - , / M i is irreducible; rn is the L q t h of the coniposition series. Proposition 5.2.2. (Jordan-Holder). Suppose M has a composition series of length m. 7 hen every cornposition series.for M has length m.

Proof. Induction on m. Suppose M = M , 3 M , 2 ... M , = 0 and M = M , 2 M ; ... 2 Mi = OarecompositionseriesforM.IfA4, = M‘, then we get M, 2 M i 3 ... 3 MI a composition series for M , of length t - 1 ; by induction, (r - 1 ) = (rn- I), so t = m, and we are done. If M I # M i , then, putting MI’ = M , n MI, 1 Q i < t, we get M, 3 M’,’ 2 M;’ 3 ... 3 Mi’ = 0 and M ’ , 3 M ; ? M ’ ; = , . . . z M ; = O . But M I ’ - , / M I ’ = ( M I ‘ - , + M I ) I M : c M i . - , / M : , so either MI’-] = MY or M ; - , / M : is irreducible. Thus, throwing out equal terms in the chains, we get a composition series M ; = MY, 3 MY2 3 ... 3 MYk = 0 for some k d t . By induction, m-1 = k 1 and t - 1 = k 1 (seen by looking at M I and M ; , respectively). Hence m = t . QED’ =I

+

+

Definition 5.2.3. If M is an R-module with a composition series of length m then M has length m.

If M is an R-module and S is a submonoid of Z = Z(R) we can form M, can be written in the form s-l 0y for suitable s in S, J in M ; we write this as ys-’, for short. Then M,y is an R,-module in the natural way (rs; ‘)(ys;’) = ( ~ y ) ( s , s , ) - ~ . M, = Z, O ZM . Every element of

Proposition 5.2.4. Suppose R is a semiprime PI-ring satisjjing ACC(annihi1ator ideals), S = {regular elements of Z(R)}, and M is a ,finitedimensional R-module. Then M,y has some length m, as R,-module.

Proof. By Theorem 1.7.34 Q,(R) = Rs has the form R , 0 . ’ .0 R, for suitable simple PI-rings R,, ..., R,. Note that if M = Ry, then M,s = R,yy,; hence M,? is finite dimensional over R,. Let M i = (R,+R,+..-+R,)M,. Then M , = M , 3 M k - , = , . . . 2 M o = 0 , and each M i / M i - can be viewed as an R,-module. But R i is finite dimensional over Z(Ri), a field, implying M , / M i - , is a finite-dimensional vector space over Z(Ri),and thus has finite length. Therefore M,s has finite length. QED Definition 5.2.5. With the notation as in Proposition 5.2.4, we call rn the reduced length of M, written as p ( M ) .

x;=,

x;=,

,

Proposition 5.2.6. Suppose R is a semiprime PI-ring sarisjjing ACC(annihi1ator ideals) and M I E M, are R-modules. Then p ( M 2 / M , ) + P W , ) = P(M2). Proof. Let S I= (regular elements of Z(R)} and let t+b: ( M 2 ) , ,-+ ( M 2 / M 1 ) , be the canonical module homomorphism $(ys-’) = (J,+ M , ) s - I . Clearly

35.2.1

The Theory of Noetherian PI-Rings

23 1

ker$ = (Ml)s,so Ic/ induces an isomorphism (M2),J(Ml)S:(M2/M1),<, and we conclude by applying the Jordan--Holdertheorem [to (M2)J. QED We would like to emphasize the following easy observation. Remark 5.2.7. Suppose M I E M are R-modules and AM G M, for some A d R. Then A E Ann,(M/M,), so M/M, can be viewed naturally as RIA-module by ( r + A ) ( y + M , ) = r y + M , . Definition 5.2.8. Suppose R is a PI-ring with ACC(idea1s) [so that, in particular, (Nil(R))' = 0 for some t], and let M be an R-module. Write MI = Nil(R)'-'M/Nil(R)'M, 1 < i < t, and viewing MI naturally as R/Nil(R)-module, define p(M) = ,p(MI).

xi=

Proposition 5.2.9. Suppose R is a PI-ring with ACC(ideafs). if M , G M, are arbitrary R-modules, then p(M,) = p(M,/M,)+p(M,). Proof. Follows easily from Proposition 5.2.6; details left to the reader. QED

Remark 5.2.10. Suppose M is an R-module and p(M) = 0. Then letting denote the canonical image in R/Nil(R), for each y E M we have r E R such that r y = 0 and f is regular in Z ( R ) . [Indeed, for each i, we have p(Nil(R)'-.'M/(Nil(R)'M)) = 0, implying, given y,eNil(R)'-' M, there is suitable r i e R such that ?; is regular in Z(R) and vjy,eNiI(R)'M. Take y1 = y , and inductively take y i + , = riyi for i 2 1. If Nil(R)' = 0, take r = r, . . . r, ; clearly ry = 0 and r = . .Fis regular in Z(R).]

<.

We also need a standard, easy fact about minimal primes. Lemma 5.2.11. Suppose P is a rank 0 prime ideal of a ring R. Then P n Z(R) contains no regular elements of R . Proof. Let S = { z a l z ~ Z ( Ris) regular and a l p } . O$S, so by Zorn's lemma there is some Ba R maximal with respect to B n S = 0. Then B G P , and B is prime. [Indeed, if A , , A, 3 B then Ai contains some z , a , ~ S , with a a,$P and z ; E Z ( R ) regular; then a , r a , $ P for some r in R, so z1z2(a1ra2) = z,a,rz,u,e(Sn A,A,), implying A , A , $Z B.] By hypothesis B = P , s o P r , S = ( Z i . QED

Theorem 5.2.12

(Principal ideal theorem: Jategaonkar). Suppose R

is a PI-ring with ACC(idea1s). For any ZEZ(R),i f f ~ S p e c ( Ris ) Rz-minimal,

then rank(P) < 1.

Proof. Suppose on the contrary that there is a chain P , c P , c P in Spec(R). Passing to RIP, we may assume P , = 0 (i.e., R is prime). Pick nonzero c in P , n Z ( R ) , and let A, = {rERIrzkERc}.Then A , 5 A, G '..

232

[Ch. 5

NOETHERIAN PI-RINGS

are ideals of R, so for some m, A i = A ; + for all i k m. Let x = Y”;for any r in R, if rx’ E Rc then rx E Rc. Let M = (Rx+ Kc)/Rx’, viewed naturally as R/Rx’-module, and let M I = (Rx2+Rc)/Rx2 z M . We claim that p ( M / M , )= 0 ; by Proposition 5.2.9 it suffices to find respective submodules M‘,M‘, of M , M , , such that M , / M ‘ , M’ and M / M ’ 2 M ; [because then, by comparing M 3 M ’ =) 0 and M , 3 M ; 3 0, we see that p ( M ) = p ( M , ) ] . Let M‘ = R.u/Rx’ and M‘, = (Rx’ Rxc), R.u’. Right multiplication by .Y yields a module isomorphism M / M ’ + M’,. On the other hand, there is a module homomorphism tj: R.u -+ M , / M ‘ , . given by + ( r s ) = rc+ (Rx’ Rxc). Clearly R.u’ c ker tj. Furthermore, if r.u E ker tj then rc = r l .u2 +r,uc for suitable ri E R, implying r 1 s 2E Re. so by assumption on s = zm,r,.y E Rc; writing r l x = r’c, we get rc = r’c3-+ rzxc = (r’+r,)xc, implying r = (r’+ rz).u and rx E R Y’. Therefore kertj = Rx’, so tj induces a module isomorphism M‘ :M,,’M’,, as desired, and the claim is established. Write - for the canonical image in R = (R/Rx2)/Nil(R/Rx2).By Remark 5.2.10 there is some r in R such that i is a regular element of Z ( R ) and r x E M , ; thus r x I= r,x2+r2c for suitable r l , r 2 in R. Then (r-r,x)x = r , c E P , . But x $ P , , so r - r l x E P l c P ; hence r E P. Therefore P contains the regular element F of Z ( R ) ,contrary to P being a rank 0 prime of R (cf. Lemma 5.2.1 1). QED

+

+

Incidentally, if we d o not assume :EZ(R) we may get into trouble; cf. Exercise 2 . Rank of Prime Ideals

Given the principal ideal theorem, it is a simple matter to prove all prime ideals have finite rank. Lemma 5.2.13. Suppose R is a PI-ring with ACC(idea1s). Given a chain P = Po =) P , 3 ... I> P , in Spec(R), and Z E P n Z ( R ) , we can frnd a chain P = Po 3 P’, =I ... 3 P k - , 2 P, in Spec(R) with Z E P k - l . Proof. Choose Po 3 P; 3 ... =) Pk-l P, with greatest possible , j such that Z E P;. I f j = m- 1 then we are done, so assume j d ni-2. (Write -~ P:, = P,.) In the prime ring R = R/P;+’, we have Pi 2 Pi+ 3 0. Thus by E Spec(R) such that is not 5-minimal, so there exists Theorem 5.2.12 z E Py.k c P;. Replacing P J + , by Py+ contradicts our choice of,j. QED

,

,

,

Theorem 5.2.14. Suppose R is a PI-ring with ACC(idea1s). Zf R B and if P€Spec(R) such that RB G P and rank(P!RB) = k in R/RB, then rank(P) d k + t . = C:=,Rb,

Proof.

Induction on t . Consider a chain P

2

P,

3

... 3 P,. By Lemma

$5.2.1

The Theory of Noetherian PI-Rings

,.

233

Now in R = R/Rb, we have a chain 5.2.13 we may assume b , E P,P 3 3 ... 2 Moreover,= = Rb,and R/= :R/RB; hence, b y i n d u c t i o n k + ( t - l ) > , r a n k ( P ) ~ m - 1 , s o m ~k + t . QED

G.

x:::

Definition 5.2.15. If B a Z ( R ) , we say B is spanned in R by t elements if RB = Rb, for suitable bj in B.

xf=,

Remark 5.2.16. If R satisfies ACC(idea1s) then every ideal B of Z(R) is finitely spanned in R. Proposition 5.2.17. Suppose R is a ring of PI-class n, with ACC(idea1s). I f P ~ s p e c , , ( Rand ) P nZ(R) is spanned in R by t elements, then rank(P) < t . Proof. Let B = P n Z(R). P is the only prime ideal of R lying over B, so rank(P/RB) = 0 (in RIRB), and we are done by Theorem 5.2.14. QED

Definition 5.2.18. Suppose R is a PI-ring with ACC(idea1s) and P E Spec(R). Let n be the PI-class of R/NiI(R) and define rk(P) by induction on n as follows: If PESpec,(R) then rk(P) is the smallest number of elements spanning P n Z(R) in R ; if P $ Spec,(R) and g,,(R) is spanned in R by t elements, then

rk(P) = t+rk(P/n{P’$ SpecJR)})

[in R/n{P‘$ Spec,,(R)}].

We are ready for a “finite rank of primes” theorem. Theorem 5.2.19. Suppose R is a PI-ring with ACC(ideals). !f PESpec(R), then rank(P) < rk(P) < a. Proof. Suppose R/Nil(R) has PI-class n. If P ~ s p e c , , ( R )then we are done by Proposition 5.2.17. Otherwise, letting N = n{P‘$Spec,(R)), we have by induction on n rk(P/N) 2 rank(P/N) (in R/N) = rank(P/Rg,(R)) [in R/Rg,(R)]. We conclude by Theorem 5.2.14. QED

The Intersection of Powers of the Jacobson Radical There is another famous result for a commutative Noetherian ring C , that

n& (Jac(C))’= 0; in fact, if C is a domain then for every A q C , nz A’ 0. This has significance in view of the fact that we can then form the A-adic completion of a commutative Noetherian ring, and one would =

like a parallel result for PI-rings with ACC(idea1s). However, there is the following counterexample of Herstein [65] : Example 5.2.20. subring ( A e , Qe,,

,+

Let A

+ Qe,,)

= { m / n E Qlm E Z, n odd E Z), and let R be the of M , ( Q ) . Then R satisfies ACC(right ideals),

234

NOETHERIAN PI-RINGS

[Ch. 5

ni..= ,

but (Jac(R))'# 0. [Indeed, let J = Jac(R). Clearly Jac(A) = (m;n E QI m is even and n is odd}, and J 3 Jac(A)e,, +Qe,,, implying J' 3 Qe,, for each i. Hence ( J' # 0.1 Another counterexample was found independently by Jategaonkar [69]. On the other hand, Jategaonkar [74a] obtained a positive result for Noetherian PI-rings; we present a pretty proof due to Schelter [75], based only on elementary properties of modules with composition series. First we need a characterization of modules having a composition series. Proposition 5.2.21. Let R be a PI-ring and let M be a .finitedimensional R-module. M has a composition series $.for some i n there are maximal ideals P , , . .,P , of R such that P , . .. P , M = 0.

Suppose M = M, 3 M , 3 ... 3 M, = 0 is a composition Proof. ( 3 ) series, and let Pi= Ann,(M,-,/Mi), 1 6 i < t. Then Pi is a primitive ideal, and thus maximal, by Kaplansky's theorem, and P ; . . P , M c M, = 0. (-=)Write M i= P i . * . P ,M, 1 < i < m, and M, = M . Then each Mi-,IMi can be viewed naturally as a finite-dimensional RIP,-module. But [R/P,:Z(R/P,)] is finite by Kaplansky's theorem, so Mi-JM, is a finitedimensional vector space over the field Z(R/P,). Therefore M 2 M, 3 ... I> M, = 0 can be refined to a composition series. QED Proposition 5.2.22. Suppose R is a PI-ring with ACC(idea1s). The following conditions are equirafent: (i) rank(R) = 0 ; (ii) every finitedimensional R-module has a composition series; (iii) R has a composition series as R-module ; (iv) eoery Jinite-dimensional right R-module has a composition series (of right R-modules); (v) R has a composition series us right R-module. Proof. (i) * (ii) Each P ESpec(R) is both minimal and maximal, and so by Proposition 4.4.13 there are only a finite number of prime ideals P,, . . .,P, of R . Then PI. . . P , c Nil(R), so for some t, by Proposition 4.4.11, (PI. .. Pk)*= 0; we are done by Proposition 5.2.21. (ii) (iii) Trivial. (iii) =-(i) Suppose P ESpec(R). By Proposition 5.2.21 for suitable maximal ideals P , of R we have P ; - - P , R = 0 c P , so some P i s P , implying P is maximal; thus rank(R) = 0. (i) (iv) (v) ==. (i) is the right-sided analog of (i) (ii)3 (iii) * (i). QED

-

There is one more module-theoretic notion entering very naturally into PI-t heory.

35.2.1

The Theory of Noetherian’PI-Rings

235

Definition 5.2.23.

A module M is an essential extension of a submodule M, if M, intersects all nonzero submodules of M nontrivially.

The next result is a consequence of the exercises to 41.1 1, but we give the proof anyway. Proposition 5.2.24. Suppose R is a prime ring ofPI-class n and M is a finite-dimensional R-module that is an essential extension of M,. Then cM s M, for some nonzero c ~ g , ( R ) .

,

Ry,, obviously we need only find nonzero Proof. Writing M = c i ~ g , ( R )such that c , y , ~ M , ,1 < i < n. Fixing i. let L = (rERlry,EM,}. For any X E R Rxyi is a submodule of M, so by hypothesis either xyi = OEM, or M, n Rxy, # 0; at any rate, L n fix # 0 for each X E R. Now let S = Z(R)-0. Clearly L, intersects every nonzero left ideal of R , nontrivially; since R , is simple PI, R, is the sum of minimal left ideals, each of which is contained in L,, so L, = R,. Thus L A Z ( R ) # 0; taking nonzero z € L n Z ( R ) and any nonzero c in g,(R), we have (cz)y,€Mi, as desired. QED Theorem 5.2.25. Suppose R is a Noetherian PI-ring and M is ajinitedimensional R-module that is an essential extension of a submodule M,. I f M , has a composition series, then so does M . Proof. Since M is a left Noetherian module, we can take a submodule N 2 MI of M, maximal with respect to N having a composition series. If N = M we are done; we shall arrive at a contradiction by assuming N c M. Take a submodule M ’ of M such that N c M’ and such that P = Ann,(M’/N) is maximal (among all possible such MI). Then P E Spec(R). [Indeed, if A , , A, are ideals of R and Ai 2 P then A,M’ $! N and, by assumption, Ann,((N + A,M‘)/N) s P , so A,A,M’ $ N.] Note that P cannot be a maximal ideal of R , because then refine M’ 3 N 3 0 to a composition series of M‘, contrary to assumption on N. Let A = Ann,N. By Proposition 5.2.21 there are maximal ideals Qi of R such that Q m . . . Q 1 E A; thus, going the other direction, we see that RIA has a composition series as RIA-module, so rank(R/A) = 0 by Proposition 5.2.22. Let denote the canonical image in the Noetherian ring RIAnn, M’. P is a finite dimensional right R-module; by Proposition 1.8.34, there are maximal ideals P j of R with 3 P P , 2 P P , P , 2 .... But P P , ... P,M’ E PM‘ c N, so each P P , “ ‘ P , is naturally a 1eji RIA-module; P is finite dimensional and thus of finite length as RIA-module, so some P P , ... P , = 0. Let N’ = P, ... P, M‘. If N’ c N then P , ... P, c P , so some Pi c P, contrary to P not maximal. Thus ( N ’ + N ) is a proper, essential extension of N ; also N ‘ + N c M and is thus finitely generated. ~~

236

[Ch. 5

NOETHERIAN PI-RINGS

Viewing ( N ' + N ) / . Y as an RIP-module, and using Proposition 5.224, we get some C E R!P such that c((N'+N)/N) = 0: thus Ann((N'+N)/N) 2 Ann(M'/N), contrary to the choice of M'. QED Schelter's proof has shown us exactly where we need ACC(right ideals), without which the result would of course be false. Jategaonkar's theorem is an immediate consequence. Theorem 5.2.26 (Jategaonkar). Ij' R is a Noetherian PI-ring, then = 0 for every finite-dimensional R-module M . I n particular, ,(Jac(R)')= 0

ni..= ,(Jac(R)'M)

Proof. Let J = Jac(R). We shall show for each y in M that yCJ"M for suitable m E Z C , and the theorem will follow instantly. Let M1 be a submodule of M maximal with respect to y f f M , . Letting M' = M/M,, a finite-dimensional R-module, we see that every nonzero submodule contains ( M , +Ry)/M,, which is thus irreducible. By Theorem 5.2.25 M' has a composition series, so by Proposition 5.2.21 P,. . . P , M' = 0 for suitable maximal ideals Piof R . Thus J"M' = 0, so J"M c M, and y f f J"M. QED

Corollary 5.2.27

ni..=,Jac(R)' is nilpotent. Proof.

(Cauchon). If R is a PI-ring with ACC(ideals),then

Pass to R/Nil(R), which is Noetherian, by Corollary 5.1.8; thus QED

0;,Jac(R)' c Nil(R), which is nilpotent.

See Schelter [76c] for further results concerning

nl.=I'M, with I d R.

Intersection of Powers of Ideals

(-)z,

I' = 0 for an ideal I of a We turn now t o the question of whether prime Noetherian PI-ring. One might expect this to be the case always, but in fact the opposite is often true, as evidenced by the following easy example. Call an ideal I of R idempotent if I 2 = I . Example 5.2.28 (Robson). A prime, Noetherian PI-ring, finitely spanned over its center, having idempotent maximal ideals. Let C be quasilocal commutative Noetherian domain [e.g., Z, for some P E Spec(Z)], let J = Jac(C), and let R be the subring ( C e , , + J e l 2 + C e , , + C e z 2 ) of M,(C). Obviously Z(R) % C , so R is finitely spanned over C and is Noetherian. On the other hand, R has a maximal ideal P = ( J e , , + J e , , + C e 2 , +Ce,,) with P 2 = P. (In fact, there are two idempotent, maximal ideals of R lying over J.)

There are a number of positive results however, which are in fact quite a

Ch. 51

Exercises

237

bit easier than Theorem 5.2.26, relying only on the easy commutative case (cf. Kaplansky [70B, Theorem 741). Theorem 5.2.29. Suppose R is a prime ring of PI-class n satisfying ACC(ideals), and P ESpec,(R). Then Pi= 0.

nl.=

Proof. Let S = Z(R)-P. Passing to R,, we may assume P n Z(R) is a maximal ideal of Z(R) and 1 €g,(R). Thus, by Theorem 1.4.32 R is finitely spanned over Z(R) and P = ( P n Z(R))R, so everything follows at once from the commutative version of Theorem 5.2.26. Q E D Corollary 5.2.30 (Robson-Small). Suppose R is a PI-ring with ACC(idea1s) and let N j = n{PESpecj(R)}for each j . Every idempotent prime ideal of R is Nj-minimal for some j ; thus there are only aJinite number of idempotent prime ideals of R.

Proof. Suppose PeSpecj(R) is idempotent. Then P contains some Nj-minimal Q in Specj(R), and by Theorem 5.2.29 PjQ = 0; thus P = Q . QED

EXER CI S ES 55.1 1. (Wadsworth-Schelter) Find an example of a field F with subfields F , , F , such that F is algebraic over F , nF,, [ F : F , n F , ] = cc, [ F : F , ] < co, and [ F : F , ] < m. The subring R = I M , ( F [ A ] ) + F , [ A ] e , , + F , [ l ] e , , of M , ( F [ I ] ) is prime Noetherian, but Z(R) is not Noet herian. 2. (Jensen-Jondrup [73] and Small) Nagarajan [68] gives an example of a commutative, Noetherian domain A with an automorphism of degree 2 such that the fixed subring C is not Noetherian. Show the skew polynomial ring is Noetherian and integral over its center. 3. (Cauchon) With the notation in Example 5.1.18, R has precisely one nonzero prime ideal J = c,M,(C), which is the unique largest left (resp. right) ideal of R. [ H i n t : By Remark 5.1.17, every prime ideal of R contains co, and thus J 2 , and thus J ; also R/J :Q(12), a division ring.] Likewise Z ( R ) is quasi-local with unique nonzero prime ideal coA,, but (coA,)R c J. Looking at lower triangular matrices, find a maximal commutative subring of R that is not Noet herian. 4. (Cauchon) Take C = R [ I , , A,], co = A,,D the partial derivative with respect to i, and , build R as in Example 5.1.16. Then INC fails from Z ( R ) to R. 5.

(Schelter) Working in A

A , = Q[,/%,d+1,,I2,A,$],

= Q(4+$)[A,,A2], and I = & A , + l , & A , .

let A , = Q [ 3 , & + i , , I Z , & h ] , Then I = 1,A = 1 , A 2 + i 2 $ A , ,

and A , n A z = Q [ J g ] + I , which is not Noetherian. Let R = A,e,,+A,e,,+le,,+~e,, c M,(A). R is a finite-dimensional A , @ A,-module and is thus Noetherian. Also R is affine and rank(R) = 2. So L O fails from Z ( R ) to R , and the “principal ideal theorem” fails for Z ( R ) ,since I is {A,}-minimal but not minimal in Spec(Z(R)). *6. (Raised by Jategaonkar and Small) Is there a Noetherian PI-ring which IS not finitely spanned over a commutative Noetherian subring? Is there a semiprime counterexample?

238

NOETHERIAN PI-RINGS

[Ch. 51

7. (Formanek- Jaregaonkar [74]) If R satisfies ACC(ided1s) and is a finite dimensional module over a commutative subring C , then C is Noetherian. [Hinr: It suffices to prove R is a left NoetherianC-module. Wemay assumethis holdsforR/l,foreveryO # IQR.1I-R issemiprime,use theproofofCorollary 5 I.8.Otherwise write R a s a submoduleofR/Nil(R) @ Nil(R),Nil(R)20 . .' , noting Nil(R) is nilpotent.]

45.2 1. Generalize Proposition 5.2.22 to a finitely generated extension R of a ring satisfying ACC(ideals), where R <: M , ( Z [ 5 ] ) . 2. (Small) Let R = M , ( H [ I ] ) and r = 2e,, +i.e,,. Then the prime containing r has rank 2. 3. If R is a PI-ring with ACC(ideals) and R is integral over Z(R) then every prime ideal of Z(R) minimal over a principal ideal has rank < 1. (This can be proved directly, using ideas of Kaplansky [70B] ; compare with Exercise 5.1.5.) 4. SupposeR has PI-classnand sdtisfiesACC(ideals),g,(R)hasaregularelement c o f R , A d R , A' = 0. and c'.$ A for all i 2 1. Then

,

In Exercises 5-8. arsume R is a finitely generated PI-extension of a ring M' satisfying ACC(idea1s). 5. Every prime ideal of R has finite rank. (Hint:Use Exercise 1.10.1.) 6. (Schelter) Assume R also is prime. With the notation as in $4.3, Rb is Noetherian and

finite dimensional over W;. 7. n ; Z I Jac(R)' is nil. [Him: We may assume R is prime; then, by Exercises 6 and 4.4.10 Jac(R)'.] 0= I Jac(Rb)' 2 8. R has only a finite number of idempotent prime ideals; if R is prime of PIclass n then 0 2 Pi= 0 for every P E S ~ ~ ~ , ( R ) .

nz

CHAPTER 6

THE THEORY OF THE FREE RING, APPLIED TO POLYNOMIAL IDENTITIES In Chapter 4 we saw how a combinational result about monomials in Z{X} led to the extremely powerful theorem of Shirshov. This chapter features a systematic investigation into Q { X } , in order to obtain a wide variety of interesting theorems, most notably Regev’s theorem that the tensor product of two PI-rings is a PI-ring. Most of the theory is due to Regev, and so this theory might well be called “Regev theory.” One feature of this theory is the explicit computation of bounds known to exist but otherwise completely inaccessible (for example, the k in Theorem 1.6.46). The methods are almost completely independent of the theory of the preceding chapters, although we use ideas related to the proof of Shirshov’s theorem. In the last section we present a theorem representative of work of the “Russian school,” due to Latyshev [76], that any T-ideal containing [XI, X,][X,, X , ] ... [ X Z n - , ,X,,,] is finitely generated.

$6.1.The Solution of the Tensor Product Question This section is spent on a proof of the following famous theorem of Regev Theorem 6.1.I (The tensor product theorem). The tensor product qfany two PI-rings is n PI-ring.

This result was discussed somewhat in $1.8, but here we give Regev’s approach, which has opened the way for further investigation of PI-rings. After concluding the proof, we shall pursue this theory of Regev, for further elaboration in $6.2. One of the key steps of the proof of the tensor product theorem has been shortened considerably by Latyshev, through the use of Dilworth’s theorem from combinatorics. We shall give the proof of the tensor product theorem backward, by reducing to a certain assertion and then verifying the assertion combinatorially. Clearly it suffices to prove that the tensor product of any two relatively free PI-rings is a PI-ring or, equivalently, if I , and I , are T-ideals 239

240

[Ch. 6

THE THEORY OF THE FREE RING

of Z{X}, that (Z{X } / I , ) O n( Z { X } / 1 2 )is necessarily a PI-ring. This leads us to a study of T-ideals or, more precisely, the multilinear polynomials in these 7'-ideals. Let C be a commutative ring; for k E Z + , let 5 be the Csubspace of C { X j spanned by .(X,,X,,...X,,ln~Sym(k)}, and let V , = C . Remark 6.1.2. Vk is the set of multilinear polynomials of degree k in X , , . . ., X, with coefficients in C. [V,: C] = k ! . Definition 6.1.3. If I is a T-ideal of C{X), define 1, c k ( I )= [i$k/lk :C ] ; ck(I)is called the k-codimension of I .

=I n

V,, and

Obviously c k ( I )d k ! , and we shall now see the importance of having c k ( I ) much less than k ! (Regev [72]). Suppose C = L and I"), I ( 2 ' are T-irleals Lemma 6.1.4 of Z{X} such that ,for some n, c , , ( I ( ' ) ) c , , ( I ( ~ )<) n!. Then R = ( Z { X } / I ( ' ]@Il ) (Z{ X)/Z(2))satisfies a multilinear, R-strong identity (with coeficients i n Z ) of degree n ;hence R is a PI-ring. Proof. For j = 1, 2 write d j = c,,(I(j)).Then suitable { h ; j ) ( X , ., . . ,X,)l 1 d i < d j } span V,/I!,J'; for each TC E Sym(n), we can find suitable m:!! in Z

such that X n l . . . X n n - C ~mi?hjj)EIy). ~, We write formallyf(X,, ..., X , ) = ~ n t S y m ( n l t n...Xn,,, X n , and aim to find suitable t , in Z such that ,f is an identity of R ;we shall be done if the t , are relatively prime (i.e., En Zr, = Z). Take arbitrary r l j ,..., rnj in Z [ X } / P . Then for all TC in Sym(n). rKl,j...rnn,j = 1% m i / ~ h ~ j ) ( r. .l ., i,,rnj),so

=

c c( c

I=

1 u= 1

neSym(nJ

t , r n ! ~ ~ r n L/ i~! ~l ) ( r l .. l , . ,r , , )

o h : 2 ) ( r l 2 .. . . ,r,,2).

Thus we need only find relatively prime integers r" such that ~ n F S , m ( n ~ t n m ~=, 0l ~for i n ~1f < ! i < d , and 1 < u < d , . But we have d l d 2 homogeneous equations in n ! indeterminates, which must have a nontrivial solution because, by hypothesis, d , d 2 < t i ! . QED The whole tensor product theorem therefore hinges on getting a suitable bound for codimensions. Now note that for any k there exists n E Z ' with n! > k". Thus we have the following reduction. Remark 6.1.5.

To prove the tensor product theorem, it suffices to

$6.1.]

The Solution of the Tensor

24 1

prove the following statement for a T-ideal I of Z{X}: If Z { X ) / I satisfies a polynomial identity of degree d, then, for some number m(d) independent of n, c , ( I ) < rn(d)” for all n. Regev concluded his proof by showing c,,(I)< (3.4d-3)”.We shall obtain a better result by following Latyshev [72], which relies entirely on a theorem of Dilworth with a pretty constructive proof by Amitsur. Definition 6.1.6. For nESym(n), define p ( n ) to be the maximal such that there exist i , < i , < . . . < i , ~ { l , ..., n } such that ~ ( i ,> ) n(i,) > . . . > n(ik).(Compare with Definition 4.2.3.)

k

For example, if n = (i:;::), then p ( n ) = 4, because 1 < 2 < 3 < 5 and ~ ( 1>) n(2) > 743) > 745). To study p ( n ) [for ncESym(n)] we shall use the Amitsur tables of n. Write two parallel tables T,(n) = ( t i j ) and T,(n) = (uij) as follows: t , = 1 and u1 = n l . Inductively, t , (if it exists) is the smallest k , t l , j - l < k d n, such that nk > ul.j- ; u I j = nk. Continue along the first row as far as possible and then start t h e second row. We build the second

,

row by taking the smallest k not appearing in the first row and letting = k, u 2 , = nk. Continue as in the first row but ignoring all k which already appear in T I .Thus, when finished, TI and T, each have n entries. t,,

Example 6.1.7.

Let n

=

(if::;:); then p ( n ) = 3 and

We are now ready for Amitsur’s formulation of Dilworth’s theorem. (Dilworth [50]).

Theorem 6.1.8

p ( n ) is the number ofrows i r i T,(n).

Proof. Let d be the number of rows in T,(n). If i , < ... < ik and n(il) > > n(i& then i,,..., ik are in distinct rows, implying p(n) < d. Conversely, we build a sequence id, ..., i, as follows: id = t,, and, inductively, given i,, from row ( m l), take i , = t,,, with j maximal, such that t,, < i,,,. Note that u,, > n(i,+,) since otherwise, when constructing the tables, we should have put i, c 1 on row rn of T,(x). Thus i , < ... < id and n ( i l ) > ... > n(id),implying p ( n ) 2 d . Thus p(n) = d. QED 1 . .

,

+

Before applying Dilworth’s theorem, we look at the connection to codimensions. Remark 6.1.9. The restriction to V , of the (partial) ordering of # ( X ) in Definition 1.1.3 coincides with the lexicographic ordering. Theorem 6.1 . I 0

(Latyshev [72]).

If I is a Tided o f C { X ) such that

242

THE THEORY OF THE FREE RING

C{X}/I satisfies a polynomial identity g of degree d, theri ( . , ( I ) {nESym(n)lp(n)< d}.

[Ch. 6

< order

of

Let S = {Xrl...X,,lr~Sym(n)and p ( t ) < d}, and let A be the C-subspace of C {X} spanned by S . We are done if A + I , = V,, so assume ) that h = X,,...X,,$A+l,; take n such that there is some x ~ S y m ( nsuch h is minimal possible (under the ordering of Definition 1.1.3). Since p ( n ) 3 d,thereexistii < ... < i,withn(i,) > ... > n(id).Writeh = / I , X , , , ~ , X , , ~ . . . hdXnidhd+, for suitable h , ~ . f / ( X Then ). h-h,g(X,,,h,,X,,,/i, ,..., X F i , h d + * ) is a sum of monomials of smaller order, and so (by induction) is in A + I , , . But h,g(X,,,,h,,. . .) E I , , implying h E A + I , , contrary to assumption. QED Proof.

To conclude the proof of the tensor product theorem, we need only estimate {IrESym(n)Ip(n)< d). Remark 6.1.11. For every nESym(n) such that p ( x ) < d we can view T,(n) [and T,(n)] as functions from { 1 , . . . ,n) to (d- 1) (since the elements of each row ascend). Since T,(n) and T2(n) determine TC, the order of {nESym(n)Ip(x)< d} < ((d= (d-1)2". In particular, in Theorem 6.1.10 c , ( I ) < (d-- l),,.

In view of Remark 6.1.5, we have concluded the proof of the tensor product theorem. Relation of Codimensions to Specht's Problem

The crux of the tensor product theorem was the estimation of the codimension and we note that Remark 6.1.11 can be improved (cf. Exercise 1 ) to give a better bound on the codimension of a T-ideal containing a polynomial of degree d. Codimensions are intriguing for the following reasons. Remark 6.1.12. Suppose JE Q(X) is completely homogeneous. Iff is not an identity of a Q-algebra R , then every multilinearization off is not an identity of R . (Just induct on the multilinearization step; details are left to the reader.) Then if I , J are T-ideals of Q{ X ) with I , = J , , I and J contain the same completely homogeneous polynomials of degree n. Proposition 6.1.13. Suppose I c J are T-ideals of Q{X} and for some k, c,(I) = c,,(J).for all n > k. ! f I i s finitely generated, then J is.finitely generated. (In fact, J is generated by I and a finite number of multilinear polynomials of degree d k).

Proof.

Ever! polynomial of J is a sum of completely homogeneous polynomials, so Remark 6.1.12 yields the assertion easily. QED

36.2.1

243

Representations of Sym ( i i )

This idea is illustrated in Krakowski-Regev [73], which, by a codimension computation. shows that the T-ideal of identities of the exterior Q-algebra is generated by [ [ X I , X,], X,]. (Another, easier proof is given in Exercises 6.3.3, 6.3.4;the result is due to Latyshev [63b].) At present it seems virtually impossible to find a method of computing c,(I) for arbitrary T-ideals I. However, for I = .f(M,(Q)),one may be able to compute c,(I) for suitable large i i by the Krakowsky-Regev method; find some upper bound, and prove that this equals c,(J), where J is the T-ideal generated by some nice identity of M , ( Q ) . Using Proposition 6.1.13 one could then conclude I was finitely generated.

46.2. Representations of Sym(n) In this section, following Regev [78a]. we extend the theory or $6.1 in the case C = Q (for sake of simplicity). We shall use the classical theory of the representations of Sym(n) to obtain a more precise description of I,, where I is a T-ideal of Q{ X) containing a "nice" polynomial. For example, this description yields important information when I contains a Capelli polynomial. (There are also applications to trace identities, but these lie outside the scope of this book.) Recall from $6.1 that V, is the Q-subspace of Q ( X ) spanned by all As a Q-vector space, V, is isomorphic to the group [ X r l... X,,17~~Sym(k)). algebra Q[Sym(k)], and we use this identification to view V, itself as a Qalgebra; i.e., ( X r l ... X,,)(X,, ... X,,) = X n r l . . . X n r kfor 71, rESym(k). [Recall m i means ~ ( 7 1 ( i ) ) . ] Remark 6.2.1. Suppose and 71 E Sym(k).Then (xrl'.. xrk),/'(xl?. . . Xk) 9

,f'(X,,. .., X,) = x.rar Xnrl

" '

=

~ - r t S ~ m ( k )" .rXi X , , Ei l

Xnrk

=

./'(x" I? '.

,

9

V,,

xr&);

consequently, for any T-ideal I of Q X! , I, is a /
To study 1, further, we want to look at the group algebra Q[Sym(k)], which we denote as A for the rest of this section. Remark 6.2.2.

x,,a,n-', where

a, E

A has an involution (*), given by ( ~ K E S y m , k , = a,~)*

Q.

244

[Ch. 6

THE THEORY OF THE FREE RING

Proposition 6.2.3.

A is semisimple.

Proof. Since .4 is finite dimensional over Q, it suffices to show '4 is & n in B we get 0 semiprime. If B a ( A , *) and B2 = 0, then for any b = = bb*, the coefficient of 1 being 0 = I n B: ; since each Bn E Q, we conclude that each Pn = 0, SO b = 0 for all h in B. Hence A is semiprime. QED

zK

(We are about to forget about the involution on A, which makes me feel uneasy because il might provide more information about V,,, especially about polynomials that correspond to the symmetric elements of A . ) At any rate, by Proposition 6.2.3, A is a finite direct sum of simple components Mn,(Di),where each Di is a finite-dimensional division algebra over 02;one is very interested in computing the number of simple components, as well as the actual idempotents e(iij)of M n i ( D i ) ,1 d j d ni. There is a very picturesque method of doing this, due to Young, which we now describe briefly (without many proofs); for a more thorough account the reader could turn to Dieudonne-Carrel1 [70B] or Miller [72B, Chapter 41. Definition 6.2.4. A descending partition ( m l , ..., m u ) of k is a sequence m, 2 m, 3 ... 2 mu > 0 for some u, such that Cr=lmu = k . To each descending partition (m,, . . . ,mu)of k we construct a ,frame consisting of u rows of boxes such that row i has mi boxes. A tableau is a frame in which we have written the distinct numbers 1 , . . . ,k, one number per box. Given a tableau 7.write nij for the number appearing in boxj of row i ; the nij are called the erttries of T. Example 6.2.5. The typical tableau corresponding to the descending partition (3, 1, 1) of 5 is

\

)I 1 2

I

n13

1

)I3 1

Obviously there are 5 ! such tableaux corresponding to this partition. il

Definition 6.2.6. < i2 and j , < j ,

A tableau is stundard if

H , , ~ ,d

whenever both

Example 6.2.7. By inspection, let us count the number of standard tableaux for the frame of Example 6.2.5. Obviously n l l must be 1, and we can put any ascending pair of numbers in the other two boxes of the first row, determining a standard tableau; thus there are )(: = 6 standard tableaux corresponding to the partition (3, 1, 1). Now each n ~ S y m ( k acts ) on a given tableau T, replacing nij by ~ ( n , ~ ) . Write T, for the new tableau obtained by the action of n on T. We say K is a

Representations of Sym(n)

$6.2.1

245

row permutation if, for each i, j, n ( n i j ) E { n i l..., r nim,} (i.e., n induces a permutation of each row); n is a column permutation if, for each i, j , n ( n i j ) E { n l j , n Z...}. j , Write P ( T ) for the set of row permutations of T, and Q ( T )for the set of column permutations of T. Obviously P ( T )and Q ( T )are subgroups of Sym(k), and P ( T )n Q ( T )= (1).

ny=

Remark 6.2.8. P(T)z Sym(m,), the product of the permutation groups of each row; likewise Q ( T )is the product of the permutation groups of each column. Definition 6.2.9.

An element

eE A

is semi-idempotent if e2 = ae for

some a # 0 in Q. The use of semi-idempotents is merely a convenience because if ez = ae, then a-’e is idempotent. Definition 6.2.10.

Given a tableau T, let 4 T ) = ~nEP(1’),itV(7.)(Sg~)7L~.

In the next three theorems we describe almost completely the structure of A in terms of tableaux.

Theorem 6.2.11. For any tableau T, e ( T ) is a semi-idempotent of A , and A e ( T ) is a minimal left ideal of A (thus lying in some simple component of A ) . Tableaux Tl and T, correspond to the same frame i$Ae(T,) and Ae(T,) lie in the same simple component of A , if e(T,) = .rr-’e(T1)7c.for suitable IT in Sym(k). Proof.

Cf. Miller [72B, Theorem 4.11 (this is fairly straightforward).

Theorem 6.2.12. Every minimal leji ideal of A is isomorphic (as Amodule) to some corresponding leff ideal A e ( T )for a suitable standard tableau T. [ I n fact, as A-modules, A z @ A e ( T ) , summed over all standard tableaux of partifions ofk.] Proof.

Cf. Miller [72B, Theorem 4.121 (much more difficult).

Theorem 6.2.13. I f T is a tableau of a frame F , then [ A e ( T ) :Q] is the number of standard tableaux corresponding to F.

Proof.

Cf. Miller [72B, Theorem 4.21 (also quite difficult).

We are led to search for a good formula for the number of standard tableaux corresponding to a given frame. The most natural formula is due to Frame, Robinson, and Thrall, and is called the “hook formula.” Definition 6.2.14. Suppose we have a frame constructed from the descending partition ( m l ,...,mu) of k. For the box in row i , column j , we

246

[Ch. 6

THE THEORY OF THE FREE RING

associate the hook number hi,, the number of {(i’,j‘)li d i’ < u, j < j’ 6 mi, satisfying the additional condition (i’ = i or j’ = j ) ) . (The hook number intuitively is the numbers of boxes in the “hook” extending to the right and down from the given box.) Example 6.2.15. In Example 6.2.5 we have hook numbers h , , = 5. h , , = 2, h , , = 1. h,, = 2, and h,, = 1. In the frame corresponding to the descending partition (3, 2) of 5, we have hook numbers h , , = 4, hI3 = 3, h l , = 1, h,, = 2, h2, = 1. Theorem 6.2.16 (Hook formula). The number of standard tableaux corresponding to the descending partition ( m , ,. . . ,mu) of k is equal to k ! / n { h i j I1 < i u, 1 < j < mi}. Proof.

Given in Exercises 2-8.

QED

Obtaining Polynomial Identities from Young Diagrams

Having raced through the theory of Young diagrams, we shall now see how to get polynomial identities from them. The main idea is that for any T-ideal I of Q{X},I , is a left ideal of V, (given the multiplication of the group algebra A ) , and thus contains a minimal left ideal isomorphic to some A e ( T ) for some standard tableau T. Let ,it, denote the element e( T), written out as a polynomial. We can write?;. explicitly; namely, let 1,. = x - ~ r r Q ( T ) ( ~ g ~‘ ”) X~r ,k l, and then .fr = L:ntP(7.).f7’(Xn1, . . - ,x m k ) . Proposition 6.2.17. Suppose I is a T-ideal of O{X} such that e( T )E I , and let n i j denote the ij entry of T. If $ is a homomorphism of Q{ X } such that $ ( X n j , )= Xi for all 1 d i < u. 1 < j < mi (notation us in Definition 6.2.4), then $ ( j > ) ~ l . Proof.

For all n € P ( T ) ,$ ( X r n , ,= ) X i = $ ( X n , , ) Hence .

( m , ! .. . m, ! )$(.ji.)=

$(fdXr

. . . ,X n k ) )= Ic/(.$)

E

I.

QED

ntP(T)

Remark 6.2.18. Suppose column j of T has length yj, If T has entries nij = i + q p for all i, j , then

xiz\

fT

= S,,(X,,

...~x,,)s,I(x,,+I,..., x4,+92)...’

a product of standard polynomials (straightforward verification). Proposition 6.2.19. Suppose I is a T-ideal of Q ( X } and for some .frame F we hai*ef,-EIfor all tableaux T qf F ; also suppose F has v columns, column j hutling length q j , 1 ,< j < u. Then S,, ( X I ,. . . , X , , ) . . . S,”(XI,.. ., x q LE )1.

46.2.1 Proof.

247

Representations of Sym(n) Combine Proposition 6.2.17 and Remark 6.2.18.

QED

We shall get optimal use of Proposition 6.2.19 by bringing in codimensions. Theorem 6.2.20

(Regev [78a]). Let I be a Tideal q f Q ( X ) . .Ifthere ( m , , . . . ,i n , ) of k who~eframeF has more than c k ( l ) stantlard tableaux, then I , contains the (2-sided) ideal of V, corresponding to F (rf:Theorem 6.2.1 1 ). i s u descending purfitiori

Proof. Let n be the number of standard tableaux of F , and let B be the ideal of V, corresponding to F. Then B is a sum of minimal left ideals, so it suffices to show L c I , for every minimal left ideal L of V, contained in B. Otherwise L n I , = 0, implying

[ h:01 3 [ L : Q] contrary to hypothesis.

f[fk:

Q], so ( . , ( I ) 3 [ L : Q] = I T ,

QED

We shall now prove that every T-ideal satisfies Theorem 6.2.20 with respect to a very nice frame; all we d o is confront the hook formula with the upper bound of the codirnensions (Remark 6.1.1 1). The computation used here is due to Amitsur. Lemma 6.2.21. Let F be the “r~~ctangular,fratne” corresponding to the descending partition (n7,. . .,m ) o f ! (i.e., mi = m, 1 < i 6 u, and k = mu). Let e be the base qj’ the natural logarithms. !/’ I is a T-ideal of O{X ) with ck(Z) G ak, where (mu)/(m+u)3 eaj2, then F has more than c h ( l ) standard tab1eau.u. Proof. Let hij be the hook numbers of F. Then, since the arithmetic mean is greater than the geometric mean, we have

=

( ~ n u ( u + l ) + n ~ ( t nI -) u ) / 2 k = (u+in)/2

It is also easy to see (through integration by parts) that k(log(k)- I ) + 1

=

1:

log(x)tl.u <

k

1 log(t)= log(k!), I=

I

where log denotes the natural logarithm, so ( I J I U / ~ ) ’< k ! . By hypothesis, a < ( m u / e ) ( 2 / ( m + u )<) ( k ! / u , , j h i j ) ’ and so, by the hook formula, the number of standard tableaux of F > ak 2 c k ( l ) . QED

248

THE THEORY OF THE FREE RING

[Ch. 6

Now, by Remark 6.1.11 we can apply Lemma 6.2.21, with a = (d- 1)2 to Theorem 6.2.20 and Proposition 6.2.19 to obtain instantly the following result of Regev (slightly improved by Amitsur). Theorem 6.2.22. Let I be a T-ideal of Q{X} containing a multilinear polynomial of degree d . If (mu)/(m+ u) > (d - 1)’e/2, where e is the base of the natural logarithms, then I , contains the ideal of V, corresponding to the rectangular frame of u rows and m columns; in particular, 6 1. ( S , ( X , , ..., Corollary 6.2.23. Suppose R is a PI-algebra (over Q) satisfying a polynomial identity of degree d , and let p = (d - 1)2e/2. I f m > p u / ( u - B ) , then ( S J m is an identity of R .

Thus we get a version of Theorem 1.6.46 with two differences: m is explicitly computed (and is not very large), but the minimal permissible value of u is the smallest integer larger than ( d - 1),e/2 z 1.359(d- 1)2 (as opposed to 2[d/2]). Regev [78a] has shown that Corollary 6.2.23 holds for arbitrary PI-rings, by means of the theory of Young diagrams in characteristic p ; he also uses Theorem 6.2.22 to strengthen his tensor product theorem. Of course, all of these results would be improved by a better estimate of the codimensions, a feat recently achieved by Amitsur (in preparation). Regev [78b]-Amitsur also used these ideas to characterize the Capelli identity (cf. Exercise 9), leading to some interesting results of Regev [78b] that (for low degrees) the standard identity implies the Capelli identity; for in our terminology. Amitsur and Regev also have example, C , ES~(&D{X}) used further results on Young diagrams to examine Capelli polynomials even more closely. 56.3. Finite Generation of Certain 7-Ideals

In this short section we give a purely combinatorial attack on Specht’s problem, following Latyshev [76], who proves that every T-ideal of Q{X) containing [XI, X,] . . - [ X 2 n - 1X2J , is finitely generated. I do not believe that this theorem directly implies the corollaries Latyshev [76] claims, for he seems to misapply Amitsur [55b], and also quotes Latyshev [66], which is also wrong (in lemma 1). (In fact, contrary to Latyshev’s papers, it is very tricky to verify that a given algebra satisfies an identity [X,, X,] ... [ X 2 , - , ,X,,] ; we shall discuss this question soon.) Nevertheless, Latyshev’s theorem has considerable interest, illustrating further the role of combinatorics in PI-theory, and possibly his method can be extended. We start by examining sets with partial order.

56.3.1

249

Finite Generation Higman's Theorem

Definition 6.3.1. Suppose S is a set with partial order Q . Let H ( S ) sequences of elements of S } , and extend Q to H ( S ) by the following rule: (sl,, . . ,s,) ,< (s;, . . . , s:) iff there is a monotonically increasfor all i, 1 d i d t. ing function rl/: Z + -,Z + such that $(f)Q u and si e = (finite

For example, if S = L with the usual order, we then have (1, 2, 4) < (5, 2, 3, 1, 7) [where $(1) = 2, 1/42) = 3, and $(3) = 51, but (1, 2, 4) 6 ( 6 5 , 1,2,3,3,3, 1). The relation Q is in fact a partial order on H ( S ) . [Indeed, reflexivity and transitivity are clear, so it remains to show that if (sl,. . ., s t ) Q (s;, . . ., s): and (si,.. . ,s:) d (s,,. . .,st), then t = u and s; = si, 1 6 i e t . With $ as in Definition 6.3.1, clearly t Q u ; likewise, u Q t , so u = t, implying $(if = i, 1 < i < f ; the rest is now immediate.]

Remark 6.3.2.

Definition 6.3.3. An ascending sequence of a partially ordered set ( S , <) is a sequence (s,, s2,. . .) such that si ,< si+ for all i. S is ascending if every infinite sequence of S has an infinite ascending subsequence.

,

The key result is the following special case of Higman [52]: Theorem 6.3.4 (Higman's theorem). Suppose S is a finite set with the discrete (partial) order e , i.e., each pair of unequal elements is incomparable. Then ( H ( S ) , < ) is ascending.

Proof. Consider an infinite sequence P of H(S). We are done unless every ascending subsequence of P is finite; so let P' be the subsequence of P formed by taking each element of H ( S ) which is the final element of a maximal ascending subsequence of P. Clearly P' is infinite; moreover, writing P' = (il,iz, ...) for suitable ii in H ( S ) , we see that ii 6 ij for all i < j . This condition shall lead us to a desired contradiction. Erasing parentheses and commas, we shall view elements of H ( S ) as words composed of elements of S . Suppose S has n elements; inductively assume the theorem is true for any set of < n elements. Choose P' such that the length of i1is minimal ;write il = si2 Bear in mind that we may replace P' by any infinite subsequence starting with i,. By induction hypothesis P' has an infinite subsequence each member of which contains s, so we may assume each fi = i,,si,,wheresdoesnot appeariniIi(andi,, isblank).Theni,,,i,,, ... has an infinite ascending subsequence ;hence we may assume i1 < i, < * .Also i, 6 ii,soiZ1 isnot b1ank;byassumptiononthelengthofthefirsttermofP',the sequencei,,,i,,, . . .has twoelements izid bzj.ThusPi = iIisSziQ iIjsizj= ij, contradiction. QED

,.

, ,

250

THE THEORY OF THE FREE RING

[Ch. 6

Definition 6.3.5. A sequence ( i l , i 2 , ..., i k ) has height t if it can be partitioned into r consecutive ascending sequences (with t taken to be as small as possible).

For example (1,4,7, 2, 5 , 9 , 3 , 6 , 8, 11, 10) can be partitioned into (1,4,7), (2, 5, 9), (3, 6. 8, l l ) , (lo), and thus has height 4 ; ( 1 , 2, 5. 3) can be partitioned into (1,2,5), (3), and thus has height 2. If i is a sequence of height t all of whose elements are in { 1 , . . .,k } , then i can be written as a t x k matrix as follows. Partition S into t consecutive ascending subsequences i I , ...,if; i f j appears in ii,write 1 in the ij position; otherwise write 0 in the ij position. This is called the matrix representation of iand is clearly well defined ;moreover, if S, S' are distinct permutations of (1,. . . ,k ) each having height t, then obviously their matrix representations are distinct. For example, the matrix representation of (1, 4, 2, 3, 5, 6) is

.):%A(

Let Pk, = {n E Sym(k)Ithe sequence (nl, n2,. . . ,nk) has height < f ] , and Pkf.The matrix representation of any element of P,, is a t x k let P, = matrix, which can be viewed as a sequence of k columns, each column in turn taken from 10, I}('). Give {0, lj.([)the discrete ordering, i.e., any two columns are either equal or incomparable; since {O, 1)'" is finite, it is ascending. Thus, we can apply Higman's theorem to H ( ( 0 , l } ' r ) ) with the induced partial ordering of Definition 6.3.1 to obtain the following observation : For any infinite sequence (n,,n2,...) of members of P,, the matrix representation of each is in H ( i 0 , l}'f));hence there are i < j such that the matrix representation of xj is obtained from the matrix representation of ni by inserting suitable columns in suitable places. For example, we might have x i = 142356 and nj = 15723468; here we obtained (AyyyAyAy) from (AyyAyy) by inserting (:) after column 1 and after column 5. Then, for ni in Sym(ki) and nj in Sym(kj), there is a monotonically increasing function $: 1 , . . . ,ki} + (1, ..., k j } such that njlnj2...xjkj = w,$(nil) x w2$(ni2)...~~,,I,h(niki)wk,+ for suitable words w,. [We obtain $ by ignoring the inserted columns; continuing the above example, I,h respectively sends 142356 to 153468, so that nj = 15723468 = $(1)$(4)72$(2)$(3h+b(5)$(6).]

uz=

(A)

Application of the Combinatoric Method to Polynomials

For the remainder of this section, we adopt the following conventions: ( f l , . . .rfi)T denotes the T-ideal of Q(X) generated by polynomials fi, . . . , A ; V, is the Q-subspace of Q{X} spanned by all I X , l . . . X , , ) ~ ~ S y ~ n ( k )and } , for 1 E Q { X ) , I , = I n V,. The height of the

56.3.1

Finite Generation

25 1

sequence (711,. . .,zk) is also called the height of X , , ...X,,. We shall use the order on monomials given by Definition 1.1.3; if gEQ{X), write L(g) for the monomial of g having largest order. Definition 6.3.6. Suppose f is a multilinear polynomial, and J ( f ) T . Say f is n-Latysheu if for every T-ideal I zJ , each k, and every g in 1,we havegEJk+{hEl,lL(h) d L(g)and L(h) has height dn}. =

Proposition 6.3.7.

[X,, X,] ... [X2n-1,X,,] is 2n-Latysheo.

Proof. Let J = ([X1,X,]~~.[X2,-,,X2n])T and take ~ E Z and + the T-ideal I zJ arbitrarily. For g in I,, write V(g) = J , { h E I,JL(h) d L ( g ) and L(h) has height < 2n). We are done unless g $ V ( g ) for some g in 1,; take such g with L(g) minimal. We may assume the coefficient of L ( g ) is 1. Obviously L ( g ) has height > 2n so, writing L ( g ) = X , , . * * X n kwe , have 1 < i, < < i,, < k such that n(iU)> n(iu+l), 1 < u < 2n. Write i2,,+, = k + l , yo = X , l . . . X n ( i l - l ) (if i , = 1, take yo = I), and, for 1 < j d n, Y 2 j - 1 = X a ( i z J - I ) , Y z j = X = ( i ~ , . l + ~ ) . . . X ~ ~ let i ~ Jg’=g-yo[y,,y,]x +l-l)~ [y3,y4] . . . [ Y ~ ~ - ~ , . Y ~ ,Since ,]. L ( g ) = y o y l y , “‘y,,, has been cancelled, we have L(g’)c L(g), so by assumption g’e V(g’) L V ( g ) .Hence g E V ( g ) + J , = V ( g ) ,a contradiction. QED

+

1 . .

Theorem 6.3.8 (Latyshev 1761). Eoery T-ideal containing a 2nLatyshev polynomial (for some n ) is finitely generated.

Proof. Suppose I contains the 2n-Latyshev polynomialf; if I is not finitely generated, then by Remark 6.1.12 we can pick an infinite set of multilinear polynomials LEI, 1 < i < a,chosen inductively as follows. , such that I,(&+,) Takef, =f;givenf, ,...,fi we takefi., ~ l - - ( f,..., is minimal, and by Definition 6.3.6 we may (and will) assume that L(&+,) has height d 2n. Taking the above interpretation of Higman’s theorem we see that for applied to matrix representations of fI,(A)J1 < i < a>, suitable i < j , writing t(f;.) = X,, ..-X,, and L(&) = X,,---X,,. [for x in Sym(k) and 7 in Sym(k’)], we have some monotonically increasing function $: (1. . . . ,k l -,(1,. . .,k‘) with U S i )= Y lXIL,,,,~2XIL(n2~...~kXILL for suitable Yu. Then L(fj-Y1f;:(xc/t(nl)yZ,XiI,(n2)Y3, xt/,(nk)Yk+l)) < L ( f i )because we have killed off the leading monomial; by assumption, fi- Y Ofi(X,(, 1)Y2 , .‘ .)I E (.f,,.. .,.rj- 1 ) T , implying fi E (f , .. fi- 1 ) T contrary to choice off,. QED

fi),

’ ‘ ‘ 9

9

.?

‘9

In view of Theorem 6.3.8, it is very interesting to see which T-ideals contain a polynomial [XI, X,] ...[XZn-l,X,,,]; these must be finitely generated. In the following examples, these T-ideals will be called good.

252

[Ch. 6

THE THEORY OF THE FREE RING

Example 6.3.9.

If R is an algebra of upper triangular matrices, then

f ( R ) is good.

Another example, far more intriguing, arises from the following observation. Assume R is an algebra over a field C. Remark 6.3.10. If M,(C) $ R , then for some m, ( X , [ X , , X Z ] J mis an identity of R . [Indeed, for any direct power R’ of R we have M 2 ( C ) R’/NiI(R’), implying R’/Nil(R’) is commutative and thus satisfies the identity X , [ X , , X , ] ; we conclude with Amitsur’s method.] In particular, in. this case, [ R , R ] E N ( R ) ;cf. Definition 1.6.19.

<

<

Remark 6.3.11. If M , ( C ) R and N ( R ) is nilpotent, then . f ( R ) is good. (Immediate from Remark 6.3.10.)

This leads to the question, “When is N ( R ) nilpotent?” First observe that L,(R) s N ( R ) E Nil(R) and Nil(R)/L,(R) is nilpotent; thus N ( R ) is nilpotent iff Nil(R) is nilpotent. So we ask, “When is Nil(R) nilpotent?” From 54.4 we know for R affine that Nil(R) is nilpotent iff R 6 M , ( C ) for some t, yielding the following result: Example 6.3.12. . f ( R ) is good.

Suppose R is affine. If M 2 ( C ) R

< M , ( C ) , then

EX E R C I S ES 86.1 1. If R has a polynomial identity of degree d, then c . ( . g ( R ) ) < ( d - I)’“-‘. number 1 (resp. n ) is fixed in the first (resp. second) Amitsur table.] *2. How far can we improve Exercise I ?

[Hint: The

96.2 (Specht [SO]) I1 I is a T-ideal of Q ( X }generated by multilinear polynomials of degrees . ,dk, then I is generated by a single multilinear polynomial of degree max(d,, . . . . dk). (Hint:I, is a sum of minimal left ideals of V. and thus has the form V p for some .Y in ln.) 1.

d

Exercises 2-8 comprise a proof of the hook formula. q. 0 < q < u--7, we have 0 define a polynomial g ( R ) = i q (tP/./’(ti))(./(~)/ii.-ti)).Theng hasurootsanddeg(g) < u - 1,sog = 0.Checkcoefficient of iu-i.1 In what follows. consider a frame F with k boxes, u rows, in which there are d ( F ) standard tableaux. Let mi be the number of boxes in the ith row of F, and nj be the number of boxes in the jth column. Then k = x i m i = 11,. Let I = ( i l m , > mi+ ,) ;for each i in I let Fi be the frame obtained by deleting the box in position ( i , m i J . Let i i = m , + u - i , I < i < u. so that t , . ..., fu aredistinct. Writef(l) -= (k-t,)...(L-t,),and write/’ for the derivative of,f

2.

For any distinct integers

t,.

..., f u and for any integer

=~.r=,(ff/nj+,(ti-ij)) [Hint:Let l ’ ( l ) = ( ~ - t l ) . -2.) . . ( i and .

-x;=i

zj

Ch.61

Exercises

xie,

253

3. d ( F ) = d(F,). (Just look where n can appear in a standard tableau.) 4. For any g(L) in Z[A] of degree < n-2, we have (g(ri)/,f’(ti))= 0 by Exercise 2. Writing ,f(,l- I ) = f ( , l ) - f ’ ( L ) + g ( , l ) , and computing C Y = , , f ( t i -l)/y(ti) in two ways, show njfi((ti-tjl)/(ti-ti)) = u. 5. IF1 ( t ~ r ] j ~ i ( ( f i - r j - l ) / ( r ~ - r j= ) ) k) . [Hint: Induction on u ; first d o the special case t, = 0 and then calculate the expression obtained by replacing each ti by t i - t , ; note k = (Eri)-u(u- 1)P.I 6. Write A(tl ,..., t.) = nl,i<Jau(tj-t,). Then d = k!A(tl ,..., tu)/(tl!...ru!) . [Induction on k , by removing the box in the (i,m,)-position for each i in I ; if i41,then t i = t i + , 1, so A ( t l , . . . , t i - l , ti- 1, t i + l , . . ., t n ) = 0.3 7. Let h r j be the hook numbers of F . Then hij = (ri-tJ). [Hint: There are ( u - i ) values of t i - f j , all distinct, and mi values of h,. Thus it is enough t o show h, # t i - t C for all j . all u z i. Two cases: j Q m, and j > m,..] 8. Prove the hook formula from Exercises 6 and 7. 9. (Amitsur-Regev [76b]) Suppose I is a T-ideal of Q{X}.The Capelli polynomial C,,is in I iff for all k and every frame F of height > m I, contains the ideal of V, corresponding to F. [Hint: (=) It suBces to take the tableau T as in Remark 6.2.18 and show Consider the the polynomial corresponding to e(T)rr belongs t o I for every B in Sym(k). (e) descending partition (m, 1, 1,. . ., I ) of 217- I.]

xy=

z;=,

+

n?;,

ti!/nr=i+l

,

$6.3 1. Any commutative Q-algebra (not necessarily with 1) which satisfies an identity nor in X,] is nilpotent. the T-ideal generated by [X,, 2. [ X IX,]’ , is in the T-ideal of Q { X }generated by [[X,, X,].X,];o n the other hand, the , ’.. (infinite-dimensional) exterior algebra satisfies no identity of the form [ X I X,]

[X,,- 1 . X,,l. 3. (Specht [SO]) Let U = Q { X ) be relatively free. A higher commutator of U will denote where/ is a higher commutator of U . Call f’dpecht iff is some [ X i , Xj]or, inductively, [Xi,/], a sum of products of higher commutators. I f f is multilinear and

f ( X l , .. ., X i - , , I, X i + ,, . . ., R,)

=0

for all

i,

then f is Specht. (Hint: Note that

bl ...b,- lab,... b,

=

bl .. , b,- 1 [a, b,]b,+ ... b,

+ bl ...blab,+

... b,

;

continuing, write f = g, +g,X,, where every occurence of X l in g1 is in a commutator. Sending rf, H 1 sends g, HO, so g, = 0. Conclude by induction.) X,],X3]is generated 4. (Latyshev [63b]) Any T-ideal of Q ( X )properly containing [[X,, and a polynomial of the form [X,,X2]...[X,,X,J.( H i n t : Look a t the by [[X,,X,],X,] , is central, every multilinear Specht relatively free ring satisfying [[XI,X,],X,];since [ X IX,] Xi,].. .[Xi,,, ,, Xi2,] for suitable n.) Consequently, the 7’-ideal of polynomial has the form p[Xi,, the exterior algebra is generated by [[XI, X J ,X,].

-,,

5. If J ( M , ( C ) ) , n 2 4, is generated byf,, . . . ,X. then at least one of thef, is not 2n-normal. (Hint: Recall the argument of Proposition 2.4.23.)

CHAPTER 7

T H E T H E O R Y OF GENERALIZED IDENTITIES In this chapter we build a theory of generalized identities (GIs) from the foundation laid in Chapter 2. Not only is this theory very pretty and natural, arising merely by extending the set of coefficients, but it provides a very useful way to encode specific information about rings and their elements. Several important PI-theorems turn out to be consequences of more general, easy-to-prove facts in the GI-theory ; Bergman’s clear proof of the first fundamental theorem on rational identities (48.2). as well as several applications, are also based on GI-theory. The heart of GI-theory lies in the structure of primitive rings. Accordingly, the “socle” of a ring is defined and described in $7.1 in sufficient generality to be applied later to the (*)-primitive case. as well as the primitive case. $7.2 contains most of the major structure theorems of GI-theory, stemming from a technical result describing the “evaluations” of a generalized monomial on a primitive ring. In 47.3 we look at an involutory analogue of “primitive,” leading to the extension of the structure theorems to the (*)-case in 87.4. One immediate application is a famous theorem of Amitsur, that (*)-PI implies PI. In 47.5 we bring in an important ultraproduct technique from logic to yield decisive results on prime and (*)prime rings; these results are tightened in $7.6 by using Martindale’s “central closure.” Various applications to PI-theory are given in the text as well as in the exercises.

47.1. Semiprime Rings with Socle

In the general study of primitive rings, and more generally of semiprime rings, the notion of “socle” arises naturally in several contexts. In this section we examine some basic properties of the socle with the specific intention of applying them to primitive and (*)-primitive rings, in $7.2 and 47.4, respectively. Intuitively, we want to see how closely primitive rings with minimal nonzero left ideals resemble matrix rings over division rings. Throughout 1hi.Jsection, R denotes a semiprime ring. Both here and in (j7.3 we shall carry over standard methods (cf. Herstein [76B, Chapter 1.21); for 254

[g7.1.]

Semiprime Rings with Socle

255

more insight into rings with socle, the reader should consult Jacobson [64B, Section 41. Definition 7.1.I. The socle of R, written soc(R), is 0 unless R has a minimal (nonzero) left ideal, in which case soc(R) = x(minima1 left ideals of R ).

First we suppose that soc(R) # 0, i.e., R has a minimal left ideal L. Note that for any nonzero idempotent e of L, we must have x = .ye for all x in L. (Indeed, L = Re, so x = re for some r in R, implying x = re = re2 = xe.) Proposition 7.1.2. Suppose U E L and La # 0. Then La = L and = 0, and L has an idempotent e such that ea = a = ae.

Ann,a

Proof. La and Ann,a are left ideals of R contained in L ; since L is minimal, we get La = L and Ann,a = 0. Thus a € La, so a = ea for some e E L. Then ea = e2a, so (e2- e ) E Ann,a = 0, proving e is idempotent ; hence a = ae. QED Corollary 7.1.3. L has a n idempotent.

Proof.

L2 # 0 since R is semiprime, so La # 0 for some U E L. QED

Of course, L is an irreducible R-module. Thus End,L is a division ring, by Schur’s lemma. Note that for any idempotent e of R, eRe is a ring (with multiplicative unit e ) . Proposition 7.1.4. For any idempotent e of R, End,Re = eRe, under which the right module operations of Re on End, Re and eRe are ident$ed.

Proof. Define $: End,Re -+ eRe by $(p) = ep(e)EeRe for p in End,Re. Then $(p) = p ( e 2 )= B(e), so $ is clearly a ring homomorphism, and ker $ = {BE End,Relp(e) = 0} = 0. Given r in R, define p, in End,Re by right multiplication by re, i.e., f i r ( r ’ e )= r’ere. Then $(p,) = ere, so $ is an isomorphism. Also, for each re in Re and /3 in End,Re, p(re) = rep(e) = re$@), identifying the module operations. QED Corollary 7.1.5. I f e is an idempotent of L, then eRe is a division ring. Proposition 7.1.6. Rr is a minimal left ideal iff rR is a minimal right ideal.

Proof. (*) Suppose 0 # x ~ r R It. suffices to prove r E x R . Rr has an idempotent e, so r = re. Write x = ra = rea. Then reaRrea # 0, so for some r‘ in R , 0 # rear’reExR. But eRe is a division ring, so ear’re is invertible (with respect to e ) , implying re E xR. So r = re E xR. (c) By left-right symmetry. QED

256

rHE THEORY OF GENERALIZED IDENTITIES

[Ch. 7

Corollary 7.1.7. soc(R) = 0 unless R has minimal right ideuls, in which case soc(R) = x(minima1 right ideals of R). Hence soc(R)a R.

Proof. If r # 0 is in a minimal right ideal (of R), then rR is a minimal right ideal, so Rr is a minimal left ideal, implying rEsoc(R). The rest is immediate. QED Proposition 7.1.8. I f R is prime, then for any 0 # AQR we have

soc(R) s A. Proof. This is trivial unless soc(R) # 0. Then for every minimal left ideal L we have 0 # AL c L, so L = AL E A. Thus soc(R) = z(minima1 left ideals) c A. QED

We come to another way of looking at the socle. For the remainder ofthis section, assume M is a faithful, irreducible R-module, and let D = End, M, a division ring. For each r in R, rM is a D-subspace of M, so we can define M-runk(r) = [rM:D]. When there is no ambiguity about M, we write rank for M-runk. Remark 7.1.9. For all r,, rz in R rank(r,r,)

< rank(r,), and rank(r, + r 2 ) < rank(r,)+rank(r,). rank}

a R.

< rank(r,), rank(r,r,) Thus {elements of finite

Our main objective is to identify soc(R) with the elements of finite rank. Remark 7.1.lo. Suppose V is a D-subspace of M . If rank(x) = m and y1,..., y, are elements of V such that xy, ,..., xym are Dindependent, then we can expand y1,...,y, to a basis (y,,y, ,... } of V

-= cc

having the property x y i = 0 for all i > m. [Indeed, by Zorn's lemma, we can find a maximal D-independent set T = {y,,y,, ...} of V having this X:' , x y , D ; writing xy = ,xyidi, property. For any y in I/, we have x y E we have ~ ( y - Z ~ ! ~ y =~ 0, d , )so V - ~ ~ = , y i disi spanned by T, by maximality of T, implying y is spanned by T.]

xy=

In the above remark, the basis of V need not be countable although, merely for convenience, the notation {y,, y,, .. .} suggests that the set is well ordered. Lemma 7.1.11. If rank(r) = 1, then Rr is a minimal leji ideal of R. Proof. Rr # 0 since r # 0, so it suffices to prove for every x # 0 in Rr that Rr E Rx. Suppose ry, = y', # 0. Let T = {y,, y,, . . .} be a D-basis of M such that ryi = 0 for all i 2 2. Writing x = a,r for suitable a, in R, we have a,y; # 0 (or else x.44 = 0, contrary to x # 0). Take rl such that rIu,y', = y,. Then (rlulr)yi= rj.; for all y i in T, implying (r1a,r--r)yi= 0, so r = r l a , r = r,xERx. QED

$7.2.1

The Basic Theorem

257

Lemma 7.1.12. ff r e R arid rank(r) = t 2 1, then there are rank 1 elements r l , . . . ,r, in Rr such that r = Xi= Iri.

xiz

y,D for suitable D-independent y i in M , and Proof. Write rM = pick xi in R such that x i y i = yi and xiyj = 0 for all j # i. Then for any y in M we get suitable di in D such that ry = & l y j d j = C:,j=lxiyjdj xi(ry) = I:=( x i r ) y ;letting ri = xir, 1 d i 6 t , we have r-Z:= ri = =O. QED Theorem 7.1.I 3.

soc(R) = {elements offinite rank}

Proof. ( 2 ) If r E R and rank(r)=t, then by Lemma 7.1.12 Rri for suitable rank 1 elements ri of R ; hence by Lemma 7.1.1 1 Rr = Rr is a sum o f t minimal left ideals. Hence r E soc(R). ( c )We are done unless soc(R) # 0. Then by Proposition 1.5.9 we may assume M is a minimal left ideal L of R. By Remark 7.1.9 and Proposition 7.1.8 we need only find an element of finite rank. Let e be an idempotent of L. Identifying eRe with End, Re as in Proposition 7.1.4, thereby obtaining an injection R --t End(Re),,, (given by left multiplication), we have rank(e) = 1, as desired. OED

x.f=

Our treatment of primitive rings relies heavily on Theorem 7.1.13; our main strategy will be to work the socle through manipulation of elements.

87.2. The Basic Theorem of Generalized Polynomials and Its Consequences

In $1.5 we commenced the study of PI-algebras by proving Kaplansky’s theorem in two parts; (i) by the density theorem, and the staircase, each primitive PI-ring has the form M , ( D ) ; (ii) by closure, [D:Z(D)] < co. Replacing PI by GI (generalized identity), we shall obtain Amitsur’s theorem [65a], in which conclusion (i) is weakened to the following: (i’) The socle of a dense subring of EndM, satisfying a proper GI is nonzero; conclusion (ii) remains intact, that is, [ D : Z ( D ) ]< co. [There is an interesting monomial condition that is weaker than “GI,” given in Appendix B, equivalent to (i’).] Actually, a more explicit result will be given, that every GI of a primitive ring can be rewritten in such a way that each monomial contains a coefficient of bounded rank. This theorem is very powerful, even yielding nice applications to the PI-theory. We start by assuming R is a primitive ring with faithful, irreducible module M . Our basic goal is to find some notion to take the place of a

258

[Ch. 7

THE THEORY OF GENERALIZED IDENTITIES

“staircase” of matric units. More specifically, writing a GI of R in the form

il sS>m(t) i,

=1

we want to find x , , . . ., st in R and y in M such that rinlxn,. ‘ . . ~ , ~ r l~y, = , ~ 0+ unless 71 = I and i = 1, and r l ,.yl . - - x f r l , y # 0. Finding such elements is extremely complicated, although one can simplify matters by reducing to the case t = 2 (cf. Exercise 1). We approach this idea inductively. working only on the rinr. We shall carry some notation through Theorem 7.2.2. D = End,M. Letting F be a maximal subfield of D, form R F as in Proposition 1.5.12; then M is a faithful, irreducible RF-module, with F End,,M. We consider a multilinear generalized polynomial f ( X I , . . ,X , ) of ( R F ) { X ) . Viewing M as F-vector space, let “subspace” denote finite-dimensional F-subspace of M . Given subspaces V,, . . . , y , we say ,f is (V,, . . . , I/;)-oalued [or just (I/)-ualued]if for all ri in RF such that ri = 0, 1 -< i < t , we have , f ( r l , ..., r,)M s V,. Remark 7.2.1 .

Iffl, J; are (I/;)-valued, then (fl

+.f2)

is (v)-valued

Theorem 7.2.2 (Notation as above) S u p p o s e , f ( X 1 ,...,X I ) is u sum of’ u monomials und is (I/,)-ualuedf o r suitable subspaces V,, . . . , V,; let u = max([v: F]IO < i < t ) . Let W, be the F-subspace q f R F spanned by the coeficients of J: 7hen f can be written as a sum of
coeficients in W,, each monomial hauing a coeficient of rank over F .

611

+iu(v-

1)

Proof. Call a generalized polynomial good if it can be written as a sum of monomials (with coefficients in W,), each having a coefficient of rank ,< u +$v(v- 1). We shall show f is good by using simultaneous induction on t and u. First note that if t = 0, then .f is a constant w ; by hypothesis, wM E V,, so w has rank < [V,:D] = u and we are done. Next, for any t if u = 0, then .f’= 0 and we are done. Thus we may assume t 2 1 and u 2 1. Write 1

f=

0 4

C C hij(X1,.. . , X i - 1, X i + 1 , . . .,Xr)Xiwij, ,= j = I 1

xi=

where each hij is a monomial [in ( R F ) { X } ] thus ; ui = u. For each i, we reorder the hij if necessary such that for some v; < ui we have w,M c for all j such that uf < j < ui. with u; chosen minimally (possibly with (L = 0). The reason for considering u; is twofold. First, for all j > u:, wij has rank <[I/:D] < u ; secondly, for all j > u:. if ri I/ = 0, then riwijM = 0, proving h i j X i w i jis (0, V,, . . , K)-valued.

<

$7.2.1

259

The Basic Theorem

Let f ’ = Cj=lZyL hijXiwijand f ” = f - f ’ . Now f ” is good, as well as (0, Vt,. . ., c)-valued. Hence, we are done i f f ” = f . Even iff” # 0, we see that ,f’ is (V,, V,, . . .,T/;)-valuedby Remark 7.2.1, and thus by induction on u is good ; hence f =f ’ +f ”is good. So we may assume f“ = 0, that is, wijM $ for all i,j. Now let fi = h i j X iwij; we focus on ,L, which is clearly the sum of all generalized monomials o f f having label X , , . . . X , ( r -l)Xr for all n in Sym(r - 1). By symmetry, we may assume f, # 0, and we shall build “one step of a staircase,” using f,. Choose y, in M such that w t l y tq! By Corollary 1.5.3 there exist a1 = 1,a2,...,au, in F such that for each y’ in M there is a corresponding r’ in R with r’ = 0 and r’wrjy, = y’aj, 1 d j 6 u,. Let g, = ajhtj(X1,.. . ,Xr-l), and put = K + ~ ; L wijy,F. For all risuchthatriI.;’=O,l < i d t - l , w e h a v e

x;;

c.

<

zy=

v’

l’,

gt(rl,...,rr- 1 )Y’

=

C hrj(rg,. .

. 3

rr- i ) ~ ‘ a j

j= 1 0,

=

C h r j ( r l ,.. . ,r,- l)r’w,jy, =,f(rl, . . .,r r - ,, r’)yrE v,. j= 1

Since y’ was arbitrary in M , gr is (Vo, Vi, ..., V,’_,)-valued. Now let gi = g,X,wtl. Since gr is (V,, V;, . . ., <‘-,)-valued, gi is (V,, V;, . .., v L l , T/;)valued. Moreover, g, is a sum of o, monomials, and max([y’: F]JO 6 i < t - 1 ) d u+max{u,(i < r} < u + ( D - u , ) . Hence, by induction on t , g, (and thus 9;) can be written as a sum of
L-d

=

C hrjxtwtj j= 1

0,

t’t

ajhrjxtwtl

=

j= 1

C htjXr(Wtj--jWtl) j=2

(since at = l), a sum of (ut- 1 ) monomials. Hence g is now a sum of u - 1 monomials and, by induction on u, g can be written as a sum of d (ti - 1)‘ monomials each having a coefficient of rank < ( u + u - l ) + ) ( u - l)(v-2) = u+$u(u1); i.e., g is good. But u‘

-(0 -

1y

= (u - (v - 1))( u f - +v‘-2(v - 1)

+ ...+ (u - 1y-

1 )

2 u‘-

Therefore f = gi+gl can be written as a sum of d u ’ - l + ( u - 1)’ < u‘ monomials and is good. QED This result serves as the source of the entire GI structure theory. The main point, by Remark 7.1.9, is that each substitution in every generalized monomial off produces an element of finite rank, which is thus in the socle; this is the key to Theorem 7.2.9 below, to which we now lead.

260

THE THEORY OF GENERALIZED IDENTITIES

[Ch. 7

Corollary 7.2.3. Suppose f is a multilinear generalized identity of a dense subring R of End MD, f is a sum of u monomials, and F is a maximal subfield OfD. Then #can be rewritten US a sum of v' monomials with coefficients spanned (ouer F ) by the coe8cients o f f , such that each monomial has a coejicient of rank < f u ( u - 1) ouer F (and also ouer D). Proof. Noting that rank(over F) 2 rank(over D ) for any maximal subfield F of D,we pass to R F and apply Theorem 7.2.2 with all = 0 (and thus for u = 0). QED

Corollary 7.2.3 generalizes Kaplansky's theorem, because if R is a dense subring of End MD, having a polynomial identity f that is a sum of u monomials, then we can rewrite f as a sum of monomials each having a coefficient in F of rank
fi

Proof.

Iff is an identity of M , ( F ) that is a sum of u monomials, then n G +v(v - I ) < f r 2 , implying u > QED

J2n.

This result can be improved considerably (cf. Exercises 2, 3, 4). Corollary 7.2.5. Suppose R is a dense subring of End M , and has a multilinear GI, f ( X . .,X l ) , which is a sum of u monomials. Then, .for every generalized monomial fn o f f and for all r l , . . .,rl in R, rank fn(rl,. . . r,) < @ + 1 ( v - 1). Proof. By Theorem 7.2.2 .fn(rl,.. . ,rl) is written as a sum of < u' terms, each of rank G f u ( u - l), so by Remark 7.1.9 rank Jn(rl ,..., r l ) < u'($u(u- 1)) = $v'+'(u- 1). QED

One could write down a proof of Corollary 7.2.5 directly and obtain rankf,(r,, . . . ,rl) 6 u'. (Detailsareleft to thereader.)However,even this bound is quite crude, and the real sign significance is that the bound depends only on u and t. Definition 7.2.6.

Iffisageneralized polynomial,GM,-(R) = u { . f ; ( R ) I

$7.2.1

The Basic Theorem

26 1

every generalized monomial o f f ) . Define Zf(R) = ideal of R generated by GM,-(R),and GI(R) = UfGM,.(R)l,fis a multilinear GI of R}. fT

Remark 7.2.7.

G I ( R ) d R . Hence GI(R) = C(If(R)lall GIsfof R).

Theorem 7.2.8. I f R is primitive, then GI(R) G soc(R); if GI(R) # 0, then GI(R) = soc(R). Proof. GI(R) c soc(R) by Corollary 7.2.5. If GI(R) # 0, then by Proposition 7.1.8 soc(R) E GI(R), so GI(R) = soc(R). QED Amitsur's Theorem

Recall f is R-proper if some generalized monomial of ,f is not a GI of R. By multilinearization, GI(R) # 0 iff R satisfies an R-proper GI. This leads to the famous theorem of Amitsur [65a] that initiated modern GI theory. A dense subring R of End M, satisfies a proper GI iff Theorem 7.2.9. soc(R) # 0 and D is PI. Proof. If R satisfies a proper GI, then GI(R) # 0, so by Theorem 7.2.8 soc(R) # 0. Moreover D has a maximal subfield F . By Corollary 7.2.3, for D][D:F ] , implying [D: F ] some nonzero x E RF, 00 > [sM : F] = [xM:

< 00. The converse is easy. Suppose soc(R) # 0. Then there is an idempotent e such that D = eRe. If D has degree n, then S , , ( e X , e , . . . ,eX2,,e) is a GI of R, one of whose generalized monomials is e X , e X , . . . e X , , , e ; since R is prime, this generalized monomial is not a GI of R, so S,,(eX,e, . . .,eX2,,e) is Rproper. QED Improper Generalized Identities

Since the above results are so decisive, we should like to take a closer look at improper generalized identities; we shall see that they are really trivial. The key step is the linear case. Lemma 7.2.10. Suppose R is an algebra over a $eld F and f ( X l , .. ., X , ) is a multilinear generalized monomial in R { X } . I f we write ,f = r i l X l f , , f o rril in R and,/; in R { X } such that v is minimal, then the { ril I 1 d i < 0) are F-independent.

x:Y=l

Proof.

Otherwise x r = l a i r i l = 0 for suitable aiin F , not all 0, so we

262

[Ch.7

THE THEORY OF GENERALIZED IDENTITIES

may assume a ,

=

1. Then

c

f=

i= 1

c

c

C rllx1jt- C

EirilXlfl

=

1 ri1X1(j;-ai,/1)9 i=2

i= 1

contrary to the minimality of v. QED Lemma 7.2.11. I f x y = l r i , X l r i 2 is a GI of a dense subring R qf End M, with r 1 2 # 0, then { r i l11 < i d v } are F-dependent. Proof. Choose yo in M such that r12y, # 0, and let y i = ri2jf0, 1 d i ,< v.ByCorollary1.5.3thereareelementsal = 1,a, ,..., a,,inFsuchthat given y in M we have corresponding r in R with ryi = yai, 1 ,< i m. Then 0= lri,rri2)yo= x y = l r i , r v i= ( x : = l a i r i l ) y since ; y is arbitrary we get

<

(xr=

Cr=lairil= 0. Q E D Proposition 7.2.12. Suppose R is dense in End M,, F is a maximal subfeld of D, and Z ( R ) is a f e l d such that a n y f n i t e set of Z(R)-independent elements of R is F-independent (viewed in RF). Then every multilinear generalized monomial # 0 (in R{X}) is not a GI of R . Proof. Suppose, to the contrary, that f # 0 is a generalized monomial that is an identity o f R ; we may assumef has label X 1 . . . X , and write f = x : f = l r i l X l f for l r i l in R such that c is minimal. Hence by Lemma 7.2.10 the ril are Z(R)-independent, and so by hypothesis are F-independent. Now the,f;.are nonzero generalized monomials and so by induction on t they are not GIs of R. Pick r 2 , .. ., r, in R such that f 1 ( r 2 ..., , r t ) # 0, and let ri2 =fi(r2, .. . ,r,). Then rilX,ri2is a GI of R, thus of RF, implying by Lemma 7.2.1 1 that the ril are F-dependent, a contradiction. QED

x;=

Corollary 7.2.13. I f R is simple, then every multilinear generalized monomial # 0 (in R { X } ) is not a GI of R . Proof.

of

Apply Corollary 1.5.19 to Proposition 7.2.12. QED

Corollary 7.2.14. D is a GI o f R .

lf[D:Z(D)]

=

00

and R is a D-ring, then every GI

Proof. First note by Theorem 7.2.9 that D satisfies no D-proper GI. W e claim that i f f is a GI of D, then f = 0 in D{X); the assertion follows immediately. f is a consequence of any generalized monomial of its multilinearization f . and f is D-improper, as noted above; thus we may assume .f is already a multilinear generalized monomial. Hence f = 0 in D{X} by Corollary 7.2.13. QED

In Exercise 10 we replace D by any simple ring of infinite dimension over its center, but Corollary 7.2.14 is enough for the important application to

47.2.1

The Basic Theorem

263

the fundamental theorem of generalized rational identities (cf. Corollary 8.2.12). Strong GIs Having examined R-proper GIs, we continue with R-strong G I s ;f is Rstrong iff f is proper for every homomorphic image of R. Note that f is Rstrong iff 1 E Zf(R) [since if 1 Z,(R), thenfis R/I,(R)-improper]. Hencefis (nR,)-strong for any direct product of copies R, of R.

e

Theorem 7.2.15. R is a PI-ring.

If R has a completely homogeneous R-strong GIf; then

Proof. By Amitsur’s method, it suffices to consider the case Nil(R) = 0. In this case R[I] is semiprimitive by Amitsur’s theorem. Viewing R[A] as a subdirect product of dense subrings R, of End(M,),”, YE^, for suitable vector spaces M y over division rings D,, and letting F , be a maximal subfield of D,, we view each R, E End(M,JF,; letting r, be the image of r in R,, we define B = { r E R [ I ] I{[ r y M y: F,] Iy E I-} is bounded). Clearly BaR[I]. Obviously we may throw out all generalized monomials offthat are GIs of R, so we assume all generalized monomials of f a r e not G I s of R. Also, 1 E Z,(R) E Zf(RII]),so f is R[I]/B-proper if B # R[A], in which case some multilinearization g off is R[I]/B-proper. On the other hand, g is a GI of R[A] and is thus R[L]/B-improper by Corollary 7.2.3. We must conclude that B = R, so 1 E B, i.e., {rank 1 Iy E r}is bounded, say by k. Hence S,, is an identity of each R,. Therefore S 2 k is an identity of R. QED

In the above proof, k is not bounded. Indeed, M , ( F ) satisfies the M,(F)strong GI [ e l l X l e l l , e l l X 2 e l l ]for all (arbitrarily large) n. This theorem is very useful for applications; we give one now. Theorem 7.2.16. Suupose there exists an element r in R and a polynomial g ( I ) in Z(R)[I], I being a commuting indeterminate over R , such that C R ( r )is a PI-algebra, g(r) = 0, and g’(r) is invertible (where g’ is the formal dertvative of 9). Then R is PI-algebra.

Let g(A) = x F = , o c t k A k for suitable ctk in Z(R); then g’(A) ~ = l c t k ~ ~ ~ ~ r i Note X l f for - l all - i x. in R that ~ ( x ) C E R ( r ) ,and f(1) = g‘(r) is invertible in R. Take a multilinear polynomial identity h ( X , , . . . , X , ) of CR(r).Then h ( f ( X , ) , .. . , f ( X , ) )is a GI of R, one of whose generalized monomials is f ( X l ) . . f ( X , ) ; specializing each X i -to 1, we see that 1 EI,(R), so by Theorem 7.2.15 R is a PIring. QED Proof.

= xr=lkakk12k-1. Definef(Xl) = ~

264

THE THEORY OF GENERALIZED IDENTITIES

[Ch. 7

The Modified Density Theorem and Its Consequences

Sometimes we need to use the following mild generalization of the density theorem : Modified density theorem. I f M is an irreducible R-module with D and if B is a lest ideal of R with BM # 0, then for any Dindependent elements y,, . . .,y, in M and for all y ; , . . . , y ; in M there exists suitable b in B such that by, = yi, 1 < i < m. = End,M,

Proof. BM is a nonzero submodule of M , so EM = M, and in the proof of the density theorem we could replace routinely R by B. QED

Thus, if R is dense in EndM, and R, is a subring of R containing a nonzero ideal of R , then R, is also dense in End MD. Our first use of the modified density theorem is to provide, by means of the regular representation, a general form of counterexample. Example 7.2.17 (Amitsur). Suppose R is an algebra over a field F . Viewing R as a vector space over F , we can use the regular representation of R to inject R into End,R. Let A = soc(End,R), and let R , be the F-subalgebra of EndfR generated by R and A . Then R, is dense in End,R, and soc(R,) # 0. Moreover, many properties can be passed from R to R,. For example, one can easily prove A is F-algebraic; if R is algebraic, it follows easily that R, is algebraic. A striking illustration of this method is given when we recall that Golod [64]-Shafarevich gave an example of an algebraic algebra that is not locally finite (cf. Herstein [68B]). Thus, we see instantly that a primitive algebraic algebra with proper GI need not be locally finite. Here is another application of modified density. Proposition 7.2.18. Suppose B is a left ideal of a dense subring of End M,, and BM # 0. l f f ( X I , ..., X , ) is a generalized polynomial wrth f ( B ) = 0, then f i s a GI ofEnd M,.

By modified density, given any x,,. . . ,x, in End MD and any y in ( X I , . . . , x , ) y = f ( b l , . . . ,b,)g = 0. (The point is that we have only a finite number of multiplications and additions of a finite number of elements, so we could view things in a finite dimensional subspace of M . ) Hence f ( x , , .. . ,x,)M = 0 for all x, in End MD, implyingfis a GI of End MD. QED Proof.

A4 we can find b , . . .., b, in B such that f

Corollary 7.2.19. Any primitive ring R is contained in an R-ring multequivalent to R, o f i h e form End M , for some vector space M over a suitable jield F.

47.3.1

Primitive Rings with Involution

265

Proof. Applying Proposition 7.2.18 to some closure of R, we see R is mult-equivalent to some End M,. Therefore End M, is an R-ring. QED

57.3. Primitive Rings with Involution

So far our specific study of identities of rings with involution (cf. Chapters 2, 3) has been for PI-rings that have an involution. I n order to obtain the best results, one has t o restructure ring theory so that the major concepts are redefined in terms of the involution. The obvious place to start is with “primitive,” in view of Kaplansky’s theorem. Accordingly, in this section we look at “(*)-primitive.” The key point is that we want to find some (*)analogue to injecting a primitive ring into EndM,. Now we know (by Remark 1.5.24) that a ring R is primitive iff it has a maximal left ideal A with Ann, RIA = 0. In this case we could take M = RIA, D = End, M , a division ring, and inject R into EndM, canonically. So, to find the (*)parallel, suppose (R,*) is a ring with involution and A is a maximal left ideal of R. Then A* is a maximal right ideal of R. Let M = RIA and M* = RIA* (as right module). The map (r+A) + r* + A* induces a 1 :1 correspondence from the irreducible module M to the irreducible right module M*, which we also call (*); in fact (*) is an isomorphism of the additive group structures of M and M*. Let D = End,M and D* = End M i . Remark 7.3.1. In the above notation, there is an anti-isomorphism (*) from D to D*, given by d*y* = (yd)* for all y* in M*; and there is an antiautomorphism (*) from End M, to End,.M*, given by y*P* = (by)* for all y in M (with given in End M,,). Hence (*) induces an exchange

involution of End M, 0 End,,M*, denoted as 0. Remark 7.3.2. In the above notation, there is a natural homomorphism $: (R, *) + (End M, 0 End,,, M*, o), taking r to (r, r)(viewing r in End M, and in End,, M*);

k e r t , b = { r E R I r M = O = M*r) = ( r ~ R l r M = o = r * M ) . We are interested in the situation where ker t,b

= 0.

Definition 7.3.3. For a module M, define Ann(R,*)M= (Ann,M)n (Ann, M)*, the “largest” ideal of ( R , * ) in Ann,M. (R,*) is primitiue if Ann(,, *, RIA = 0 for some maximal left ideal A of R. Remark 7.3.4.

Ann(R,.)RIA

Proposition 7.3.5.

=

(Ann,R/A) n (AnnXR/A*).

The following statements are equivalent for (R, *):

266

T H E THEORY OF GENERALIZED IDENTITIES

[Ch. 7

(i) ( R , * )is primitive; (ii) R has an irreducible module M with Ann(,.*,M (iii) R has a primitit e ideal P with P n P* = 0.

= 0;

Proof. (i) =. (ii) is immediate from Definition 7.3.3 (with M = R / , 4 ) . (ii) s (iii) is obvious with P = Ann,M. To see (iii) (i), let A / P be a maximal left ideal of RIP such that 0 = Ann,,,,((R/P)/(A/P)) : Ann,,,(R/A). Then Ann,(R/A) G P , so Ann(R,+, R I A G P P, P* = 0. QED =j

Corollary 7.3.6.

If ( R ,*) is primitive, then ( R ,* ) is prime and Jac(R)

= 0.

Usually it is more convenient to focus on statement (iii) of Proposition 7.3.5 for proofs about (*)-primitivity, but the original definition is useful when we need to work more closely with general structural properties involving (*). A (*)-Analogue of the Density Theorem

We turn now to “(*)-density.” Even for matrices, matters are quite complicated, as we saw in $2.5 while attempting to build the longest possible “staircase.” Fortunately, we could manage the computations because every involution was either of transpose type or symplectic type. In the case of primitive rings, in general, I do not know of any such classification of the involutions, but fortunately this can be done without too much difficulty when the socle is nonzero. Assume throughout that R is semiprime with involution (*). Remark 7.3.7. Suppose L is a minimal left ideal with idempotent e. Either L*L = 0 or e = eae*be for suitable a , b in R . (Indeed, if 0 # L*L = e*Re then 0 # (Re*Re)2= Re*ReRe*Re, implying, for some a E R , 0 # eae*Re, a right ideal of the division ring eRe; thus eae*Re = eRe.) Proposition 7.3.8. Suppose soc(R) # 0. Then either (i) R has u minimal left ideal L with symtnetric idempotent, or (ii) R has a minimal lejt ideal L with L*L = 0, or (iii)for some minitnal left ideal L, X*X = 0 for all x in L and End, L is afifirhi.

Proof. Case I. For some minimal left ideal L and some x in L, we have Lx*x # 0. Let a = Y*X, a symmetric element of L. By Proposition 7.1.2 L has an idempotent e with ea = ae = a. Now e*e is symmetric, and e*ea = e*a = (ae)* = a* = a, proving e*e # 0 and (e*e)2- e*eE Ann,a = 0, yielding (i). Case 11. For ever) minimal left ideal L‘ and for each .Y in L‘, LX*Y = 0. We claim that some minimal left ideal L has the following property: Property P. X*.Y == 0 for all .Y in L.

47.3.1

267

Primitive Rings with Involution

Note. If L lacks property P, then LL*

= 0.

Indeed, taking x in L such that

.x*x # 0, we have Rx*x = L, and L(Rs*.u)* = L x * x R = 0.

Now take a minimal left ideal L‘ of R , and x # 0 in L’. Then x R is a minimal right ideal of R by Proposition 7.1.6, so R.u* = (xR)* is a minimal left ideal. If Rx* has property P for some x in L‘, then we are done. Otherwise, for all x in L‘ Rx* lacks property P, so 0 = ( R x * )(Rx*)* = Rx*xR, so x*x = 0 for all x in L.!, proving the claim. So take L having property P and take a nonzero idempotent e in L. For any element r in R we have e*r*re = (re)*re = 0. Hence e*r*e+e*re = e*(r*+r)e = e * ( r * + I ) ( r + I ) e - e r * r e - e * e = 0, proving e*r*e = --*re. By Remark 7.3.7 we are done [having (ii)] unless e = eae*be for suitable a, b in R . Hence for all r I , r z in R we have e*ber,er,e

(e*r?e*b*e)r,e = -e*r$(e*b*er,e)

=

-

=

e*rre*rTe*be

=

-e*b*er,er,e

= e*berler2e,

yielding

0 = e*be[er,e,er,e]

= eae*be[er,e,er,e] = e[er,e,er,e] = [er,e,er,e];

thus eRe is a field. QED Definition 7.3.9. Suppose V is a right vector space over a division ring D. A map (,): V x V 3 D is a sesquilinear form if ( Z i ~ i , C , i ~ > d j ) = uJ)dj for all ui, c) in V and all d, in D. For Vl s V , define V: = { U E V l ( V l , u ) = 01, a subspace of V. (,) is nondegenerate if VL = 0. (,) is alterriating if ( u , u ) = 0 for all u in V. If D has an involution (*), then (,) is Hermitian when ( u , , u 2 ) = ( u z , u l ) * for all v l , v2 in V.

xi,j(vi,

Multilinearizing shows ( u , , u 2 ) + ( v z , u l )

=

0 if (,) is alternating.

Definition 7.3.10. If R has an irreducible module M with D End, M and if R has an involution (*), we say a form (,):M x M + D is (*)-compatibleif ( x y , , ~ , )= (y1,x*y2) for all x in R and y,,y, in M . =

Remark 7.3.11. is in End V,.

Given yo, y 1 E V, the map y + y o < y , , y ) (for all y in V )

Proposition 7.3.12. Suppose M is an irreducible R-module, (*) is an involution of R , and (,):M x M + End,M is a nondegenerate, (*)compatible, Hermitian (resp. alternating with End, M a field) sesquilinear form. I f X E Rsuch that xy = y o ( y l , y ) for all y in M , then X * J J = y , ( y o , y ) (resp. -yl(y,,y))for a l l y in M . Proof. For Y’ in M , (.Y*Y,Y’) = ( y , x y ’ ) = ( y , y o ( y l , y ’ ) ) = ( y , y 0 ) . (yl, y’). If (,) is Hermitian, we get ( x * y , y’) = ( y o , y)*(y‘, y,)*

268

THE THEORY OF GENERALIZED IDENTITIES

[Ch. 7

= ((Y’,YI)(YO,Y))* = (Yl(Y0,Y)~Y’) for all Y‘, implying .y*4’= Yl(Y0,J) (since (,) is nondegenerate). Theother assertion is proved analogously. QED

Theorem 7.3.13. Suppose R is semiprime with (*)and soc(R) # 0. Then there is a minimul Ieji ideal L satisfying one of‘ the following three properties (viewed as right End, L-vector space):

(1) L*L= 0; (2) L has a nondegenerate, (*)-compatible Hermitian,form ; (3) L has a nondegenerate, (*)-compatible alternating form, and End,L is a-field. In fact, we ma.v assume in Cases (2) and (3) that for any yo,yl in L therr is some r E R such thar ry = yo(vl,y) for all y in L. Proof. We apply the conclusions of Proposition 7.3.8. If (ii) holds then we have conclusion (1) here. If (i) holds then take L = Re such that e* = e (e idempotent) and define (,) by (xle, x,e) = ex:x,e, a nondegenerate Hermitian form that is obviously (*)-compatible, yielding (2); if y o = xoe and y, = xle, then yo(yl,y) = (.u,ex:)y, so we are done. Finally, suppose (iii) holds ; i.e., we may assume L = Re, x*x = 0 for all .x in L, and e = eae*be for suitable a, b in R (or else L*L = 0 by Remark 7.3.7). Define (xle, x,e) = eae*x:x2e. Clearly (,) is sesquilinear, (*)-compatible, and alternating. Moreover, if re€ Re’, then 0 = eae*Rre, so 0 = eae*beRre = eRre, implying (Rre)’ = 0, so Rre = 0 and re = 0. Thus (,) is nondegenerate, yielding (3); if yo = xoe and y1 = xle, then yo(yl,y} = (xoeae*x:)y, so we are done again. Q E D

Note that under conclusion (1) above, R cannot be prime. Example 7.3.14. In order to understand Theorem 7.3.13 better, consider (R, * ) = ( M , ( F ) , *). If (*) is the transpose, then we take e = e l l . a symmetric,rank 1 idempotent,setL = Re,,,anddefine(y,, y,) = t r ( ~ : y , ) ~F for y,, y, in L. Obviously (,) is Hermitian, nondegenerate, and (*)-compatible. The element eij of R corresponds to the transformation yweil(ejl,y); likewise eji corresponds to y + + e j l ( e i l , y ) , illustrating Proposition 7.3.12 (since eji = e t ) .

If (*) is the canonical symplectic involution, then we have n = 2m for some m and again take e = ell, but note this time, for all x in Rell, that x*x = 0. [Indeed, if x = laieil,then m

x* =

c aiem+l,m+i-

i= 1

i: aiem.1.i-m

i=m+l m

ai-mem+l,i -

= i=m+l

c ai+mem+l.iy

i= 1

$7.3.1

269

Primitive Rings with Involution

so n

m

x*x =

C (-aiai+m)em+ + 1 i= 1 1,1

ai-maiem+l,l

= 0.1

i=m-Cl

Thus, we can define ( y 1 , y 2 )= t r ( e l , m + l y ~ y 2for ) , y,,y, in Re,,, and (,) is alternating, nondegenerate, and (*)-compatible. When trying to construct elements in primitive rings with involution, it is very useful to keep Example 7.3.14 in mind. Definition 7.3.15. If V is a given right D-vector space and Vl is a Dsubspace, then the codimension of V, (in V ) , written codim(V,), is [( V W l ):Dl.

If V, is a D-subspace of I/, if t = [Vl :D] < and if D is a sesquilinear form, then codim(V+)< t . [Indeed, if V, luiD, then V: 2 , ( v i D ) ’ ; for each i, clearly codim((v,D)’) = 1.1

Remark 7.3.16.

(,):V x =

I/+

n:=

Theorem 7.3.17. Suppose ( R ,*) is primitive and soc(R) # 0. There is a minimal left ideal L, and D = End, L, with a sesquilinear, (*)-compatible form (,):L x L + D having the following property for any left ideal L, with L, L # 0, and f o r anyfinite dimensional D-subspace V of L . Given y‘ in L - V and y” in ( V + y‘D)’, one has x in L , with x V = x* V = x*y’ = 0 and with xy’ = y”. In fact (fL*L = 0 for some minimal lefi ideal L, then we may take (,>to be trivial; i.e., y“ can be arbitrary in L. Proof. Using the modified density theorem we have x1 in L, such that x,y‘ = y’ and x1 V = 0. We need to find x2 such that x2y’ = y“ and x:(V+y’D) = 0; we are then done by setting x = x2xl. We shall find x2 by using Theorem 7.3.13. First assume (1) holds, i.e., L*L = 0 for some minimal left ideal L. L2 # 0, so by the modified density theorem we have some x2 in Lsuch that x,y’ = y”; x f L E L*L = 0, and we are done. Now assume (2) or (3) holds in Theorem 7.3.13, including the assertion at the end. Let L be a minimal left ideal with the desired nondegenerate, Hermitian, or alternating (*)-compatible (,):L x L + D. Then there exists some element y , in L with ( y , , y‘) # 0. Multiplying y1 by a suitable element of D, we may further assume ( y l , y ’ ) = 1. Now we have x2 in R such that x2y = y ” ( y , , y ) for all y in L. Clearly x2y’ = y”. By Proposition 7.3.12, for all y in V, xTy = p y l ( y ” , y ) for p = f 1. I f Y E V+y’D, then ( y ” , y ) = 0, so xf(V +y’D) = 0. QED

Corollary 7.3.18. W i t h the notation as in Theorem 7.3.17, given y , , ...,yc arbitrary in L, and a finite-dimensional D-subspace V of L not containing y , , there exist d , = 1,. . . ,d,, in D having the ,following property:

270

THE T H E O R Y OF G E N E R A L I Z E D IDENTITIES

[Ch. 7

For any element y in ( V + y i D)’ there exists un element r in R such that rV = r* V = 0, ryi = yd,, and r*yi = 0, 1 < i 6 1).

Proof.

Mimic proof of Corollary 1.5.3. QED

Matrix Algebras with Involution Using Example 7.3.14 as motivation, we shall now obtain an isomorphism theorem [with respect to (*)I that contains Theorem 3.1.61 and is free of restriction i)n the ring (i.e., that it contains $). Since the classification of involutions into orthogonal and symplectic types degenerates in characteristic 2, we start with a new description. Definition 7.3.19. Suppose R is primitive, with soc(R) # 0. An involution (*) o n R is Hermitian (resp. alternating) if some minimal left ideal L of R has a nondegenerate, (*)-compatible Hermitian (resp. alternating) form. (By Theorem 7.3.13 (*) must be Hermitian or alternating.) Theorem 7.3.20. Suppose R = M,(F), with two inuolufions (*), ( J ) of thefirst kind, which are both Hermitian (resp. both alternuting). Then there is a jnite-dimrnsionul extension j e l d K of’F such that, extending (*) uiid (J) naturallj’ to M , ( K ) = M , ( F ) O I . K ,we haue ( M , ( K ) , * )z (M,(K),J).

Proof. Assume (*) and (J) are both Hermitian. (A similar argument works when they are both alternating.) Take minimal left ideals L,,L, of M , ( F ) such that for u = 1,2 L, has a nondegenerate, (*)compatible (resp. (J)-compatible), Hermitian bilinear form, written (,),. By a Gram-Schmidt process. working in a suitable finite extension field K of F , we have an orthonormal base y 1”,..., ynu of L, [with respect to (,>,I, u = 1,2, i.e., ( y i u ,J , ~ , ) = d i j . Write riy’ for the element of R = M , ( K ) such that r$’y = y i , ( y j , , y ) for all y in L, (cf. Theorem 7.3.13). If ccijr$’ = 0 for r i j in K , then for each y in L, and for each i

xy,j=

n

0 = (~i,.

C i.j= 1

n

mijrir’y)

=

C j= 1

n

aij(.vju,Y) =

< 2 aijYju,Y), j= 1

implying each ccij = 0. Thus [r!r)ll < i,j < n} are K-independent and by a dimension count must be a base of R over K . Then it is clear that the map C a..r!?) IJ V -+ x a i j r $ ’ is a homomorphism ( R , *) + ( R ,J), which is an isomorphism because R is simple. QED

$7.4.1

Identities of Rings with Involution

271

57.4. Identities and Generalized Identities of Rings with Involution

In this section we extend the results of $7.2 to the involutory case. The theory is motivated by a much more special question asked by Herstein in response to Kaplansky’s theorem: If R is a simple ring, having involution (*), and if some classical polynomial vanishes under all substitutions of symmetric elements, is R central simple? After several years, Herstein [67] answered his own question affirmatively, and Martindale [69a] extended the result to primitive rings. Amitsur [69] found a beautiful generalization in the context of (*)-identities, that every ring having a (*)-identity with coefficient 1 is a PI-ring. In this section, we further generalize Amitsur’s theorem to (*)-GIs, obtaining Amitsur’s theorem as a special case. In order to understand better what is going on, let us lower our sights a bit, focusing on the most important part of Amitsur’s theorem, a (*)-version of Kaplansky’s theorem. The proper conjecture in the (*)-structure would be, “If ( R ,*) is primitive and satisfies an ( R ,*)-proper identity, then (R, *) is simple, and [R :Z(R, *)] is finite.” The proof would be to use some sort of density argument to build a staircase. Now we have a suitable (*)-density theorem (Theorem 7.3.17). Unfortunately, the hypotheses include “soc(R) # 0.” This condition is more related to GI-theory, so the easiest way of extending Kaplansky’s theorem to rings with involution is probably by GItechniques. This is the main step in our program, and is trivial in one special case. Definition 7.4.1. If f ( X , , X y , . . . , X , , X : ) is a generalized polynomial, then GM,(R, *) = { ,f,(R, *)[every generalized monomial .f, o f f ) , Z,(R, *) = ideal of R generated by GM,(R, *),and GT(R, *) = [GM,(R, *)I every GI .f‘ of ( R ,*)}.

u

Remark 7.4.2.

If (R, *) is special, then GI(R, *)

u

=

GI(R).

It is useful to show that a given ring with involution satisfies a special GI because then the results of $7.2 apply. Proposition 7.4.3. ( R ,*) is special.

l f ( R , * ) is primitive and R is not primitive, theri

Proof. Let P be a nonzero primitive ideal of R such that P n P* = 0 (cf. Proposition 7.3.5). Po P* is a special subring (without 1 ) of (R, *); hence, for any f ( X , , X:, . . .,X , , X : ) that is a GI of (R, * ) , f ’ ( X , , X , , . . .,X,, - X 2 , ) is a GI of P O P * , and thus of P* I( P o P*)/P, an ideal of the primitive ring RIP. By Proposition 7.2.18 , f ( X l , . . . , X 2 t is) a GI of RIP; by an

272

.THE THEORY OF GENERALIZED IDENTITIES

[Ch. 7

analogous argument f ( X , ,..., X 2 , ) IS a GI of RIP* and thus is a GI of R. QED This result, due to Baxter-Martindale [68], is one of the rare examples of an important structural result whose proof relies intrinsically on passing to rings without 1, and is very useful because it enables one to look at primitive rings, which are better known than (*)-primitive rings. Nevertheless, it is more natural not to mix categories so blatantly; we postpone the main application of Proposition 7.4.3 a little. Here is a cute corollary. Corollary 7.4.4. Zf(R, *) is primitive and R is not primitive, then ( R , *) can be injected into some mult-equivalent (End MF @ (End M,)OP,0 ) fiw a suitable vector space M over afield F. Proof. Take a nonzero primitive ideal P such that P n P* = 0. Identifying RIP* with and letting R‘ = ( R / P ) F be a closure of RIP, we can inject

( R , *) + ( R / P 0

0(R’)OP,0 ) + (End M , 0 (End M,)OP,0 ) . 0 ) -, (R’

Sinceeachidentityof(R, *)isspecial,weuseProposition 7.2.18 toseethat multequivalence holds at each step. QED Corollary 7.4.4 provides the ideal sort of injection for GI-analysis. We should now like to find a closure of primitive (R,*), in the context of the involution. This can actually be done quite easily. As in 47.3, let A be a maximal left ideal of R such that Ann(,,,,(R/A) = 0; let M = R / A and M* = RIA*, and consider the injection t,b: ( R , *) -, (End M,, 0End,, M*, 0 ) of Remark 7.3.1. Take a maximal subfield F of D. Let T = End M , and T * = EndF,M*. We define an anti-isomorphism (*) from T to T * in the obvious way ; giveii fi in T , define fi* by y*fi* = (By)* for all y* in M * . Then T O T * has an exchange involution ( 0 ) induced by (*), and (End M, 0EndDIM*,0 ) is naturally injected in T 0T * . We inject F into T O T * by taking a (in F ) to the map &, given by dCy,,y:) = (yla,a*y:). Now F is a symmetric subfield of T O T * that centralizes $ ( R ) , so we can form (t,b(R)F,o)5 ( T @ T * , o ) ;to be suggestive, we write (RF. *) in place of ($(R)F,o). M acts on the first component of each element of ( R F , * ) [viewed in ( T @ T * , o)], thereby becoming an irreducible RF-module with Ann(RF,t,M = 0. Also, as in Proposition 1.5.12, we have F = Z ( R F , *). Call ( R F ,*) a closure of ( R , *). Proposition 7.4.5. Assume ( R , * ) is primitive and not special. Then, passing to (RF.*), we have RF is closed primitive, and ( R F , * ) is multequivalent to ( R , *)

47.4.1

Identities of Rings with Involution

273

Proof. Applying Proposition 7.4.3, we see that R is primitive. Hence R is prime, implying Ann,(R/A) = 0. Now Proposition 1.5.12 applies, so R F is clearly closed. Obviously ( R F ,*) is mult-equivalent to ( R ,*). Q E D

This simple observation clarifies the GI-structure of primitive rings with involution; to begin with, we can toss away the “stumbling block,” described earlier, to extending Kaplansky’s theorem. Theorem 7.4.6. If(R, *) is prirnititle with proper GI, then soc(R) # 0. Proof. Take some multilinear,f(X,, X:, . . . , X , , X : ) which is a proper GI of R, and take primitive P Q R with P P, P * = 0. If,fis ( R , *)-special then soc(R) # 0. (If P = 0 this is Theorem 7.2.9; otherwise RIP has some minimal left ideal LIP, and then L n P* is a minimal left ideal of R.] Thus we may assumefis not ( R ,*)-special. Thus by Proposition 7.4.5 we may assume R is a dense subring of End M,, where K = Z ( R ) is a field. Moreover, we may clearly choosef’such that no subsum of generalized (*)monomials of,f’is a GI of (I?,*). Since,fis not (R,*)-special, we may then assume that ,f(X,,XT, ..., X , - , , X;C- X,,O) is not a G I of ( R , *). Thus there are elements r l r:, . . . ,r,- 1 , r:- of R such that , f ( r l ,r:, . . . ,r,- rF- I , X , , 0) is , = , f ( r , ,r:, . . . , rF- X , , X:), we see that ,fl not a GI of R. Writing . f l ( X r XF) is a proper GI of ( R , *). I ril X r r i 2+ ril X:ri2 for suitable Now write out ,f,( X t ,X:) = rij,rij in R. By Lemma 7.2.10 we may assume that { r i l l 1 < i d u ) are ri2yoK. By K-independent. Choose yo in M arbitrarily and let I/ = Corollary 1.5.3 there exist elements a, = 1, a 2 , ..., a, in K such that given y in M there is a corresponding element r in R with rrizyo = ya,, 1 < i < u. Thus, for all .Y in R such that s V = 0, we have (substituting x*r for X , )

x,Y=

x;=,

xf‘=

o=( /

c rilx*rri2+ 1 r i l r * x r i 2 ) y o u

\

i= 1

i= 1

U

=

I’

C ril.u*(rri2yo)+ C rilr*(?tr~2yo) = i= I

i= 1

1

riIx*a, (i=”l

J’.

This is true for each y in M . Hence, for every x in R such that x V = 0, we have 0 = ril.Y*ai= (xx.Y= airi:)*, implying xx,Y= airi: = 0. Thus, by the density theorem, a j r 5 ) M c V, implying air; E soc(R) and is I air:)* ;= = Cf=a i r i l ,contrary to the Fnonzero because otherwise 0 = ( independence of the r i l . Q E D

xy=

(x:Y=

We could extend Amitsur’s theorem (7.2.9) and Kaplansky’s theorem immediately at this point, but without much extra work we can even extend Theorem 7.2.2. Actually, we shall repeat the proof of Theorem 7.2.2 using

274

[Ch. 7

THE THEORY OF GENERALIZED IDENTITIES

the partial density provided in Theorem 7.3.17 in place of the usual density theorem. Suppose ( R , *) is primitive, and in fact suppose that soc(R) # 0 and R is a dense subring of End M,. Letting "subspace" denote finite-dimensional Fsubspace of M . we say f ( X , , X : , . . . ,X , , X : ) is (&)-valued for subspaces V,,. . ., if for all ri in R such that r i & = rTF = 0, 1 < i 6 t, we have ,f(r l , ..., r,)M c V,. In order to apply Corollary 7.3.18, we further assume that M is a minimal left ideal L satisfying the conclusion of Theorem 7.3.17 and (thus Corollary 7.3.18). Thus we have the nondegenerate sesquilinear, (*)-compatible form (,>:M 0 M + F.

v

.

Lemma 7.4.7 (Notation as above) SupposeJ'(X,,X : , . . . X , , X:) is a sum of' v monomials and is (v)-valuedf o r suitable subspaces V,, . . ., J( : let u = max{[ : F] 10 < i < r } . Ler W, be the F-subspace of' R F spanned by the coeficients off: Then f can be wrirten as a sum of < I + monomials with coeBcients in W,, each monomial having a coeficient of' rank < 4'"'max(u,v).

v

Proof. (Modeled after Theorem 7.2.2, with some duplications omitted.) Call a generalized polynomial good if it can be written as a sum of monomials (with coefficients in Wo), each having a coefficient of rank <4'+"max(u,v). We go by simultaneous induction on t and u, and may assume r 3 1 and 11 > 1. Write r

f=

c (c h , j ( X , , X ?,..., X i * _ , , X i + ] , X i * , 1 ) . . . + c hjj(X1,x ; , .. ., xi- x:-l, xi, X?*,,, ...)X*W!. i t

)XiWlj

xi - 1 .

j: 1

;=I

0:

1,

V)?

1,

j= 1

where each hij is a monomial in (RfXf,.). We may assume w i j M $ & and wijM $ for all i.j by an inductive argument on v. Write& = 2;;h i j X i w i j and = I$ hIiX,?wii By symmetry, we may assume ,f, # 0. Choose y, in M such that w r 1 j ~ ,5. $ Let U = W , ~ ~ , F +w j~j y r$F~; note that [ U : F ] < u + v. By Corollary 7.3.18 there exist m 1 = 1, mz, . . . ,ac, in F such that for each y in U' there is corresponding r in R satisfying r*U = 0 = r v and rwrjyr= y u j , 1 < j < v,. Let g, = a j h I j ( X l ,X : , . . . , X , - X:- ,),and put &' = I ( + ~ > L wijyIF wijy,,F, 1 -<, i ,< t . For all ri such that ri = rT y.' = 0, 1 < i < t - I, and with y, r as in the last paragraph, we have

,

s.'

+x;i

v+x;~~

xy=

,,

v'

,

$7.4.1

Identities of Rings with Involution

275

thus there is Vd 2 V, with [ Vd : F] < 2 u + v < 3 max(u, v), such that gt(rl,rT,..., r f - l , r T - l ) M g Vd for all ri with rjv’=rF&’=O, 1 < i < t - I . Hence gt is (Vd, V;, . . ., y- ,)-valued so, by induction on t , g f is good. Let g; = g t X t w r 1 , 9 = f - g i = x:ii htjXr(wtj-Ujwt1). Clearly g; is good ; moreover, g is (Vd, V;, . . ., K’- I/;)-valued and is a sum of (v - 1) monomials and so, by induction on v , is good. Hence f = g +g’, isgood. QED

(fi+fi’)+ft’+x:yf=2

Our bound on the rank of coefficients is not as good as in Theorem 7.2.2, although with a bit more care in the proof it could be lowered from 4’+”max(u,v ) to 2’(u+[v(v- 1)/4]). It would be interesting to determine the best bound. Theorem 7.4.8. Suppose ( R , * ) is primitive with proper GI f and is not special. Then passing to (RF, *) we can rewrite f as a sum of monomials with coeficients spanned (over F) by the coejicients off, each monomial having a coeficient of rank < k over F (where k is a function of deg(f ) and of the number of monomials o f f ). Proof. Multilinearizing, we may assumefis multilinear and pass to RF. By Theorem 7.4.6 soc(RF) # 0, so we are done by Lemma 7.4.7. QED

Corollary 7.4.9. If (R,*) is primitive with a proper identity f , then either R is central simple of degree < k (where k is as in Theorem 7.4.8) or else ( R ,*) is special. Proof. If (R,*) is not special, then 1 has rank
Theorem 7.4.10.

I f (R,*) is primitive and GI(R, *) # 0, then GI(R, *)

= soc(R).

Proof. If (R,*) is special, then this follows easily from Theorem 7.2.8 and Theorem 7.1.13. If (I?,*) is not special, then R is primitive and we have GI(R, *) c soc(R) by Theorem 7.4.8, implying GI(R, *) = soc(R). QED Lemma 7.4.11. I f (R,*) is a ring with involution and R is semiprimitive, then (R,*) is a subdirect product of (*)-primitiverings. Proof. Let { P , , l y E r } be the set of primitive ideals of R. Then 0 = n { P ; . J y ~= r }n((P..nP*)lyEr},so(R,*)isasubdirectproductofthe (R/(P;. n P*), *). QED

We are ready for the main theorem of this section; cf. Rowen [76a]. Theorem 7.4.12. Proof.

I f ( R ,*) has a homogeneous strong GI, then R is PI.

B y Amitsur’s method, it suffices to consider the case NiI(R) = 0.

276

THE THEORY OF GENERALIZED IDENTITIES

[Ch. 7

Then REI] is semiprimitive and, extending (*) to RLI], we may write (R[A], *) as a subdirect product of primitive (R,,,*), y E r. Let (R),F,, *) denote a closure of (R?,*). If R,.F , is not primitive, we use Corollary 7.4.4 to embed (Ri,Fy,*) into mult-equivalent (End(M,)F70 (End(M,)F:)OP,0 ) ; if R,F;, is primitive, view R,F, as a dense subring of some (EndM,,)Fy. Letting ry be the image of r in (R,,*), define B = {~ER[A](([~;,M,.: F ; . ] l y ~ r )is bounded}. Clearly B a R[A], so B n B * d (R[A], *). Let f be a strong GI of ( R , * ) . We may assume that all generalized monomials offare not GIs of R (by throwing out all generalized monomials off which are GI5 of R ) . Also, 1 E Zf(R, *) c Z,(R[A], *). If B n B* # R[3.], thenf is (R[A]/Br-) B*, *)-proper, implying some multilinearization g off is (R[I]/BnB*,*)-proper. On the other hand, g is a GI of (R[A],*),so Zg(R[A],*) c B (bc Corollary 7.2.3 and Theorem 7.4.8); likewise g* is also a GI of (R[A], *), so Z,,(R[A], *) E B, implying Z,(R[A], *) c B*. Thus I,(R[A], *) c B n R*. We conclude B n B* = R, so 1 E B, implying R is a PIring. QED Corollary 7.4.13

(Amitsur [69]).

ff (R, * ) is PI, then R is PI.

57.5. Ultraproducts and Their Application to GI-Theory This section is spent on developing the logical tools that reduce the study of GI-theory of prime (resp. (*)-prime) rings to results on primitive (resp. (*)-primitive) rings known from the last sections. The basic idea, due to Amitsur r67] and first utilized for GIs by Martindale [72a], is to inject R [resp. (R,-*)] into some mult-equivalent, primitive R' [resp. (R'. *)I,via an ultraproduct technique. This technique is so important that we shall derive it in its full generality. Definition 7.5.1. A $her .F of a set S is a family of subsets of S satisfying the following three properties: (i) 0$ 3 ; (ii) if A , , A , €3, then A , n A 2 E 3 :(iii) i f A , E .Pand A , c A , , then A , E 9. Example 7.5.2. If S is infinite, then [ A c S I S - A is finite\ is a filter, called the Frechet3lter. Example 7.5.3. Suppose R is prime and is contained in a semiprimitive ring R , written as a subdirect product of primitive images (R,/P,,lyeT}.For any r in R , write ry for the canonical image ( r + P J of r in R1/P;.and define r, = ( y E T ( r i# 0). Then 9 = ( A G rlr, E A for some nonzero r in R'I ib a filter of r, called the Amitsur filter. [Indeed, (iii) is = 0. Finally, obvious; (i) is clear because if r: = 0 for all r, then r E njlETP,,

$7.5.1

Ultraproducts and Their Application

277

to see (ii), given nonzero a, b in R , take r in R such that arb # 0; then

r, n rb 2 r a r b E

c 9 . 1

Example 7.5.4. Suppose ( R , *) is prime and is contained in some ( R , , * ) with Jac(R,) = 0. By Lemma 7.4.11 ( R , , * ) is a subdirect product of primitive { ( R , ,* ) l y E r } . Given r in R , let r, denote the canonical image of r in R:, and define rr= {ylr; # O } . Then .F= { A E rlr, G A for some nonzero r in R } is a filter of r. [Indeed, (iii) is obvious and (i) is clear. To see (ii), given nonzero, a , b in R , we have (by Proposition 2.2.29) some or element I ef R such that arb # 0 or a*rb # 0; thus r a nrb 2 rorb I-,* n r b 2 r O r r b . But, for each y, a, # 0 iff 0 # ( a , ) *= (a*),;consequently r, = Tat,implying r, n r b E 9.1 Example 7.5.5. Let S be an algebra, and let r = {finitely generated subalgebras of S } . For each finite subset T of S , let rT= { R E F I T c R } . Clearly rTlvT, = rTl n rT,,so it follows immediately that { A E rlsome rTE A } is a filter of I-. Definition 7.5.6. Remark 7.5.7. ultrafilter.

A maximal filter is called an ultrufilter. By Zorn’s lemma every filter of S is contained in an

Proposition 7.5.8. A filter 9 of S is an ultrafilter ifl for all A either A E or ~( S - A ) E . F .

cS

Suppose A , G S and A , $ F. Let 9‘ = ( B E SIB 2 A n A , Proof. (a) for some A in S}. Obviously 9 c .F (since A , EF), so by hypothesis 9‘ is not a filter; but 9’ satisfies properties (ii), (iii) of Definition 7.5.1, so we implying S - A , E F . conclude A n A , = @ for some A E 9, (t=) Suppose 9‘is any filter containing 3, and A E F. Then S - A $ F, so S - A $ 9 ; thus, by hypothesis, A E 3.Hence .9= 9’, proving .Fis an ultrafilter. QED We are now ready for the important construction based on ultrafilters. We use the notion of “similar” (algebraic) systems from $2.6. Definition 7.5.9. Suppose S,, y E r , are sets, and 8 is a filter on r. Define the reduced product of the S , with respect to 8,written nS,/.F,to be the set of equivalence classes of the Cartesian product nyerSy, under the By Definition 7.5.1 equivalence given by (s,) (s;) iff {y E rls, = s;} E 9. is really an equivalence. Moreover, if we have sets Siyr1 < i < t + 1, y E r, and operations F , :Slvx ... x S , , -, for each y, then writing S , = nSiv/.F we get an operation F : S1 x .. . x S , + S, + defined by F ( [ ( s , , ) ] , . . . , [ ( s t , ) ] ) = [(F,(s,,, . . .,s,,))]. ( F is well defined by Definition 7.5.1.) Writing nFJ8 for F , we now can define naturally the reduced

-

-

,

278

THE THEORY OF GENERALIZED IDENTITIES

product of similar systems .Y,

=

[Ch. 7

( S l y , . . ,SfyrF,,, . . . , F k y )to be

(n.S,,/.F, . . . ,nS17/.F; nFl;./S, ... , nF,:./.F), denoted as

n.Y,'3.

Definition 7.5.10. an ultraproduct.

If 9 is an ultrafilter o n

r, then n . Y ' ; / S

is called

The example 1 hat will interest us is that in which we take each system to be a ring R7 coupled with a faithful, irreducible module M , . Of course we want the ultraproduct to be a ring R = n R , . / S with faithful, irreducible The key is the following result: module M = nM,,/.F. Theorem 7.5.11. (LoS [SS]). Let { Y , l y E r}be a family of similar systems, and let .F be an ultrafilter of r. A formula $ holds in (n,,rccP,)/9, $ { y e T J $ holds in Yy} €3. Proof. The theorem is obvious for atomic formulas, so we argue by induction on rank (+) in Definition 2.6.2. Let Y' = nYEr.Yy.

Case I . $ = ( -I+~). Then $ holds in .Y iff $, does nor hold in 4" iff holds €3. in ,Y'J €9(by Proposition 7.5.8) iff { Y E rl+holds in 9,.) Case 11. $ = (IL, A +z). Then I!, holds in .Y iff and t j 2 both hold in 9 iff { y ~ r lholds $ ~ in .Y'y)€ 3and { y e T ( y z holds in Y,)E riT iff {;,€I-[ $, and t j 2 both hold in :fY} E 3, iff {y E rI($]A t j Z ) holds in Y',) E 3. Case 111. $ == ( 3 , ~ ~ )...,. $ ~ xi, ( ...). Then $ holds in 9' iff for some (si) E S i , $ l ( . . . , ( s i r )., . .) holdsin .(Piff {y E I- I$](. . . ,si ,. . .) holdsin .V,for some si,)E .F iff [ y er l ( 3 ~ ~ ) $ ~.(,.x.i , . . .) holds in ,YY}E .$. QED

{ y ~ r l $holds , in .'/',}#.F(by induction hypothesis) iff I - - ( y E T I $ I

Note that the property that 9 is maximal enters in the above proof only in Case I. This theorem has several important applications. including the following method of passing from prime rings to primitive rings: Theorem 7.5.12. (i) (Amitsur [67]) I f R is a prime ring contained in a semiprimitivr ring R,, then R can be injected into a primitive ring R' that satisjies all efemeiitary sentences holding in each primitive homomorphic image of R l . (ii) I f (R,* ) is prime and is contained in a semiprimitive ( R , ,*), then ( R ,*) can be injected into a (*)-primitive ring with involution that satisfies all elementary sentences holding in each primitive homomorphic image of ( R1, *). Proof. (i) L.et { P;.I y E I-} be the set of all primitive ideals of R and, enlarging the Amitsur filter on (Example 7.5.3) into an ultrafilter #, form 9 . map T : r -+ (r?),where each rp the ultraproduct R' = ( n 7 E r { R 1 / P ) , j ) /The = r P,, E R,/P.,.,is obviously a homomorphism from R to R', and ker(z)

+

47.5.1

Ultraproducts and Their Application

279

= { r E R l { y l r ) ,= O } E F } . But if r E R and r # 0, then {ylr),# O } E F , by definition of the Amitsur filter, so we cannot have rcker(z); therefore z is an injection. By definition of ultraproduct, every elementary sentence holding for each primitive homomorphic image of R, also holds for R‘. Moreover, letting M ; be a faithful, irreducible R,/P,-module, we take the ultraproduct (n,.,{R,/P,, M , } ) / 3 .Writing M‘ for ( n , s r { M , } ) / F ,we see that M‘ is a faithful, irreducible R’-module, because “faithful, irreducible” are elementary concepts (cf. Example 2.6.5). Thus R’ is primitive. (ii) Parallel to proof of (i), using the filter of Example 7.5.4 in place of Amitsur’s filter. QED

This theorem is very nice, especially in view of the fact that although “primitivity” of a ring is not intrinsically an elementary sentence, we have “made” primitivity elementary by throwing the faithful, irreducible modules into the algebraic system. Theorem 7.5.12 enables us to replace “prime” [resp. “(*)-prime”] by “primitive” [resp. “(*)-primitive”] in many assertions, and is crucial to the remainder of this chapter. Injecting Prime [and (*)-Prime] Rings into Nicer Rings

Theorem 7.5.13. (i) Every prime ring R can be injected into a multequivalent R-ring of the form End M,. (ii) If (R, *) is prime, then either (R,*) is special or else (R, *) can be injected into a mult-equivalent (R, *)-ring with involution (R‘,*) such that R’ is closed primitive. (in the latter case, R is necessarily prime.) Proof. (i) We use a series of injections, and check that multequivalence holds at each stage. First let R, = n,,,R, for an infinite number of copies R, of R, and let R, = Rl/Nil(Rl).By Corollary 1.6.26 and Theorem 1.6.21 there is a natural embedding from R into R,, and so obviously R, is equivalent to R. Now Nil@,) = 0, so by Amitsur’s theorem R2[A] is semiprimitive and is surely mult-equivalent to R,. But viewing R E RJA], we can now use Theorem 7.5.12(i) to inject R into a primitive ring R’ that satisfies every GI of each homomorphic image of R,[A] ; hence R‘ < R,[A], implying R‘ is mult-equivalent to R. But then by Corollary 7.2.19 R‘ can be injected into End M , , which is mult-equivalent to R’ (and hence to R ) . (ii) We just add (*) to the proof of part (i), inducing involutions at each stage until (R’,*), which is primitive by Theorem 7.5.12(ii). Then we conclude the proof, using Proposition 7.4.5. (Note that R is necessarily prime if R’ is primitive by Remark 2.1.39.) QED

The point of Theorem 7.5.13 is to enable us to apply Corollary 7.2.3 (or

280

[Ch. 7

7 HE THEORY OF GENERALIZED IDENTITIES

Theorem 7.4.8 i n the involutory case) as follows: Suppose R is prime and { ( X I ..... X , ) is a multilinear GI of R which is a sum of i' monomials. Injecting R into some mult-equivalent R-ring R' that is dense in End M,,, we get by Corollxy 7.2.5, for all r , , . . . , r, of R and ever) generalired monomia1.f; of,/; that rank,/;(r!, ..., r , ) d + r l + ' ( r - 1 ) (as transformation in End M " ) . Thus /,.(R) consists only of elements of bounded rank in End MF, and GI(R) E soc(R'). Let us state this result formally. Theorem 7.5.14. //' R is prime with proper GI uiid I / R is ir1;cvreti into soiiic' niulr-equrraleiit, printitioe R-ring R', theu 0 # GI(R) c R r\ soc(R').

Let us see how this theorem "works." (Jain). I t ' R is p r i i w w i t h proper GI. rlieii R Corollary 7.5.1 5 iioiizcro riil lefi or right iL1eLiI.s. [ I n partrcufar. NiI(R) = 0.1

litrs 1 7 0

Proof. By symmetry, we need only prove R has no nonzero nil left ideals. Inject R c End h l , with0 # R n soc(End :M,).Let A = R r~soc(End M,),and suppose B is ii nil left ideal. For every u i n A , [ u B M :F ] < [OM: F ] = I for suitablerdepending ona. But uBactson aBM by multiplication.so wecan view each trb as a nil mairix in M , ( F ) . By proposition 1.3.20. (tiB)'aB.Z4 = 0. Thus (uB)'+'= 0. implying Bu = 0, yielding BA = 0, so B = 0. QED

In the above proof. we used the socle to move us to a finite-dimensional setting (actually of matrices); we shall now develop this method systematically. in order to improve Theorem 7.5. I3 substantially. The main point is to note. for rER, that Ann',,r is an ideal of Rr (viewed as ring without 1 ) and to consider Rr/Ann',,r; we let - denote the canonical homomorphic image in = Rr/Ann',,r. Typical elements of Rr will be written as Ur,hr for a, b in R. Note that ii? = 0 iff rur = 0. Proposition 7.5.16. ( i ) If'Rr is prim, tlieii i s prime. (ii) ! f M is u ,fiiitl!ful (resp. irreducible) R module, theii M/Ann',r is a ,fuitl?firl(resp. irreducible) %wiodirle. -~

Proof. ( i ) If Z R r br = 0, then rarRrbr = 0, implying = 0, so 7iF = 0 or hr = 0.

rlir =

0 or rbr

~~

(ii) First note {hat (Ann;r)M c Ann',,r, so the action o f RI on R induces a well-defined module action of M/Ann',r on %. If M is R-faithful and urM E Ann;,r, then rarM = 0, so= = 0 ; thus k is Ri-faithful. If M is irreducible and 7 G M such that T is ii nonzero submodule__ of then RrT is a nonzero submodule of M , implying RrT = M , so ,GI= RrT E 'T: hence $?is %irreducible QED

a.

$7.5.1

Ultraproducts and Their Application

28 1

Call R strongly primitive if soc(R) # 0 and R is dense in End M , for a suitable vector space M over a division PI-ring D ; Theorem 7.2.9 says that a primitive ring is strongly primitive iff it satisfies a proper GI. We use Rr to characterize “strongly primitive” in terms of central simple algebras. ’ Proposition 7.5.17. then % is simple PI. (ii) If primitive and r E soc(R).

(i) !f R is strongly primitive and 0 # rEsoc(R), % is simple PI and ifR is prime, then R is strongly

Proof. (i) Suppose R is dense in End M , , and deg(D) = d. Also, suppose rank(r) = t. By Proposition 7.5.16(ii) is primitive, with faithful, irreducible module M = M/Ann’,r. But [ M : D ] = t, so & satisfies the standard identity SZd,,and is thus simple. (ii) Let L be a minimal left ideal of %. We claim that RrL is a minimal left ideal of R. Indeed, if L , is a left ideal of R with 0 # L , c RrL, then - _ _ - _ _ 0 # L , c RrL c L , implying L , = RrL = L. Thus, for every .Y in L there exists .Y, in L , such that .Y = . f l , i.e., r ( x - x , ) = 0. Hence rL c L , , implying RrL = L,, proving the claim. But this means soc(R) # 0, so by Proposition 1.5.9 R is primitive; by Theorem 7.2.2 r E soc(R).Write R as a dense subring of End M,. Clearly % has a subring isomorphic to D,implying D is a PI-ring. Q E D

Remark 7.5.18. If A G R and ~ E Athen , no ambiguity in the notation.

% = Aa/Ann’,,a,

so there is

Suppose R = End M , The GI-theory produces a cute result concerning 6. is a W-ring, for somesubring Wof R that is mult-equivalent (over W )to R. For any r in W n soc(R) we saw 6is simple PI and by Exercise 1.1 1.3 has a unit element e. Moreover, F(Ann’r) c Ann‘r so 6is an F-algebra in the natural way, and Fe c Z ( 6 ) . Proposition 7.5.19. Given the above, 0 # Z ( E ) c Fe, and Z(%) Fe. Proof. Write e = G. Suppose Wr E Z ( W r ) .Then r[.Yr, wr] = 0,for all .Y in W, implying rX,rwr-rwrX,r is a GI of W, thus of R. By Lemma 7.2.1 I rwr and r are F-dependent, so r w r q , and r x 0 are F-dependent ; thus Wr and e are F-dependent. Thus Z ( 6 ) c Fe. Note that 6 is a prime PI-ring without 1 , so Z ( 6 ) # 0. Putting W = R we get Z ( g ) E Fe, so Z ( 6 ) = Fe. QED =

Theorem 7.5.20. (i) If R is prime and satisfies a proper GI, then some central extension of R is strongly primitive and is closed (as a primitive ring). (ii) I f ( R , *) is prime and satisjies a proper GI, then either ( R ,*) is special or (*) extends to some closed, strongly primitive central extension of R.

282

THE THEORY OF GENERALIZED IDENTITIES

[Ch. 7

Proof. ( i ) Inject R into somemult-equivalent R-ring R , = End M,,and let R’ = R F E R , . By Theorem 7.5.14 we have some Y # 0 in R n soc(R,).By Proposition 7.5.19 Z(R’r) is an F-subalgebra of Fe (for the unit element e of Hence Z(R’rl= Fe, a field. k.r is thus a prime PI-ring whose center is a field, so R’r is simple, implying R’ is strongly primitive. We are done by passing to a closure of R‘. (ii) Analogus to (i), by means of Theorem 7.5.13(ii). QED

F).

Theorem 7.5.20(i ) should be called Martindale’s theorem (cf. Martindale [69,72]), although in fact Martindale’s full result was slightly stronger and will be obtained in $7.6; Theorem 7.5.20(ii) is due to Martindale [72]-Rowen [75b]. 87.6. Martindale’s Central Closure

A pretty construction, the “central closure” of Martindale [69], provided the original form of Martindale’s theorem; it has the advantage of being quite explicit, giving rise to several new theorems (and sharper statements of previous theorems). This construction works just as well for (*)-rings (and actually for rings with any finite group of automorphisms and antiautomorphisms); to keep the exposition simple, we shall mostly restrict ourselves to the case without involution, making parenthetical comments about the (*)-case and leaving greater generality to the reader. Let R be prime, and let .f = {nonzero ideals of R). [In the (*)-case,let (R. *) be prime, and let .9= {nonzero ideals of ( R , *).I Define an equivalence on = {(.L A ) I A E f and f ’ : A R is a module homomorphism: as follows: ( f , ,A , ) (.f2, A 2 ) if.f, and,f, have the same restriction to some ,4 E A , n A , in .f. Write [ j ;A ] for the equivalence class of (f, A ) . The set of equivalence classes has a ring structure, given by [ f,, A , ] + [ j i , A , ] = [.f, +f,, A , n A,] and [fl, A1][f2, A L ] = [Ill2,( A , n A 2 ) ’ ] ; we call this ring Qo(R). Let j i denote the right multiplication map,f,(r’)= r’r.

-

-+

Remark 7.6.1. There is a ring homomorphism R Qo(R) given by. r [ j i , R ] , whose kernel is { r e Rl A r = 0 for some A in . f ) = 0; in this way we view R E Q o ( R ) . -+

-+

Definition 7.6.2. If A E .f and . f : A -+ R is a module homomorphism, we say (1;A ) is admissible iff: A R is also a right module homomorphism. [In the (*)-case, we also require f ( a * ) = f ( u ) * for all Q in A.] The extended centroid Z of R is { [:JA] E Qo(R)l(,f,A ) is admissible}. -+

Proposition 7.6.3. Z(Q,(R)).

The e.utended centroid Z OJ’R is afield contaitied in

67.6.1

283

Martindale’s Central Closure

Clearly 2 is a subring of Qo(R). Suppose ( J A ) is admissible. For in Qo(R) and all ~ 1 9 in ~ 2 A n A , , f ( f i ( a i a 2 ) ) =.f’(aifi(a2)) = f(a,)f1(a2) = .f,(f(a,)a,) = f l ( f ( ~ I a 2 ) ) 7 implying “.f,fll, ( A n A d 2 ] = 0 ; consequently 2 c Z ( Q , ( R ) ) . Moreover, if [,f, A] # 0, then f ( A ) E .Iand ( k e r f ) f ( A )=f‘((kerf)A) z . f ( k e r j ) = 0, implying kerf’= 0 ; then [ J A]-’ [ f - ’ , , f ( A ) ] E Z , proving 2 is a field. QED Proof.

[.fi,Ai]

We are now ready for the key definition. The ceritral closure of R, denoted by R , or RZ, is the

Definition 7.6.4. subring R Z of Qo(R).

Remark 7.6.5. . R is a central extension of R, and Z ( R ) = 2 [since

2 G Z ( R ) c CQo,R,(R)= 21.

Remark 7.6.6. In the (*)-case, R is a (*\-ring. [Define ( z r i z j ) *= x r T z i for ri in R, zi in Z . To see this is well defined, write C:= rizi = 0 for zi = K,Ail with (fi, A i ) admissible and let A = A , ; we may assume for all a E A that Zf=&(a)ri = 0, implying for all a,, a, in A

,

k

1

C .fi(a,)r? a2 = 1jXa1rTa2) = C.lr?.f;(a,)

= a , CrT./l(a*)

i=1

= q ( ~ . / i ( u ~ ) r=i f0 ;

thus

xf=l , / i ( u , ) r E~ Ann, A = 0.1

For every .Y in R there exists A in .Y such that 0 Remark 7.6.7. # A u G R. It follows easily that R is prime. [In the (*)-case, R is (*)-prime.] Thus we have transformed R into a closely related ring whose center is a field. The term “central closure” is justified by the following result. Proposition 7.6.8.

R is its owti central closure.

Proof. It suffices to show that the extended centroid of R is Z . Suppose we have 0 # A‘ a R and f ’ : A’ + R, with (,/”, A ’ ) admissible. Then, putting A = A’ P, Rand lettingf’bethe restriction of,/”to A , we have A E -9,and (,/; A ) is admissible, i.e., [,/; A ] E 2.Conversely, given admissible (,J A ) , we can definea bimodule homomorphism,/“:R A R + R, by k

I.!(1 .Yi,a,,i2) i= 1

k

=

c

.Yiij(ai).Yi2?

i= 1

and (,/”, R A R ) is admissible. These two correspondences identify Z canonically with the extended centroid of R . [The (*)-case is proved analogously.] QED

284

THE THEORY OF GENERALIZED IDENTITIES

Definition 7.6.9.

[Ch. 7

R is centrally closed if R = R.

The central closure has the following remarkable property due to Martindale [69]-Bergman [74bP], generalizing Corollary 1.5.19. Recall Theorem 1.8.18. Theorem 7.6.10. J f R is centrally closed with F = Z ( R ) ,thenfor every prime extension R , 2 R we have R , R O FH , where H = C,,(R). (Note that H is not assumed to be commutative!) Proof. Otherwise we have F-independent elements x,,. . . ,.x, in H and rixi = 0 ;choose such a situation with t nonzero r l , . . .,rl in R such that minimal. 0 # r,R,x, = Hr,x,R, so t > 1. Define a mapf:Rr,R -+ Rr2R by

xi=,

f ( x u j r l bj) = xair2bj \J

for a j ,bj in R .

i

If 2.a.r J J l bJ. = 0, then letting ri = x j a j r i b j ,1 < i ,< t, we see that 0 = C:= ri.yi = C:=,rixi, implying each rf = 0 (by minimality of t ) ; in particular r; = 0. Therefore f is well defined, and obviously ( J R r , R ) is admissible. Since r2 = f ( r , ) , we get rz = arl for some a in 2 = F. Therefore r l ( s l +ax,)+ rixi = 0, contrary to the minimality of t. QED Corollary 7.6.11. I f R , is a prime, central extension of centrally closed R , then R1 = R O Z c R , Z ( R , )If. a,,. . .,a, are Z(R)-independent elements of Z ( R , ) and x i = l r i a i= Ofor ri in R, then {r,, ...,rt) are Z(R)-dependent.

Now we look at some examples that show that the central closure is a good non-PI analogue to the ring of central quotients of a prime PI-ring. Example 7.6.12. Every simple ring is centrally closed. [Indeed, R is the only nonzero ideal of R, and every admissible V; R ) can be identified with the element ~ ( I ) E Z ( R J . ] Example 7.6.13. If R is prime PI, then the ring R , of central quotients of R is central simple, from which it follows quite easily that R , = R.(Proof is left for the reader.) In this case Theorem 7.6.10 is part of a famous theorem of Wedderburn (cf.Jacobson [64B, p. 1181).

Wedderburn's result just quoted is needed for Exercise 7. Example 7.6.14. Suppose R is a dense subring of End M,. There is a canonical injection J / : Z -+ Z ( D ) .[Proof. Take 1.1; A ) admissible, noting that A M = M and Ann', A = 0 because 0 # A a R and M is faithful, irreducible. Thendefinef': M -+ Mby,f"(ay) =f(a)yforanyain A,yin M.Ifa,y, = a2y2, then for each u in A, 0 =f(a)(a,y, -a2y2) = f ( a a l ) y ,-,f(aaz)y2 = a(f(al)yl -,f(a,)y,), so ,f'(alyl )-f'(u2yz) E Ann', A = 0, proving .f' is well

57.6.1

Martindale’s Central Closure

285

defined. Moreover, clearly f ’E End, M = D, and f ’ commutes with all elements of D, so f ’ E Z(D). We define $ :Z + Z(D) by $([A A ] ) = f’; it is immediate that $ is a well-defined injection.1 Thus, identifying Z(D) with Z(End M , ) canonically, we see that R c RZ(D) G End M,. In particular, if Z(End M,) c R, R must be centrally closed. A special case of this result is that every closed primitive ring is centrally closed, but the converse is far from true. (If D is not commutative, End M , is centrally closed but is not closed as a primitive ring.) This example gains significance from the following theorem : Theorem 7.6.15. (i) If R is prime with proper GI, then the central closure R is strongly primitive, with nonzero socle. (ii) ’1 (R, *) is prime with proper GI, then either every GI of’(R, *) is special or else, letting ( R , *) be the central closure of (R, *), we have R is strongly primitive with’nonzero socle. Proof. By Martindale’s theorem we can embed R in some closed, strongly primitive central extension R’. By Theorem 7.5.14 there exists nonzero r in R n soc(R’). Now let F = Z ( R ’ ) and consider R’r = R’r/Annk.,r, a central simple Fe-algebra by Propositions 7.5.17 and 7.5.19, where e is the multiplicative unit of R’r. Now 6 can be viewed as a 2-algebra without 1. Nevertheless, setting t = [%:Fe] 1, we claim every t elements of 6 are Z-dependent. (Indeed, if r l , . . . ,rr are elements of R, then rTr,. . ., are F-dependent, implying rrl r , . . . ,rr‘r are F-dependent, and so rrl r, . . . ,rr,r are Z-dependent, implying cr,. . . , are Z-dependent.) But % is a prime PI-ring without 1, and thus has a nonzero central element c. Since {c, . . .,c’} are Z-dependent, we have aici = 0 for suitable q2 1 and ai in 2 such that aq = e. Hence a i C q = e, so eE Rr. Now Rr is finite dimensional over Z e , so 6 is simple. Thus R is strongly primitive by Proposition 7.5.17. (ii) Proved analogously to (i). QED

+

xt=

Actually, there is a formulation ofTheorem 7.6.15(ii)that lies more correctly in the (*)-theory;cf. Exercise 8. Theorem 7.6.15(i) is the full formulation of Martindale’s theorem. The central closure is a very natural construction, and has been used by Bergman r74bPI to prove INC for finite extensions (cf. Exercise 11); there are probably many more uses for it, still to be discovered. Let us conclude this chapter by reformulating Theorem 1.4.34 in a very general setting. Actually, we shall present a magical proof that is completely GItheoretic, independent of Lemma 1.4.33! Theorem 7.6.16. For any prime ring R with central closure RZ, elements r l , . . . , I , of R are Z-dependent i$ C 2 ‘ -l ( r l r ...,rtr X , , . . ,XZtis a GI ofR.

286

THE THEORY OF GENERALIZED IDENTITIPS

[Ch. 7

Proof. Passing to R Z , which is mult-equivalent to R, we may ;issume R is centrally closed, with Z = Z ( R ) . If r l , . . .,r, are Z-dependent. then C z f - l ( r l ,.., . r,, X i , I , . . ., Xz,-l) is a GI of R since C z f - is I-normal. Conversely, suppose C2,- l ( r l . . . . ,r,, X,,l , . . . ,X Z I - is a GI or R . Itiject R into some mult-equivalent R-ring End M,; (Theorem 7.5.I.3). Then Czf-l(rl,...,r,, X , + l,:. . ,X z f - is a GI of End M F .But

CZ,-l(rl,. ..,r1. x,+1,. ... xz*1) f

=

C ( - t ) i + l r i X , + l C Z , - . l O,..., . l ~ ; - ~ , r,...., ; + r i , ~ i ,t . L. . , x ~ , - , ) . i= 1

By Lemma 7 2.1 1 either r l . . . . , I ' , ;ire F-dcpciideiil o r each Czf-3(rl,.... r i - l , r i + r , , X r i z,..., X 2 , , ) is a ( ; I . in which case. by induction on L, r l , . . ., ri- rit . . ,rf are F-dependent, so certainly r l , . . ., rf are F-dependent in either case. But then by Theorem 7.6.10 r l , . . . ,rf are Z dependent. QEC)

,,...,

EXERClS ES

87.2 1. If R has a proper GI, then R has a proper multilinear GI ofdegrce 2. ( H i i l l : You may assume that R has a proper GI with the property that no subsum ofgenerali7c.d moiiomials isa GI of R ; then specialize.) This fact can be used to simplify several proofs. 2. Using the notation and hypotheses ofTheorem 7.2.2.prove that,/can be wriltrn iis a sum ofmonomials with coefficientsin W,one ofwhich has a nonzero coefficient ofranh ::U + rr(log, L,), where n = [W: f]. (Hirrt: You may assume ti, < $.) 3. Every multilinear identity of M,(F) is a sum of at least 2k monomials. *4. Can the bound o f Exercise 3 be improved? [A plausible conjecture based on S,, is (2k)!] 5. IfS,(a,X, ,..., rr,X,)isa GI o f R = E n d M , , t h e n e a c h a , E s o c ( R ) . 6. IfR isdense in End M,, where D has degreed, and if L is a sum of u minimal left ideals o f R , then S,,+l is a n identity of L . 7. IfA,Bareleft idt-alsofasemiprimeringR,havingrespectivepl-degrees u , ~thens,,,,,,,. , is an identity of the left ideal A + B ; in fact, GI(R) = x{left ideals of R that satisfy a standard identity). 8. In any ring, the sum of two left ideals that are P1 is PI. 9. If R is simple with R-proper GI. then R is PI. (As usual, I E R . ) 10. If R is simple and [R:Z(R)] = xj, then R is equivalent to every R-ring containing R. 11. If R is semiprime with f and satisfies the GI [r.[r.X,]]. then rEZ(R).

,

87.4 1. If (R, *) has ii symmetric (resp. antisymmetric) element r satisfying a polynomial y(L) in Z(R)[L] such that C , ( r ) is PI and y'(r) is invertible, where g' is the derivative of y. then R is PI. 2. If ( R , * ) IS semiprime with and satisfies the GI [r.[r.Xl +X:]] with r* = r. then r E Z ( R , *). (This exercie is useful in studying Jordan algebras.)

287

Exercises

Ch. 71 $7.5

I . Suppose R is prime [resp. (R, * ) is prime] and has a homogeneous R-polynomial/that is central [resp. (R, *)-central]. Then R is a PI-ring whose PI-degree is bounded by a function of the number of monomials off; or,fis constant. 2. Suppose ( R , *) is prime,fI ( X , , . . . X,) is a homogeneous, generalized (*)-polynomial,and j i ( X , + I ,.... Xu)is an ( R , *)-proper, classical (*)-polynomial such that [ is a GI of(R,*). Then either (i),/, is a constant in Z(R, *), ( i i ) / I is a GI of (R, +), or (iii) R is a PI-ring [ofdegree bounded by a function on deg(,f,) and the number ofmonomials off,]. (Hirrr; Prove the result first for the special case ,I; is a constant and obtain the general result through specializing using Exercise I.) 3. Generalize Corollary 7.5.15 to rings with involution. 4. Using ultrafilters, inject an algebra R into an algebra R' such that every elementary sentence holding for all finitely generated subalgebras of R also holds for R'. Generalize this to arbitrary algebraic systems. 5. Here is a fact useful in the papers of Passman [71b, 72a. 72b] on group rings with polynomial identity. If every finitely generated subgroup of a group G has an abelian subgroup of finite index
.

-

1. Suppose R is prime with proper GI, and 0 # X E S O C ( R )then ; Z ( R z ) :Z . Hence, for B R, there is a in Z such that ax E B. 2. If R is prime, then every completely homogeneous, R-proper GI of R is a GI of R .

a

Here is an outline of the structure theory of semiprime rings with GI ; cf. Rowen [77b]. If R is semiprime, call R a GI-ring if GI(R) is a large ideal of R.

3. Every semiprime GI-ring has no nonzero nil left or right ideals and is left and right nonsingular. 4. If R is a semiprime GI-ring and satisfies ACC(left annihilators), then R is a PI-ring and the ring of central quotients of R is semisimple. 5. A primitive GI-ring with ACC(ideals) need not be PI. 6. Say R has bounded iridrz k i f ? = 0 for every nilpotent r E R. Any algebraic, semiprime GIalgebra of bounded index is PI. ( H i n t : Pass to the prime case and then to the central closure.) 7. (Montgomery-Smith [75]) I f R is an algebra over a field F and .4 E R is finite dimensional over F (not necessarily with the I o f R ) such that A 0 ,K is semisimple, where K is the algebraic closure of F , and if C , ( A ) is a PI-ring, then R is PI ; if, moreover, R is semiprime, then PIdeg(R) is bounded by a function of PI-deg(.4) and PI-deg(C,(A)). [E\-terisitv h i r l r : In rapid succession, we may assume F = K , Nil(R) = 0, and R is prime: letting e bea central idempotent of .4 such that eAe is simple, we have S,, is an identity of e,4e for some d and eRe = eAe @ Z,eRe,C,,,(eAe). If C , ( A ) satisfies an identity of degree t , then D , , ( e X , e , .. . ,eX,,e) is a G I of R, so we conclude by using theorems from 657.2 and 7.5.1 8. I f (R,*) is prime with proper GI. then the central closure (R,*) is primitive, and GI(R, * ) G soc(R, *); moreover, if M is an irreducible R-module with AnnlR,*, M = 0. then deg(End, M ) is finite.

288

THE THEORY OF GENERALIZED IDENTITIES

[Ch. 71

9. Prove Theorem 7.6.16 using Lemma 1.4.33 instead of Lemma 7.2.1 I . 10. (Bergman [74bP]) I f S is a finitely spanned extension of a prime ring R, then R E

s

canonically. 11. (Bergman [74bP]) INC holds for all finitely spanned extensions ofarbitrary rings. (This result generalizes Theorem 4.1.8.) [E.
5s ,

CHAPTER 8

RAT10 N A L I D E NTlTl ES, G EN E R A LIZ E D RATIONAL IDE NTlTlES, AND THEIR APPLICATIONS The theory of polynomial identities was initiated by Dehn [22], who was trying to determine by algebraic means how to classify Desarguian projective geometries. Fortunately for us, he chose the wrong path-the correct algebraic theory to consider was rational identities, which, roughly speaking, are identities that also involve inverses. Accordingly, Amitsur [66a] developed a theory of rational identities that enabled him to dispose of Dehn’s problem (with one minor exception), proving that the only intersection theorems possible in a given Desarguian projective plane arise from the underlying division algebra having finite dimension over its center. (For example, Pappus’ theorem is true when the underlying division algebra is commutative.) Amitsur also found general theorems about division algebras, resulting from this theory. In this chapter we shall present the theory of rational identities, in an improved form due to Bergman [68P, 701, Featuring the following two fundamental theorems about generalized rational identities: (1) there is no “nontrivial” generalized rational identity of a division algebra of infinite dimension over an infinite center; ( 2 ) for division rings D, and D, having finite respective degrees d, and d,, we have d,ld, iff every (generalized) rational identity of D, is also a (generalized) rational identity of D,.The second theorem is presently much harder to prove than the first, relying on the difficult theorem of Bergman-Small [75]. The proofs of the two fundamental theorems are given respectively in 48.2 and $8.3 ; applications are given in $8.4. The most important applications rely only on the first fundamental theorem and Theorem 8.3.5.

48.1. Definitions and Examples

Before defining a rational identity formally, we would like some motivating examples. Example 8.1 .I. [ [ X , , [ X , , X , ] X , [ X , , X , ] - ’ ] 3 , X , ] vanishes (where

289

290

[Ch. 8

RATIONAL 1DENTlTlES

defined) for all substitutions in division rings of degree 3 (cf. Theorem 3.2.2 1 ). Example 8.1 2. ( X + Y ) - ' - l ' - ' ( X - ' + Y - ' ) - ' X - ' vanishes (where defined) for all substitutions in every ring. [Indeed, y - ' ( . ~ - ' + V - ' ) - ' Y - ' = ( . Y ( . Y - ' + y - l ) y ) - ' = (y+s)-'.] Example 8.1.3 (Hua's identity) ( X - ' + ( Y - ' - X ) - l ) - l - ( X - X Y X ) vanishes (where defined) for all substitutions in every ring. [Proof. If .Y-', y - 1 , ( y - ' - . ~ ) - ' , and ( . ~ - ' + ( y - ' - x - ' ) - ' are defined. then in example

8.1.2, substituting

Y-'

for X and ( y - ' -

Y)-'

for Y, we get

(x- +(I' - I - x)- ')-

= (y-'-X)(.Y+(y-'-.~))-'.u

= (J-'-S)V.Y

= .Y-X)'.K.

Hua's identity illustrates well the added complication here-- the rational expression is built up in several stages using inverses, and the expression is defined only if each "subexpression" is defined. Thus our definition must be made inductively in terms of what we call "levels," and the evaluations made in a ring R will actually take values in R u { ?}, where ? is a formally undefined element satisfying the following properties for all r in R : (i) ? = - ? = ? + ? = ? . ? = ? - ' = ' ? + r = r + ? = (ii) if r is not invertible in R, then ? = r - ' .

?.r=r.?;

Let us now define formally the set of rational expressions in a set of noncommutative indeterminates X , with coe@cients in a ring W, written W ( X ) ;we also define the level of a rational expression, and its evaluation in a W-ring R. Definition is by induction on level, as follows: Definition 8.1.4. (1) If h(X,, ..., X,)EW(X), then h E W ( X ) and level(h) = 0; given elements r l , . . . , r, in R, define h ( r l , .. . , r,) as usual. (2) If f ( X l ,..., X , ) E W ( X ) and level(f)= k , then h = ( - , ~ ) E W ( X ) , level(h) = k + 1, and h ( r l , .. . , r,) = - j ' ( r l , . . .,rt).

(3) If f ( X l , ..., X , ) E W ( X ) and level(f)= k , then h = ( . f - - ' ) ~ W ( x ) , level@) = k + 1, and h ( r , , . . .,r,) = f ( r , , . . . ,r,)- ' . (4) If f ( X l , . ..,Xt), q(X,,...,X,)EW(X) then h = ( . f + q l ~ W ( X ) , level(h) = max{level(f), level(g)} I , and h ( r l , . . . ,r,) = , f ( r , , . . ., r , ) + 9('17 . rt). ( 5 ) Iff(X,, . . . X,), g ( X l , . . . , X , ) EW ( X ) ,then h = (,fg)E W ( X ) ,level(h) = max{level(f), level(g)} 1, and h ( r l , .. . , r,) = , f ( r l , .. ., r t ) g ( r l , .. . , I,).

+

'

. 9

.

+

Although this definition is written as if X were a countable set, we could take X of any cardinality. If X = [ X I ,...,X , ) , write W ( X , ,. . . , X c ) for W ( X ) . Until further notice we fix t and work in W(Xl,..., X , ) . The

48.2.1

Generalized Rational Identities

29 1

.sube.upressions of h are the expressions used in the inductive definition of h in the various steps of Definition 8.1.4. Remark 8.1.5. Suppose h ( X , . ..., X , ) E W ( X ,,..., X I ) and r l ,... , r , are elements of a W-ring R . If h ( r , ,. . . , r,) = '?,then for some subexpressionf ot' / I , f ( r , , .. . ,r , ) IS noninvertible in R. (Indeed, the only step producing additional undefined evaluations in R is Step 3 . )

The most important special case of Remark 8.1.5 is when R is a division ring, because then the only noninvertible element is 0. In general, we say a rational expression h is R-nonrlegriierare if h(r, ,. . . ,r,) is defined for some r l , ,..,r, in R . In other words, we exclude expressions such as (,f- I ) , where f is a GI of R. Definition 8.1.6. If h ( X , , . . . , X , ) EW ( X , ,..., X I ) , .Y = ( r , , ..., r , ) ~ R('), write h(.u) for h ( r , , .. ., r , ) ; h ( R ) = (h(s)l.u E R"'}. If h ( R ) L (0, ?}, then h is called a generalized rational identity of' R ; if, moreover, W is a field, then h is a rarional identity of R . Remark 8.1.7. If R c R , are W-rings and h~ W ( X ) , then h ( R )E h ( R , ) u as seen by induction on level; thus, every generalized rational identity of R , is a generalized rational identity of R.

{?I,

$8.2. Generalized Rational Identities of Division Rings

In this section we prove the famous theorem of Amitsur [Ma] that infinite-dimensional division algebras over infinite centers all satisfy the same rational identities. Our treatment follows Bergman [68P], proving a more general theorem for generalized rational identities. (Cohn [77B] has also presented Bergman's proof.) Throughout this section D is a division ring with infinite center F , and D is also a W-ring. We focus on D('),whose elements are called poinrs. If Y, = ( d , , , .. ., d,,) are points, 1 < i < 2, and ~ E F ,then Y , + a x 2 denotes ( d , , +aii,,,....d,,+crd,,), and the line determined by points x, and s2is {.Y, + ~ ( . Y ~ - . Y ~ ) ~ ~ E F } . Given a rational expression / ' ( X I , .. . X I ) and points .Y,, x 2 of D'", we can define $: F-. D u { ? } by $ ( a ) = f(.u, +a.u,). Now $ can be viewed as a rational expression built as in Definition 8.1.4, from the polynomial ring D[A]. In other words, define D ( A ) formally as follows: If $ED[A], then $ ~ D ( A ) a n d l e v e l ( $ ) =O;if$,,tI/,~D(A),then(-$,), ($;'),($,+$,),and ($ $ 2 ) E D ( I ) ,where the level increases by I . The nature of D(A)becomes much clearer after a brief digression concerning D[A].

.

292

[Ch. 8

RATIONAL IDENTITIES

Principal Left Ideal Domains and the Ore Condition Definition 8.2.1. A left ideal L of a ring R is principal if L has the form Rr for some element r of R . R is a prirtcipal lefr ideal ring if every left ideal is principal. A domain which is a principal left ideal ring will be called a PLID (principal left ideal domain). The theory of the PLID parallels the classical theory of the commutative PID, as we shall now see. Proposition 8.2.2.

D [ i ] is a PLID.

Proof. Suppose L is a left ideal of D[A], and f ( A ) is an element of L of minimal degree in A. For any element g ( A ) in L we have g(A)= q ( A ) , f ( A ) + r ( i )for some 4. r in D[A], where either r ( A ) = 0 or deg(r(A))< deg(f(1)). However, the latter is impossible by assumption on f ( A ) , since r ( i ) = g ( A ) - 4(A).J(A) E L. Thus r(A) = 0, so L = D[A] f.(A). QED Proposition 8.2.3. If R is a PLID, then R r , n Rr, # 0 f'or any nonzero elements r , ,r2 of R.

Proof. The assertion is trivial unless R r , $tRr,, in which case we write R r , + R r , = Rr, for some r 3 4 R r , . Then r 3 = a i r l +u,r2 for suitable a , . a2 in R with a 2 # 0. Now rl = a3r3 for some a 3 # 0 in R , so r , = u3r3 = a3a,r, + a 3 a 2 r 2 .implying 0 # a3a2r2= (1 - a 3 a 1 ) r ,E R r l n R r , . QED Corollary 8.2.4. !f R is a PLID, then R r , ~ " ' Rri n # 0 jbr any nonzero elements r , ,. . . ,ri of' R.

Induction

Proof.

on

i (using

Proposition

8.2.3 for the case

i = 2 ) . QED

A domain R is called left Ore if R r , n Rr, # 0 for all nonzero elements r , , 1, of R . It is well known that if R is a left Ore domain then R has a classical left ring of quotients R, which can be described as follows: The set R = ( ( R- {O) ) x R ) / - , where is the equivalence relation, defined by ( . x , , r , ) (.x2, r 2 ) iff there exist s;,. x i in R - ( 0 ) such that .x',.ul = .x;.u2 and .x',r, = s i r r . Writing x - l r for the equivalence class of ( . ~ , r )we , define respectively addition and multiplication of two arbitrary elements s ; ' r l and.(;'r,oCRasfollows:

-

-

Take .xi, x i in H - { 0 } such that x;xl = s i x z , and define . x ; ' r , +.u;'r, = (x', xl)- (.xi r , w;r2).

'

+

Take .Y' in R - - { O ) and r' in R such that x'r, ( x ; ' r l ) ( x i ' r 2 )= ( x ' s , ) - ' ( r ' r 2 ) .

= r'x,,

and define

We leave it to the reader to check that these operations are well defined and, under these operations, is a ring (with 1 = 1 - I 1 and 0 = 1 -lo). Also

88.2.1

Generalized Rational Identities

293

there is a canonical injection R -+ R, given by r -+ l - l r , under which we have the following important injection property: If $: R S is a ring homomorphism such that $ ( x ) - ' E S for all x in R - { 0 } , then Q induces a homomorphism $ : R -+ S , given by $ ( x - ' r ) = + ( x ) - ' $ ( r ) . Although these facts are not needed until the construction in Lemma 8.2.13, they motivate the whole development of this section. A domain R is called right Ore if r , R n r , R # 0 for all nonzero elements r l , r2 of R . -+

Remark 8.2.5. Every domain that is a PI-ring is left and right Ore. (Immediate from Theorem 1.7.9.)

The Division Ring Structure D on D(A) "Almost all" means "all but a finite number of." Lemma 8.2.6. Suppose $ is a rational expression built from D[1], acting on F , with $(a,,) definedfor some a. in F . Then: (i) there are p(1) # 0, q(1) in D[A] such that $(a) = p(a)-'q(cr) whenever both sides m e defined, which happensfor almost all a in F ; (ii) i f $ ( x 0 ) # 0, then $ ( a ) # Ofor almost all a in F. Proof. Induction on level($). Note that when $ is put in the form of (i), then (ii) follows by a Vandermonde argument. If $ED[A], then (i) is trivial ($1$2), or ( t j 1 ++b2) then, by (with p = 1). If $ has the form (-$'), ($;'), induction, there are p i , qi in D[1] with $i(a) = pi(a)-'qi(a) whenever both sides are defined, which is for almost all CL in F . The lemma is proved by taking p , q as follows: if $ = (-tbl), then p = -pl(A) and q(1) = q l ( l ) ; if tj = ( $ ; I ) , then $ ( m o ) # 0, so $ , ( a o ) # 0 and, by induction, $ I ( a )# 0 for almost all a in F. Thus q l ( a ) # 0 for these values of a, and we take P(J.1 = q , @ ) and =Pl(1). If $ = W1 + $ 2 ) , takef;.(l) in D[A] such that 0 #fl(l)pl(l) = f 2 ( 1 ) p 2 ( 4 , and define P(1) =fl@)pl(A) and 4(A) =fl(A)4l(n)+f2(1)42(n). If $ = take f ( A ) # 0, g ( A ) in o[A]such that f @)41@) = g(1)p2(1) and define p ( A ) = f ( A ) p , (A) and q(1) = g(1)q2(A). Q E D

Lemma 8.2.6 has a wide range of applications, to be obtained throughout this section. For example, calling an element rl/ of D(A) nontrivial if $ ( a ) is defined for some a in F , define an equivalence on the nontrivial elements by setting $ 1 4b2 iff = $ 2 ( a )E D for almost all a in F . By inspection, is indeed an equivalence; writing [$I for the equivalence class of ($), we define - [$I = [ - $1, [$I - = [$- '1, [$,I + [$21 = [ $ I + $21, and = Clearly these operations are well defined and give a

-

-

'

294

RATIONAL IDENTITIES

[Ch. 8

division ring structure, which we call D, to the set of equivalence classes of nontrivial elements of D(A). By Lemma 8.2.6 every element of D can be written in the form [ p ( A ) - ' q ( A ) ] for suitable p(A), q(A) in D [ A ] . We shall now make an observation that identifies D with the classical left ring of quotients of D [ A ] , motivating the proof of Proposition 8.2.9. Lemma 8.2.7. [ p l ( A ) - ' q l ( A ) ] = [ p 2 ( A ) - ' q 2 ( A ) ] in D' $for suitable elernetitsj;(A).f ; ( A ) in D[A] we haue.f;(A)p,(A) =,fz(A)pz(l.)# 0 uiid l ; ( A ) q l ( A ) =,f2(44,(4.

Proof. If .fl(A)Pl(4=.f;(A)pz(A) # 0 and fl(A)ql(A) =S2(i)q2(A) f 0, then for all a where defined p , ( a ) - ' q , ( a ) = p 2 ( a ) - ' q 2 ( a ) , implying [ P i ( A ) - 'qi(A)] = [ P 2 ( A ) - ' q z ( A ) ] . Conversely, if [pl(A)- ' q , ( A ) ] = [pz(A)- ' q 2 ( A ) ] , use Proposition 8.2.3 to obtain .1;(A), /;(A) in D [ A ] such that .fl (A)pl(A) = f 2 ( A ) p z ( A )# 0. Then . f l ( a ) q l ( a ) = f 2 ( r ) q 2 ( a ) for almost all a in F, implying by a Vandermonde argumentf;(A)q,(A) = ji(A)qz(A). QED Identifying D with the Division Ring of Laurent Series over D Recall the division ring of formal Laurent series D ( ( A ) )from 43.3, which can be viewed as the localization of the ring of formal power series D[[>.]] at the multiplicative set {A'lk 2 0). Definition 8.2.8.

Define a:D ( i ) + D ( ( A ) ) inductively (on level) as

follows: if $ E D [ A ] , then D ( $ ) = $; if $ = (-$'), then a($) = -a($'); if $ = ($;I), then a($) = C T ( $ ~ ) - ' ; if $ = ($, + $2), then 4$)= a($' 1 a($z); ( 5 ) if $ = then a($) = 4$1)4$z).

(1) (2) (3) (4)

+

Proposition 8.2.9. I f [$'I = [ $ 2 ] in D, then a($') sequently, a induces an injection 5:D -+ D ( ( A ) ) .

= o ( $ ~ ) .Con-

Proof. Suppose the first assertion is proved. Then a induces a homomorphism 5: D -,D((A)), and ker 5QD; thus ker 5 = 0, proving the proposition. Thus we need only prove the first assertion; in view of the fact that [$,I = [ p ( A ) - ' q ( A ) ] = [~,b~] for suitable p(A), q ( A ) in D [ A ] , it suffices to prove a($') = o(p(A)-' q ( i ) ) (because then, likewise, C T ( $= ~ )o ( p ( A ) - ' q ( L) ) ,implying a($') = a($z).)The straightforward proof, by induction on level($,), is left to the reader. QED

$8.2.1

Generalized Rational Identities

295

Remark 8.2.10. Any element p ( A ) of D[A] has the form d+q(i)A for suitable d in D and q(1) in D [ i ] . If d # 0, then the inverse of p ( A ) in D( (A)) is d - 1 + x g d - 1 ( -q(A)d- 1 y A i .

,

Lifting Generalized Rationalized Identities We are ready for an important special case of the first fundamental theorem. Recall Definition 2.1.14. Theorem 8.2.11. Suppose F = Z ( D ) is injinite, and (viewing W = D), D , is a D-division ring satisfking every GI of D. Then D , satisjies every generalized rational identity of D.

Proof. Consider a generalized rational identity h ( X , , . . . ,X , ) of D, and let xo be a point of D(') such that h ( x o ) = 0. We want to show that h(.u)E 10, ?} for all .x in DT), viewing D E D, canonically [such that Z ( D ) G Z ( D , ) ] . For each .Y in DT' define $, in D , ( A ) by $,(a) = h(.uo + a(.Y -.yo)), and consider the Laurent series a($,) as in Definition 8.2.8. Note that $,(O) = h(.uo) = 0. Since h ( x , ) is defined, we have f ( x o ) ~ D - { O }for each subexpression (f-')of h. Hence, in view of Remark 8.2.10, we see that in building up the formal Laurent series of o($J from D[A], the only inverting comes from elements of D that do not depend on the choice of x,and g($J is a power series whose coefficients (of A) are generalized polynomials (with coefficients in D ) evaluated at x; i.e., for .Y = (x,, ...,.Y'), a($x) has the form d + xg ,.fi(.u,, . . . ,.u,)A', where,/; E D{ X 1 , . . .,X , } and d = h ( x o ) = 0. Suppose that XED(').Then, for all a in F , .u,+a(.u-x,)~D, so $,(a) E h ( D )E (0, ?}. By Lemma 8.2.6 $,(a) = 0 for almost all a in F . Thus writing $,(A) = p ( A ) - 'q(A), we must haveq(A) = 0, implying a($J = 0, so each .f](.Y) = 0. Thus each ,fi is a GI of D, implying by hypothesis each fi is a GI of D , . Therefore o($,) = 0 for all x in DY', implying by Proposition 8.2.9 and Lemma 8.2.6 that $,(a) = 0 whenever defined. It follows immediately that h is a generalized rational identity of D,. QED Corollary 8.2.12. I f Z ( D ) is i n ~ n i t eand [ D : Z ( D ) ]= co, then every generalized rational identity of D is a generalized rational identity of each Ddivision ring D .

Proof. Every GI of D is a GI of D' by Corollary 7.2.14, so we are done by Theorem 8.2.11. QED

296

[Ch. 8

RATIONAL IDENTITIES

A Change of Division Rings In order to utilize these results to their fullest, we resort to an idea from Amitsur [66a]. Lemma 8.2.13. There is a general construction taking an arbitrary division ring D into a D-division ring D' such that Z(D') is infinite and [D' :Z(D')] = XI ; i f D is a Do-ring, then D' i s a Db-ring. Proof. Inject D into the division ring T = D ( ( A ) ) ,in order to make the center infinite. Note that T has the endomorphism (T defined by c r ( ~ ~ , d J=' )xz,,,diA2i (for rn in Z, di in D ) , and A$rr(T).Now we define T' = T [ p ;(TI, the skew polynomial ring of T (in the indeterminate p) with respect to (T, discussed in $3.3. Recall that the elements of T' are written in the form ~ ~ = o a i ai p ' in , T, and multiplication is given by the rule pa = cr(a),u for all ti in T. Hence there is the natural degree function on T', given in Remark 3.3.3, which shows that T' is a domain. We shall prove T' is a PLID. Indeed, any nonzero left ideal L has an element f = Cf= aipi of minimal degree k ; since T is a division ring, we may multiply on the left by a;', and assume ak = 1 . We claim that L = T'$ Otherwise, take g = alpi in L - Tlf, having minimal degree j. By assumption, j 2 k , but then deg(g - a ) p j - r f ' ) :. j - I , contrary to assumption. Thus L = T'f'afterall, and T' is a PLID. Now we prove T' is not right Ore. Indeed, we claim p T ' n APT' = 0. Indeed, on the contrary, assume pj'= ;.pg for some n0nzero.A 9 in T'. Then afpi, we have pakick deg(f) = deg(g); writing f = x f = o a i p iand g = = Ipa;pk [by taking the terms of degree (k + 1) in p j = Apg], implying a(ak) = Ao(a;), so A = a(aka;- I). which is absurd. Now let D' be the classical left quotient ring of T'. If D' had finite degree then D' would be a PI-ring, so T' would be a PI-ring, contrary to Remark 8.2.5. Thus [D': Z ( D ' ) ] = ,x.It is easy to see that D is a Db-ring, as well as a D-division ring, bq checking the construction at each step; details are left to the reader. QED

xf=o

Incidentally, the mere existence of left Ore domains that are not right Ore, constructed above, points out the wide difference between PI-theory, which so often uses the center, and general ring theory, in which the center plays a very small role. The First Fundamental Theorem

Definition 8.2.14. Two Do-rings are rationally equivalent if they satisfy the same set of generalized rational identities (with coefficients in DO).

88.2.1

Generalized Rational Identities

297

Theorem 8.2.15 (First fundamental theorem: Amitsur [66a] ; Bergman [68P]). I f D , is an arbitrary division ring, then all D,-dioision rings that are injinite dimensional oiler an illfinite center are rationalljl equivalent. More generally, if D , and D , are D,-dirision rings with Z ( D , ) injnite and [ D , :Z(D,)] = co, then D , sarisfies ewry generalized rational identitj. of D , (with coeficients in Do). Proof. Using the notation of Lemma 8.2.13, we have 0; is a D,-division ring, and D', and 0; are Db-division rings, and all of these division rings have infinite dimensions over infinite centers. Thus, by Corollary 8.2.12 D , is rationally equivalent to D',,which is rationally equivalent to Db, which is rationally equivalent to 0;. Since D , c D ; , we conclude that every generalized rational identity of D , is a generalized rational identity of D,. QED

Corollary 8.2.16. Let Do he u giuen division ring, and for any Dodivision ring D, let . Y ( D ) = {generalized rational identities of D (with coeficients in D o ) ! . I f Z(D,) is irlJjnite and [Dl :Z(D,)] = d) .for u Dodivision rihg D , , then . Y ( D , ) = n{Y'(D)I[D:Z(D)] < 35'3. Proof. Clearly Y ' ( D , ) G (7{.'/(D)I[D:Z(D)] < co) by the first fundamental theorem. But if deg(D) = n < co, then for each k by Corollary 3.1.47 we can find an extension D , of D with k < deg(Dk)< 00. Taking the ultraproduct D = Dk)/ F,where .9is an ultrafilter of E+ containing the Frechet filter, we see that D satisfies every elementary sentence which is satisfied by all but a finite number of D, (cf. $7.5). In particular, .Y(D)2 n , . v ( D k ) . Each D, satisfies (Yx)((.u = 0) v (3y)(.uy = y.u = l)), so D satisfies this sentence and is a division ring. Moreover, by Wedderburn's theorem, for each D,,Z ( D , ) is infinite and thus, for every n, D, satisfies the sentence

(nkEH+

L,

= (3.Ul)".(3X,)(VX)(([Xl,.U] A (XI

#

X,)

A (X2

#

X3) A

=

0) A

" '

" '

A (.V2

A ([.Y,,.Y]

#

X,)

A

= 0)

A (,Y1

#

.Yz) A

" '

"').

Hence D satisfies every sentence L,. But L, says that Z(D) has at least n distinct elements. Since this is true for all n, Z ( D ) is infinite. Also, for each m and for all k > m ((3xl)~-~(3x2,)(S,,(x,, ...,.u), # 0 ) holds .for Dk, implying S,, is not an identity of D for each m, so [ D : Z(D)] = a. Therefore .Y(D1)= . Y ( D ) 2 n k . ( / ( D k 2 ) r ) { . v ' ( D ) ( [ D : Z ( D ) ]< a], thus proving.V'(D,) = n { - ' f ( D ) l [ D : Z ( D ) < ] a).QED The "Generic" Division Ring W o ( X ) Proposition 8.2.17. Suppose L is L( line of D"' with a g i r w point so, and b c W ( X , ,. . . , X , ) with h ( y 0 ) defined (in D). Then h ( r ) is defined for

298

RATIONAL IDENTITIES

[Ch. 8

almost all x i n L ; moreover, i f h ( x o )# 0 then h(x)ED- ( 0 )for almost all x in L. =

Proof. Write L = { s 0 + a ( s l- . x , ) l a ~ F } for some .xl in L. define $ ( a ) h(xo+ a ( x , - x o ) I, and apply Lemma 8.2.6. QED

Proposition 8.2.17 says that domains of rational expressions are large enough to be manageable ; for a thorough discussion of topologies arising from Proposition 8.2.17, see Exercise S and Bergman [68P]. Corollary 8.2.18. If j; g are D-nondegenerate rational expressions, then there is somt point x ofD(" such that f ( x )E D and y(x) E D. QED Proof. Supposef(x,)ED and ~ ( x , ) ED. Letting L be the line connecting s1and .Y,. we are done by Proposition 8.2.17. QED

Let X be an arbitrary set of indeterminates. Using Corollary 8.2.18, we can define an equikalence on the D-nondegenerate rational expressions of W ( X ) ,by saying,f g iff ( f +(- 9 ) ) is a generalized rational identity of D.

-

Remark 8.2.19. Any finite number of D-nondegenerate rational expressions f l , . . . ,j; have a common point of definition. [Indeed, by Corollary 8.2.18, one sees inductively that (fl + ( f , + ( . . . +1;)...)))is D-nondegenerate.]

If.&,g i are D-nondegenerate rational expressions and Remark 8.2.20. gi, i = 1-2, then ( f 1 + f z ) ( g l +gdr and (fif2) -c ( y l g d ; if, moreover, .f, # 0, then (,f;) ( 9 ; ') are D-nondegenerate.

.&

-

'

-

-

Define W,(X) = {[f]If is a D-nondegenerate rational expression of W ( X ) ] where , [ ] denotes the equivalence class under -. Proposition 8.2.21. Suppose Z(D) is infinite. Then under rhe oprrations [.fl] +[.f21= [(Il +f2)] and [fl][f2] = [(fl.fi)], W,(X) is a dir.isim ring.

Proof.

Immediate from Remark 8.2.20. QED

Theorem 8.2.22. Suppose W is a divisiori ring and D , , D, are Wdivision rings of infinite dimensions over injinite centers. Then W,, ( X ) 5 WD2(X),under the isomorphism [.fll + [f],, where [ Ii denotes the equivalence class in LVDi(Xl,.. .,X , ) . Proof.

All we need to show is that the given map is well defined, i.e., if QED

[.fIl = 0, then [fI2 = 0. But this is the first fundamental theorem.

Thus, we see that for every division ring Do, there is an associated division ring ( D o ) D ( . Y l , . . ., X , ) , which is the same for every D,-division ring

$8.3.1

Rational Identities of Division Rings

299

D infinite dimensional over an infinite center. Thus we have a very interesting rational analogue to the relatively free PI-ring.

$8.3.Rational Identities of Division Rings of Finite Degree In this section we restrict ourselves to rational identities, i.e., D, (used in the fundamental theorem) is a field F . Let us start with one interesting rational identity. Example 8.3.1. Let D be an arbitrary division algebra of degree 4. In Theorem 3.2.28 we saw that

. f ' ( X ,X , ,,X,)

=

[[XI,

x,]2,x3][[X1,X,]rX31-1

had the property that for every c l , , d z , d , of D,.f'(d,,d , , (I3)' was ofdegree 1 or 2 overZ(D).Thus by Theorem 1.4.34,C,( I,,/ (t1,,d2, d,)',,/ (d,, d,, d3)', X,, X , ) is a generalized identity of D,so we have a rational identity of D, written a bit X , , Xj)', X,, X,).(HereC,isaCapelli sloppilyasC,( I,,f'(X,,X,, X3)2,/'(X1, polynomial.) Remark 8.3.2. C , ( l , f ( X , , X , , X , ) , , . f ( X 1 , X2,X3)4,X4,X,) is not a rational identity of the generic division ring D = Z,(Y). [Indeed, it is easy to see even thatf(Yl, Y,, Y , ) # Z ( D ) ,by specializing Yl, Y, respectively to the matrices y , = (e,,+2e,,+3e3,) and y , = ( e , , + e 2 , + e 3 , ) . But then /'( Y,, Y,, Y,) has degree 3 over Z(D), implying ,f(Y , , Y,, Y3), has degree 3 Over Z(D), so C,(L.f(Yl, Yz, Y312, , f ( Y l ,Y,, Y3l4, Y,, Y5)z 0, again by Theorem 1.4.34.1 Thus we have a rational identity for all division algebras of degree 4, which is not a rational identity for any division algebra of degree 3. This is quite different from the situation for polynomial identities. O n the other hand, the rational expression of Example 8.3.1 is a rational identity of all division algebras of degree 2, because [[X,, X,],, X,] is already an identity. This situation motivates a beautiful theorem of Bergman [76], built on Amitsur [66a] and Bergman-Small [75], which completely classifies the interrelation of rational identities among different finite dimensional division algebras, namely that D, satisfies all rational identities of D , iff deg(D,)Jdeg(D,). The proof would also work for generalized rational identities, and readers are invited to check this at their leisure. Rational Equivalence of Simple Algebras of the Same Degree

To initiate the study of rational identities of division rings of finite degree, we consider F,(X) with F a field c- Z(D).

300

RATIONAL IDENTITIES

[Ch. 8

Theorem 8.3.3. lj' deg(D) = n, then F , ( X ) is canonically isomorphic to F,( Y ) , the algebra of central quotients of the algebra of generic n x n matrices.

Proof. Z ( D ) is infinite by Wedderburn's theorem. We claim that for each D-nondegenerate rational expression h ( X , ,. . . ,X,) there are , f F,( ~ Y,, . . ., k;} and g E Z ( F , { Y,, . . ., k;}) - (0) such that h( Y , , . . ., Y,) =,fg-'. We proceed by induction on level. If h E F { X J ,takef= h(Y,, ..., Y,) and g = 1 . If It = ( - h , ) , h = ( h ; ' ) , h = ( h , +A,), or h = (h,h,), take ~,EF,{Y ,..., ~ and g i E Z ( F , { Y l ,..., Y,})-{0} such that hi(Y, ,..., Y,) =Lg;', i = 1,2, and put respectively .f= -fl and g = g,, j ' = g, and 9 =.1;,.f =.f19,+.f,91 and 9 = 919ZI o r f = . f J z and 9 = 9192. Thus the first claim is proved, and we define a map a: F , ( X ) -+ F,( Y ) by a(h) =,fg-'. withfand g as in the claim. Now we claim that if h is a rational identity of D, then a(h) = 0. Writing o(h)=fg-', we see that gcr(h) =J; implying easily that ,f(.uo) = 0 for every xo ED(" on which S, g, h are all defined. But then, by Proposition 8.2.17, for each line L passing through .yo, ( ( g ( X , ,. . . ,X , ) h ) + t -f(X I , . . ., X,))) is a rational expression defined for almost all points in L, implying .f(.u) = 0 for almost all x in L. By a Vandermonde argument , f ( L )= 0 for every line L passing through xo, implying f ( D ( , ) )= 0, i.e., f ( X , , ..., X , ) is an identity of D, so f ( Y , ,..., Y,) = 0, proving the second claim. Thus a induces a homomorphism F , ( X ) + F , ( Y ) , which are division rings, so F,(X) a F,( Y ) . QED

x}

Corollary 8.3.4. A n y two division F-algebras of the same degree are rationally equioalenr (over F ) .

We can extend Theorem 8.3.3 with little difficulty. Theorem 8.3.5. For every commutative domain C 2 F ( ( ) and every division F-algebra D of degree n, D is rationally equivalent to M,(C).

Proof. By Corollary 8.3.4 we may assume D = F , ( Y ) . Then c M n ( F ( < ) )E M,,(C), so it suffices to prove every rational identity J ' ( X , , . . .,X , ) of D is also a rational identity of M,(C). Passing to the field of fractions, we may assume C is a field. Now take xl,. . .,x, in M,(C) arbitrary. There are generic matrices XI,. . . , yi, whose entries d o not appear in xl, . . .,x,. Clearly the entries of ( yiL .uI), . . . , ( yi, + x,) are algebraically independent over F;specializingeach q + yij+sjshowsf'(YI., +x, ,..., y . c + X , ) = O .But now by an argument paralleling Lemma 8.2.6 one coricludesj'(.u,, . . . , x t ) = 0. QED D

+

48.3.1

Rational Identities of Division Rings

301

The Second Fundamental Theorem

Theorem 8.3.5 and the first fundamental theorem are sufficient for most applications, but to complete the theory, we present first an important theorem of Bergman, which depends on the Bergman-Small theorems. Bergman's theorem will be presented in two parts, to pinpoint exactly where the difficult Bergman-Small theorem enters. (Second fundamental theorem, part I : Amitsur [66a] ; Theorem 8.3.6 Bergman 1761). If Dl and D, are division F-algebras with deg(D,)ldeg(D,), then D, satisjes all rational identities of D,.

Proof. Suppose deg(D,) = m and deg(D,) = mk for suitable integers m and k . By Theorem 8.3.5 D , is rationally equivalent to M,(F[(]), and D , is rationally equivalent to M m k ( F [ 9 ] ) .But the mapping taking an m x m matrix ,4 into k copies of A along the diagonal induces an injection from M,(F[<]) into M , , ( F [ < ] ) ; thus M , ( F [ < ] ) satisfies all rational identities of M,,(F[<]), so D, satisfies all rational identities of D,. QED We now want to get the converse of this result in the strongest possible form. Remark 8.3.7. If D = Q , , ( Y ) then Q J X ) equals its subring Z , ( X ) . [Indeed, every integer is inuerted ~ i aDefinition 8.1.4(3), so the assertion is ohv ioU S .] Theorem 8.3.8 (Second fundamental theorem, part I1 : Bergman [76]). Zf D l and D , are division rings o f j n i t e degree and deg(D,)!deg(D,), then D, has a rational identity with coe@cients in Z that is not a rational identity o f D , . Proof. We prove the contrapositive. Let m = deg(D,) and n = deg(D,), and suppose every rational identity of D , (with integral coefficients) is a rational identity of D,. Then, let F be the subfield of D , generated by 1, and let D = F , ( Y ) and D' = Q,,(Y). Identifying D with F , ( X ) and D' with Q,.(X) = Z,.(X) by use of Theorem 8.3.3, we define a map $: D' + D u { ?) by sending the equivalence class in Z D . ( X ) of a rational expression (with integral coefficients) t o the corresponding equivalence class in F,(X). Our supposition implies $ is well defined. Letting R = $-'(D) c D', we restrict $ to R, obtaining a ring homomorphism :R + D. If r E R - k er $ ,, then $(r)-'ED, so r-'€$-'(D), implying r -'ER. Hence, for all r in ker$, we have (1 -r)-' E R, so ker$, s Jac(R). But R/ker$, = D is simple, so ker$, is a maximal ideal of R, proving ker$, = Jac(R) is the unique maximal ideal of R. We now need the

302

RATIONAL IDENTITIES

[Ch. 8

Bergman-Small theorems. By Theorem 1.10.69 PI-deg(R)Jn. By the difficult Theorem 1.10.70,mJPI-deg(R).Hence mln. QED Bergman [76. $85, 61 contains some explicit theorems concerning existence of rational identities for (division algebras of) degree m that are not rational identities for suitable degrees ,< m. The reader should note that the PI-theorem corresponding to the second fundamental theorem is a triviality, and conclude that the theory of rational identities is just at its beginning. It promises to be very difficult, but could provide much insight into the workings of central division algebras. The current body of knowledge is essentially given in Cohn [77B, Chapter 71.

58.4. Applications of the Theory of Rational Identities

Amitsur [66a] gave three main applications of his theory of rational identities. One of them, the construction of the division rings F,(X).was discussed already in $8.2 and $8.3, so we shall focus on the other two applications, namely, to the multiplicative group of division algebras and to Desarguian projective geometry. The first application is rather straightforward, and so we start with it; the second application, as remarked earlier, was a major achievement in geometry. Group Identities

We commence with the,fieeg r o u p Y(X)on a set X = { X l ,X , , . . .}, which Here the elements are strings of X iand is defined very much like .//(X). formal elements Xi.-’,with the understanding that we shall reduce a word whenever possible hy replacing each occurrence of XiX;’or X ; ’ X i by the symbol 1,foreveryi b 1,andreplacinghlor 1hbyh.Multiplicationisdefinedon 9(X)by juxtaposition and reduction. Using the free group, one can define a theory of identities of groups in much the same manner as we did for rings; in-fact, this was done some time ago by H. Neumann [67B] (although the point of view is quite different). z , and say an element g of %(XI is a g r o u p Namely, take X = ( X l , X ...) identity of a group G -if g is in the kernel of every group homomorphism 9(X)--+ G. Note that “1” is the only group identity of % ( X ) , and is called the “trivial” group identity. If K is a field, write G L ( n , K ) for the set of elements of M , ( K ) having determinant # 0: clearly GL(n, K ) is a group. Note that as groups there is a canonical homomorphism G L ( n ,K ) -+ G L ( m ,K ) for m > iz, sending X;.j= aijeij-,(Zf.j=aijeij+E7=,,+eij).

68.4.1

Theory of Rational Identities

303

Lemma 8.4.1. I f F is an irlJiriitefield and A,, A,, arid 1, are commuting indetermiriates over F , therijor each 17 2 2 the free group on a countable set X = { X , , X , , . . .} can be identijied with a subgroQp qf'GL(ri,F(Al,A,, A 3 ) ) .

Proof. Without loss of generality, we may assume n = 2. Identify Y ( X ) with a subgroup H of G = 9 { X l ,X , } , by the homomorphism taking X i + X l X , X : . Thus, every element of H has the form X F X * X t IX* .. . X y for suitable k and suitable nonzero integers i, j,, . . .,i , . In fact, it is clear that the set of all elements of this form is a subgroup H , zH ; we shall find a homomorphism a: G -+ GL(2, F(Al,A,, 1,)) such that (ker a) n H , = 1. In fact, the homomorphism is very easy; just send ~

and extend this map to a group homomorphism a: G Note that o(X;')

= A;'(

"

")

-1, 1

and

o(X;')

--f

=

I

~

GL(2, F(A,,

12, A3)).

-;'),

so, letting K = F(A3),we have a(G) zM,(K[A,, A,]). Thus, given g in G, we can view a(g) as a matrix gijeij,where gijE K[1,, A,]; we shall write g i j for the sum of monomials of g i j of largest degree in ,Il and A,. For example, for all i > 0, a ( X l ) = ( e l +a,;l,e,, +13e,,) and . ( X i ) = ( e l +aiA2e12+ l i e , , ) , where a; = # 0; also, for all i < 0, where 4 X f )= (ell +ai11e,, +&e,,) and o ( X i ) = (ell ai = - ~ , ~ ~ , A ;#J 0. Thus, for all nonzero integers i,j.

x:j=

,

,

xiiiA\

o ( X ' , X j , )= ( e l

,+ a i A , e 2 ,+ ajA2e,,+ (A?'+

ctiaj1,1,)e2,);

setting g = X i x i , we get q l l = 1, g, = aiAl, q I 2 = a j l , , and g2, = a,aj1,1,. We claim that hzl # 0 for all h # 1 in H , ; in fact, we shall show inductively (on k) that if h = X F X $ . . - X v , then h l l = P1(11A2)k-1 and &, = p21t1:-' for suitable nonzero /I1,/I2 (in K ) depending on h. The claim is true f o r k = 1, so we write i = i,, j = jk, h' = X F - I X t - ' . . . X i , l ,h = X ; X ( h ' , and a(h') = x h i j e i j ; by induction hypothesis h',, = /?11t-2A:-2 and hil = P21:-'Ak,-2 for suitable PI, p2 in K . Since o(h) = a(X',X:)cr(h'), we see h , , = h ' , l + a j l , h ; , and h,, = u i A 1 / l ~ , ~ + ( l ~ J + a i a j l , l , )thus h~,; h l l = aj12fi21:-'A$-2 = (ajp2);lt-1A$-1and h,, = a i a j 1 1 1 2 p 2 A ~ - ' A = ~-2 ( a i u j p , ) l t l i - proving the claim. But now for all h # 1 in H , , h,, # 0, so a ( h ) is not even diagonal; hence H , n (ker a ) = 1, as desired. QED

',

The multiplicative group of a division algebra is its group of nonzero elements (under multiplication).

304

RATIONAL IDENTITIES

[Ch. 8

Theorem 8.4.2. I f D is any noncommutatiiie diuision algebra with injnite center, the multiplicative group of D does not satisfis m y nontrivial group identity. Proof. Suppose g ( X , , . . .,X,) is a group identity of D - { O j , and let j ’ = X , ...X,(g- 1). Obviously ,f is a rational identity of D, and thus by Corollary 8.2.16, of some division PI-algebra D,.But if [D,:Z(Dl)] = n, then f is a rational identity of M,(F(()) [where F = Z ( D , ) ] by Theorem 8.3.5, implying by Lemma 8.4.1 that g is a group identity of % ( X ) ; therefore g = 1 . QED

Corollary 8.4.3 (Hua). The multiplicative group oj’ a noncommutative diuisiori ring with ir$finite center is not solvable.

A cute consequence of Theorem 8.4.2 is given in Exercise 3 for PI-rings. Also, the group algebra of %(XI can be embedded in a suitable division ring (cf. Exercise 4). Desarg u ian Geometry

Now we turn to Amitsur’s pioneering solution to Dehn’s problem of characterizing Desarguian geometries. Let us start with a brief description of Desarguian (projective) planes. Let D be an arbitrary division algebra and let V = LY3), the 3-dimensional vector space over D with basis {(1,0,0), (0,l,O), (O,O, 1)). We define the Desurgztian plane P to be a collection of points and lines, where “point” (of P) means 1-dimensional D-subspace of L’ and “line” (of V ) means 2-dimensional subspace of 1’. D is called the underlying division ring of If A , , A, are distinct points, then A A 2 span a 2-dimensional subspace of V. i.e., a line of P, which is denoted AlA2;ifLland L,aredistinctlinesthendim(L, r L 2 ) = (dimL, +dimL,)dim(L,+L,) = 2 + 2 - 3 = I , so L , r , L , is a point. So we see that each pair of points spans exactly one line, and any two lines intersect at exactly one point. To satisfy our geometric intuition, we write ,4 E L when the 1-dimensional space A is contained in the 2-dimensional space I-, and we say A lies on L. The usual treatment of projective geometry is to define the projective plane axiomatically and to prove, under the assumption of the Desarguian property (defined below), that one can introduce coordinates as we have done (with respect to a suitable underlying division ring)-cf. Artin [57B]; as long as we are only looking at Desarguian planes, our definition is much more straightforward (for algebraists) and is certainly more direct for our present purposes. The way we have written the coordinates, we make some canonical choices

v.

88.4.1

305

Theory of Rational Identities

in representing points. Suppose we are given a point A , = D ( a , . u,, u j ) for ( u l , a 2 , a 3 V. ) ~If u3 # 0, rewrite 4 , = D(u~;'u,,a;'u,, I ) and call A , jriite. If u3 = 0, then A , = D(u,.u,, 0) and A , is called irfiriite; if a, # 0, rewrite A , = D ( a ; ' a , , I , ( ) ) . and otherwise, A , = D(l,O,O). Note that the set of infinite points is a line, called the lirie ar irfiriify. The point D(O,O, I ) is called the origiri. I n one sense, we can "choose" the line of infinity and the origin, as well as another point. by the following crucial result : Definition 8.4.4. A set of four points is trdriiissihle if there is no subset of three collinear points.

Proposition 8.4.5. [ f ' ~ A , . A , . A , , A4j uriil :A',, 4. A ; . A ; UP4 udr~ii.s.sih/e,rheti there is an i r i i w t ihlr lirieur trari.~fbrrmtiorip of' V such [hut p ( A i ) = A ; , 1 < i 6 4. Thus, ric.wd g e o r i i ~ t r i c u / l ~p * takes , /iric..\ to lines urirl poirits fo poirits. Proof. Let ,4; = D(1,O. 0). A; = D(0, I , 0). A: = D(O,O, l ) , and AI; = D(1, I , I ) . We shall find p , such that p , ( A : ) = ,4i, 1 d i < 4. This will prove the first assertion because if then we take p , such that p,(Ay) = A ; , I < i < 4, we conclude with p = p ; I p , . Write A i = D(ui,,aiz,ui,) for a i j in D. Since A , , A , , and A , are not collinear they span V, so u , ~= diuij for suitable 0, in D, I < , j 6 3. Then define p , by pI(sl, s,,s,) = ~ ~ d , a, , , .\-,diui,, sidiui3). p , is clearly an invertible linear transformation taking A)' to A ; , 1 6 i < 4.

x:= , (z:= ,

xf=,

x:=,

Now the second assertion is immediate.

QED This is a trivial part of the fundamental theorem of projective geometry, but is all we need here. Incidentally. the deeper part is proved most easily using the rational identity of Hua (cf. Artin [57B]). As an example of these methods, we can easily prove the Desarguian property (for Desarguian planes).

,,

A,, A , (resp. Proposition 8 . 4 . 6 (Desargue's theorem). Suppose AB,,B,, B 3 ) ure noricol[inear arid ull distiricr, md the lines { A,B,l I < i < 3 ) hrc. coricurrerit (i.e.,their inrersectiori is u point). Takirig subscripts modulo 3, dqfiriePi= A i . , l A i + 2 n B i . + , B i . ,I, ,< i d 3 : t h e r i P , , P , , P , u r e c o l l i r i r a r .

Proof. In view of Proposition 8.4.5, we may assume P I = D(l,O,O), P , = D(0, l , O ) , and A , B , n A , B , n A , B , = D(O,O, I ) . A i = D ( u j , , a i 2 1, ) and B, = D ( b i l ,hi,, l), 1 6 i d 3. Then one immediately has bij = diaij, 1 < i d 3, 1 d j d 2, for suitable d , E D. Since A,, A,, P I (resp. B,, B,, P ,) are collinear, we get a,, = a,, (resp. d,a,, = b,, = b,, = d , a , , ) ; hence d , = d,. Likewise d , = d3. Therefore d , = d , , implying P , = D ( a , , - u 2 , , u 1 2 - u 2 2 , 0 ) , which is o n thelineat infinity. Q E D ~

-

_

_

_

306

[Ch. 8

RATIONAL IDENTITIES

Here is another property which sometimes holds in V . Definition 8.4.7. V satisfies the Pappian property if, whenever A,, A,, A , (resp. B , , B,, B 3 ) are collinear points, then the points _ _ _ _ -~ P , = A 2 B 3 n A3B2, P , = A,B, n A,B,, and P , = A , B , n A,B, are collinear. ~

~

Proposition 8.4.8. A Desarguian plane satisfies the Pappian property $the underlying division algebra is commutative. __

__

Using Proposition 8.4.5, we may assume ,4, A 2 t?BIB, = D(O,O, I), P , = D(l,O, O), P = D(O, I , O), and .4 I = D(1, I , I ) . Write B , = D ( b , , h , , I). I n view of the collinearity of the A i , we have d z , d , in D such that Ai = D ( d i , d i ,I ) , i = 2 , 3 ; similarly Bi= D(djb,, djb,. I ) for suitable dj in D. Since P,, A , , B, (resp. P,, A,, B , ) are collinear, we get 1 = d;b, (resp. d , = b,), so b , = d 3 = (a,)-’.Likewise b , = d , = (d\)-’. It follows easily that P , is on the line at infinity iff D ( d , - d ? , ’ d , , d , - 1 , O ) = D ( d 3 - 1, d,-d;’d,,O), i.e., D(d,d,-dz,d,dz-d,,O) = D(d,d3-d,,d,d,-d,,,O), which certainly holds if D is commutative, but fails if d, and d3 do not commute, as is eahy to see. QED Proof.

,

v

Thus, we have a geometric property of which is equivalent to an algebraic property of D, namely, the polynomial identity [ X I .X , ] , i.e., deg(D) = 1. Intersection Theorems

We wish to generalize this principle, but first we have to be more specific about which geometric properties we wish to characterize algebraically. Definition 8.4.9. An incidence relation is a sentence “ A E L ” for a suitable point A and line L ; a nonincidence relation is a sentence “ A e L . ” Y ,.f) of a Desarguian plane A configuration (d, Definition 8.4.10. is a finite set .d of points, a finite set Y of lines, and a finite set .B of incidence and nonincidence relations involving points in .ci/ and lines in Y’. A morphism ofconfigurations (,dl, Y , ,Y , ) (..d,, Y 2 Y,) , is a pair (a,T) of set-theoretic maps a: .dl ,cr(, and T: Y , -+ Y2 such that if “ A EL” (resp. “ A 4 L ” ) is in X I , then “ ~ ( A ) E T ( L(resp. ) ” “a(A)$r(L)”)is in 4,. (,d,, Y,,.f,) and (,d2, Y,,./,) are isomorphic. if there exist morphisms ( 0 , T ) : ( d l , Y’I, .g,) ( . d 2 . 2 ’ 2 , . f 2 ) . (Of, T ‘ ) : ( . d z , 2 ’ 2 , 9,) ( d ~ .F ~ ) such that 04,d o . TT’, and T’T all are the identity mappings; in this case (a, T ) is called an i.\ornorphistn. Actually, we are only interested in those configurations that can be constructed in the following sense: -+

-+

-+

-+

y

1

9

98.4.1

307

Theory of Rational Identities

(@,a,

Definition 8.4.11. The configuration 0) is constructible. By and 9,call induction on the sum of the number of elements in &, 9, (.&, 9,9) constructible if one of the following conditions holds:

(i) for suitable A in d,( . d - { A } , Y , Y )is constructible. (A point has been added to a constructible configuration); (ii) for suitable A in d,L , # L , in Y’, we have { “ A E L,,” “ A E L , ” ) G .f, and ( c & - { A } , Y , ~ - { “ A E L ~ , ” “ A E L , ”is} constructible ) (the intersection of two lines has been labelled as a new point); L in Y’, we have { “ A , EL,”“ A , EL”}E .f, (iii) for suitable A , # A , in .d, and (cd, Y - { L } ,4 - { “ A , EL,” “ A , E L ” } )is constructible (the line spanned by two points has been drawn); (iv) for some A in d, L in Y , we have ( & , Y , Y - { A $ L } ) constructible (nondegeneracy) ; Note that to assume two points A , , A , are distinct, we can draw some line L through A , [using ($1 that misses A , , and add “ A , 4 L.” Also, every configuration isomorphic to a constructible configuration is constructible. Definition 8.4.12. An intersection theorem INT(L,, L,, L , ; % ) is a Y ,3 )is sentence “ L , n L , n L , # 0” that holds for Li in Y ,where % = (Ld, a constructible configuration. An intersection theorem INT(L,, L,, L , ;% ) is a universal intersection theorem (of V ) if, for every configuration W isomorphic to %? and for every isomorphism ( o , r ) : %-+%’ we have T ( L , )n T(L,)n T ( L J # 0.

Let us now show that each rational identity of D corresponds to a universal intersection theorem in Y . To d o this, we need to find geometric interpretations of the algebraic operations +, *, -, - i.e., “plus,” “times,” “additive inverse,” and “multiplicative inverse.” To begin, we fix four points: A , = D(0, 0, 1) (the “origin”), A , = D(1,0,1) (to give the “unit interval”),

’,

~

Ab, = D(1, l,O), and A: = D(0, 1,O). Let L , = A , A , (the “ X axis”) and L , = Ab, A: (the “line at infinity”). Let A , = L , n L, = D(1,0, 0), and on L , nor L,. Also define the points pick any point A riot lying __ __ I = L , n A , A , I ’ = L , n A , A , and the lines L = A , A and L ‘ = A , A . Now A = D(a,,a,, 1 ) for suitable elements a,, a, of D, implying I = D ( a , , a,, 0) and I’ = D ( a , - I , a,, 0). Suppose we are given two __ arbitrary points B , = D(b,,O, 1) and B , = D(b,, 0 , l ) on L , . Let B , = B , A n L , = D ( a , - b,, a,, 0). We are ready to construct “-B,,” “B; B , + B,,” and “El . B,,” respectively. ~

~

’,”

“

~-

Construction 8.4.13. Let B = A , B , n L and “ - B , ” [Indeed, B = D(a, - b,, a,, l), and so “-B,” = D(- b,, 0, l).]

=G

nL,.

308

[Ch. 8

RATIONAL IDENTITIES ~

Construction 8.4.14. Let B’ = A B , n L‘ and ‘‘By I” = I’B’ n L , . [Indeed, B’ = D ( a , , a 2 ,b 2 ) ,so “B; I ” = D( 1,0, b,) = D(b; I , 0, 1 ).] ~

Construction 8.4.15. Let B , = B,I n L, ”B,+B,” = L, n B,B,. [Indeed, B , = D ( a , +b,, a,, 1 ) and “B, +B,” = D ( h , + b z , 0, l).] ~

Construction 8.4.16. Let B; = B , I’ n L‘, “ B , . B,” [lndeed,B; = D(b,a,,b,a,, I ) a n d ” B , . B , ” = D(b,b2,0, I).]

__ p,.

= L , nB

Constructions 8.4.13 and 8.4.15 are pictured in Figure 1. An integral rational expression means a rational expression with coefficients in Z.

“-8;‘

A0 A1 8 2

81 ‘&I+8;‘

FIG. 1. Construction of”-B,” and ” B , +B,.”

Remark 8.4.17. These constructions obviously are “constructible configurations;” and, in fact, starting with A,, A , , A ; , A;, and any finite __ number of points lying on L , = A , A , , we can define the corresponding configuration “evaluating” any integral rational expression on these points.

Each construction was independent of the particular Remark 8.4.18. choice of A . Thus if we replaced A by another point A” we would get the same result. Restating this in the simplest construction, that of “-B,,” we have the configuration (,d, Y’, X),where .d= { A o .A , , A;, A:, A,, A , I , A‘, I “ , B,, B,,, B a , B , B “ } , -Y = { L , ,L , L , , L , , L J >L4, L , , L;, L;, Lk, L ; ) ?

and 9 is the set of incidence and nonincidence relations of points of J/ on lines of Y pictured in Figure 2. Then we have the intersection theorem L , n L , n L; # 0. We would like to “universalize” this theorem. Proposition 8.4.19. Suppose an intersection theorem T holds on the configuration ( . d ,Y..Y), where A,, A , , A ; , and A& E .d.I f T holds for all

48.4.1

309

Theory of Rational Identities

Using “-B,” to get the intersection theorem L , n L , n L; # 0.

FIG.2.

isomorphic images (a,T ) ( ~ & , Y , 9 )such that a(A,) = A,, a ( A , ) = A‘,, and o ( A k ) = A:, then T is universal. Proof.

Suppose

(,2,9, 7)

is an

isomorphic

=

A , , a(A’,)

configuration

to

(cd, 9, .a), where the given isomorphism (cd, 9, .Y) (,>, P , 3 ) sends

,>.

--f

each point A in to A in By Proposition 8.4.5 there is a n invertible linear transformation p of V such that p ( A , ) = A,, p(%) = A , , p(A’,) = A’,, and p ( A 2 ) = A:. Thus T holds in ( p , p ) ( . d ,9 , . 7 ) , so, applying p - l , we get the intersection theorem in (.g,P, 3). QED c&

~

Corollary 8.4.20. The intersection theorem of Remark 8.4.18 is a universal intersection theorem holding for all Desarguian geometries.

O u r first goal is the next result. Theorem 8.4.21. For each integral rational expression f there is a corresponding statement Tf,which is a uiiiversal intersection theorem of #f is a rational identity of the underlying division ring D. Proof. Take the configuration in Remark 8.4.17. By hypothesis the last line L, constructed in the “evaluation” off will intersect L , at A , iff f is a rational identity of D. In this case, L , n L, nA,A’, is an intersection theorem which, by Proposition 8.4.19, is a universal intersection theorem of V . QED Theorem 8.4.22. Suppose V arid c/. are Desarguian planes with respective underlying division rings D and D’ such that deg(D) is finite. Then either deg(D’)ldeg(D) or there is u uniuersal intersection theorem T of V that does not hold for F. Proof.

Suppose deg(D’)(deg(D).By Theorem 8.3.8, we have an integral

310

[Ch. 8

RATIONAL IDENTITIES

rational identity of D that is not a rational identity of D’,so we are done by Theorem 8.4.21. QED This is the only place where the “hard part” of the second fundamental theorem enters $8.4, and so it is worth observing that the standard polynomial (applied to Theorem 8.4.21) yields the following portion of Theorem 8.4.21: If deg(D) < dep(D‘), then there is a universal intersection theorem T of P (corresponding to the minimum standard identity of D) that is not a universal intersection theorem of 7. Before proving the converse of Theorem 8.4.22, we should like to make a few minor observations which cast further light on the constructive procedure. Remark 8.4.23. Constructions 8.4.14 and 8.4.16 are obtained respectively from Constructions 8.4.13 and 8.4.15 by replacing throughout A , by A , , and A by A,. The intuitive interpretation of this is by taking exponentials (since = eSey),viewing ,4, = D(I , 0,O) = D(- I , 0 , O ) as the point at minus infinity. Nevertheless, one always gets “Bl +B2” = “B, B,,” yielding a corresponding universal intersection theorem for all Desarguian planes, but usually “B, .B,” # “B,.B,.”

+

Remark 8.4.24. Suppose B, = D(b,, 0 , l ) with b, E Z ( D ) . For any configuration V containing A,, A , . A A , A:, and B,,we get “B,-B,” __ = “B,.B,”for every B, E A , A , ; thus we have something which could be theorem, which holds in every called a generalized uniiiersal intersection isomorphic configuration B such that A , = A,, A , = A , , A’n = A ; , A: = A:, and B, = B,. We do not want to complicate matters and so do not treat generalized universal intersection theorems here, but the proofs of this section can be generalized (cf. Exercise 6). ~

Let us now look for a converse of Theorem 8.4.22. The idea is to express a given configuration algebraically (in where V = D‘3)).A point A can be written as D ( a , , a,, u3).Naively, we could write a line L as a 2-dimensional vector subspace of V and write out when ( a , ,a,, a 3 )E L, to describe when A lies on L. However, it is much more to the point to describe L in terms of its dual, as follows: Let p = Horn,( V, D). Clearly is a right D vector space of dimension 3 + d i , 1 < i d 3. There is an inner with basis e , , e , , p 3 , where e,:(d,,d,,d3) product V x v + D, given by = 2-21d;1d;~, where d j j cD, so for a subset S of V we can define Anni S. If S is a t-dimensional subspace of V, then Anni S is a (3-r)-dimensional (right) where subspace of VI One can use this to build a “dual” geometry in points (resp. lines1 of correspond to their annihilators in which are

v,

31 1

Theory of Rational Identities

58.4.1

lines (resp. points); we leave this for Exercise 6. All we need here is the fact that Ann V Ldetermines L ; if Ann pL = ( e l x l + e , x , + e , x , ) D we shall write x, x,)D, for short. (We wrote Ann in place of Ann‘, to avoid Ann L = (*b, cluttering the notation.) Remark 8.4.25. Suppose A = D(a,,u,, a3), Ann L = (x1,x2,x 3 ) D . aixi = 0 for all d in D.

A lies on L iff

Proof.

x:=,

A lies on L iff (Ann L)A = 0, which means

x:=l u i x i

= 0.

QED

We are ready for Amitsur’s major contribution to the theory of Desarguian projective geometry, which is the converse to Theorem 8.4.22. Theorem 8.4.26 (Amitsur [66a]). Suppose D, and D , are division F-algebras and & = Dj3), 1 < i < 2. !fZ(D,) is injnite and [ D , :Z ( D , ) ] = 00, or ifdeg(D,)(deg(D,), then every universal intersection theorem is also u universal intersection theorem oft.

of’E

Proof. We shall show that every universal intersection theorem corresponds to an integral rational identity, yielding the result by Theorem 8.3.6. Take an arbitrary division algebra D. Given the sentence T = INT(L,, L,, L , ;%),weshall proceed inductively on %, attaching individual rational expressions in Z,(X) to the points and lines of %,as well as one big rational expressionftied to the incidence and nonincidence relations ;at the last step we shall get the desired rational identity corresponding to T. First note by Definition 8.4.11(ii) we may assume L , n L , is some point A in d ,where % = (d, Y ,.f), so T becomes mere1y“A E L,”.Our strategy therefore is to throw in “ A EL]’’at the end. So we start by attaching the expressionJL = 1 t o % = (0, 0, @), and apply Definition 8.4.11 inductively as follows: Y ,9)is constructible and d = d’u { A } , take inde(i) If (6‘ = (d’, X i ? ,Xi,, XkI,x,., not appearing in J 6 , ;correspond A to terminates x,,, D(Xi,, Xi>, X i 3 )and let JL = Xit,Xktt)-1fi6,. (The idea, to be used again below,is thatanyspecializationoftheXit4 yieldsapoint except whenall threeare sent to 0, in which case + ?.) - { A}, Y ,.P - { “ A E L,,,” “ A E L,”}) is constructible for L, (ii) If %’= (,.PI # L , and if, for j = p , 4, Ann L j corresponds to (h,,, h,,, h,,)D, then take X,,, X k 2 ,X , , not appearing in.f;,.; with g i to be determined indeterminates Xi, shortly, correspond A t o D(g,, g,, g 3 ) . and let j6 = (xi=g u X k , a ) - v L t . Weshalltakethegiasgeneralsolutionsto~~=, g , h , = 0,j = p , q.Todo this, take g1 = X i and write a, = hP,<,b, = hqL,,1 6 u < 3. Solving xguau= 0 and x g , b , = 0 gives us

x,,

(x:=

,

g 1 ~ 1 a ; ’ + g , a 3 ~= ~ ’- 9 2

= glblb;

1

+g3b3bi1,

312

[Ch. 8

RATIONAL IDENTITIES

so g3(a3a;’ - b 3 b ; ’ ) = gl(blb;l-ala;l), yielding 9 3 = X,(b,b;’-ala;’)(a,a;’

-b3bi1)-l,

and g2 has already been solved in terms of 93. The reader can check that this solution is nondegenerate since L, # L,. (iii) If %’= (d. 9-{ L } ,9 - { “ A , E L,”“ A , E L ” } )is constructible for A, # A , and if A,, 4,, correspond to D(gp,,gpl,gp3),D(gn,,g,,,gq3), then proceeding dually as to (ii) we solve for h, to get the general solution to I,= 3 1 gp.hu = 0 = K-3,u-lgq,hu, and, taking new indeterminates X k Y , let jt, =

(CL 1 huxk”)- !I/,,.

(iv) If %‘ = (d, Y ,9 - ( “ A # L”})is constructible, if A is identified with guhU)-lfi6,. D(g,, g,, g3),and if L is identified with (hl, h,, h3)D,let J6 =

(x.,”=

So far .fi6 has numerator 1, and was used to counter degenerate conditions. Nowwethrowin thelast sentenceofT,“A E L,”;Aalreadycorrespondstosome D(gl,g2,g3),andL,tosorne(hl,h2,h,)D.Writef= (I.,”=, g,h,)J;,.Now Tisa iff f is a rational universal intersection theorem of the projective plane identity of D. This ISwhat we wanted. QED Having proved the theorem that motivated the PI-theory, we conclude the main body of the book. EXERCISES

$8.2 1. Construct a division ring D with Z ( D ) finite. (Note D must be infinite.) *2. If Z ( D ) is finite, is Theorem 8.2.1 1 still valid?

*3. Iffis a rational identity of a division ring of infinite degree, can j b e demonstrated to be a formal consequence of the axioms of a division ring? 4. Prove an involutory analogue of the first fundamental theorem. 5. (Bergman) Detine the rational ropologv on D“’, based by the domains of definition of rational expressions. (I.c., S is in the basis iff S = {xED(‘)lh(x)is defined} for some rational expression h.) If D, is a D-ring, define the (generalized) polynomial topology based by sets { x ~ D ‘ ’ ’ I f ( x#) 0) for suitable f i n D { X ] . If S is open in the polynomial topology, then S is open in the rational topnlogy. ( H i n t : Take h =f-’.)

98.3 1. (Bergman) Write f ’ for [ X , , / l in H ( X , , X , ) , and P(j)= ( ~ “ ’ ) ‘ ( ( f - ~ ) ’ ) Then -~. P ( X ; ) is a rational identity of every simple ring of degree 2, but is central for every simple ring of degree 3. [ H i n t : For R simple of degree 2, ((X;)’)’ is already an identity. For R simple%f X;, and 1 in R are Z(R)-dependent. Divide by X‘,.and degree 3, all substitutions of

Ch. 81

Exercises

313

commute with X2.] Bergman [76] uses this fact t o construct a rational expression h such that h is a rational identity of every simple ring of degree 2, whereas h - 1 is a rational identity of every simple ring of degree 3. 2. (Amitsur-Bergman) If F is infinite then every simple F-algebra of degree n is rationally equivalent to M , ( F ) . *3. If deg(D,)(deg(D,), can there be a D,-central. rational expression that is also D,central?

98.4. 1. (Amitsur) W,jX} is ordered if W is ordered. (Him: ultraproducts.) 2. (Amitsur) If G is any group and F is a field, then FF{X} has a group of outer automorphisms isomorphic to G whenever X has infinite cardinality (as a set of indeterminates) at least as large as the cardinality of G. ( H i n f : Represent G as a permutation group of the indeterminates of X ) . 3. If R is a ring satisfying a PI .f‘(X,,X,) having only 2 monomials, then [R, R] is containedinN(R).[Hi,ir: By PI-structuretheory,wemayassumeR isprime;passing toQ,(R),we may assume R is simple. Now translate the identity into a group identity.] 4. (Amitsur [66a]) The group algebra F I ( X ) of the free group Y ( X ) can be injected into F,(X) for every division F-algebra D that is infinite over an infinite center. [ H i n f : Take an element of F S ( X ) t o its equivalence class in F , ( X ) . Using the regular representation, it suffices for any given,/(X,, . _ . ,X,) in F‘q(X) to find a finite-dimensional F-algebra for which./ is not a rational identity. Take B = F{X, ,..., X,,i/l, where I is the ideal generated by all X : and = all X , b . . . X i p ,q large and fixed. Write Y” = ( l - X u ) ( l - X , + u ) , I < u < f . Then (X:+.+X,+.+I)(R,Z+X,+I), so, viewing / ( u l ,...,.Y,) as a polynomial in X I,..., X,,, conclude the leading monomial is nonzero.] 5. Write the configuration of Desargue’s theorem as a constructible configuration, and conclude that Desargue’s theorem is a universal intersection theorem. 6. Every generalized rational identity of D yields a generalized universal intersection theorem in the corresponding Desarguian plane. 7. (Amitsur) “Dualize” the geometric theory to characterize Desarguian planes in terms of theorems involving collinearity of points.

.\;’

APPENDIX A

CENTRAL POLYNOMIALS OF FORMANEK In this appendix we develop in detail the original central polynomials of Formanek [72], which settled the important 25-year-old question raised by Kaplansky and revolutionized PI-t heory. Since we have already a “second generation” central polynomial (gl,),we now d o the (associative) PI-theory without them. However, these polynomials have merit beyond their considerable historical interest, and are developed in this appendix for the following reasons: 1. Formanek’s polynomial has the lowest degree of any known central polynomial for M , ( Q ) ; 2. Formanek’s polynomial gives rise to a 2-variable central polynomial for M,(C), which has applications to nonassociative algebras ; 3. the proof of Formanek’s polynomial gives us a “generic” form of the characteristic polynomial of a generic matrix, due to Amitsur [73]; 4. the proofs involve matric unit computation, related to graph theory, a fruitful method in PI-theory worth presenting in detail.

The treatment of Formanek’s polynomial follows Formanek [72], Amitsur [73],and Jacobson [75B],witha few additionalobservationsinorder toobtain a ’-variable central polynomial. Razmyslov [73a] independently discovered a class ofcentral polynomials, which were also multilinear; the reader can find a good treatment (based o n Markov [73]) with embellishments, in Amitsur [76]. In particular, Amitsur found an interesting M,(Q)-central polynomial which is an identity of M , , ( F ) whenever char(F) divides i i . (Compare with Exercise I .4.10.) Remark A . l . If f’is an identity of M n ( 4 ( c ) ) ,then ,f is an identity of M,(C) for every commutative algebra C. [Indeed, f is an identity of M , ( 4 [ < ] ) L M,(&(<)),so apply Proposition 1.1.8(ii).] Remark A.2. Suppose a classical polynomial f is M,(F)-central for every field F. Thenfis M,(C)-central for every commutative ring C. [Indeed, [ X J is an identity of M,(cj([))and thus of M,(C), implying,f(M,(C)) E C . 1 ; moreover, for every maximal ideal P of C, since C / P is a field we have 0 f / ( M , ( C / P ) ) = / ( M l , ( C ) / A 4 , , ( P )so ) ? f (M,,(C))# 0.1 315

316

LAPP. A

CENTRAL POLYNOMIALS OF FORMANEK

Of course, [X,,X2]2 is M,(C)-central for any commutative ring C. The question for 25 years was, what happens in M,(C) for n > 2? Here is Formanek's answer, somewhat elaborated. Given a commutative polynomial p ( t l , ...,t,) = ,...,i,,,)ai[y..-tkm, we associate the (noncommutative) (m - 1)-linear polynomial fp(Xl, ..., X , ) = ~ a i X ~ X l x ~ x , ~ ~ ~ X ~ - ~ x , _ Let us examine (m- 1)-linear polynomialsf(X,, .. .,X,), with the notation as in $1.3. Let B = {eij(l< i,j < n>, and let ro = C1=l
E Z,{

Proof. First, it suffices to provef(H,{ Y'})= 0 by Proposition 1.3.15. But then, we need only show thatf(Z,(Y')("-')x { r ,}) = 0, so it suffices to prove f(M,(Z[<])('"-l' x {r,}) = 0. Now apply Lemma 1.1.31 to conclude that this is true iff f(B("-') x [ r o } )= 0. The converse is trivial. QED

COrOllarY A.4. With the notation as in Proposition A.3, a classical, (m- 1)-linear polynomialffl,, .. . ,X,) has the propertyf(A4,fC)) C Cfor all commutatioe rings C i$S(B("-') x (1,)) c a((). Proof.

Apply Proposition A.3 to [Xm,f(Xlr...,Xm-,, X,,

Thus, we want to show that fp(ei!jl,ei2j2,. . .,eim_I j , under suitable conditions on p. Write tifor
,)I.

QED

r,) is always scalar

Remark A.5.

- a j l i , 'sip,-2in,- ,p(ti,,. . ., ti,*_t j m .*

1)

.

l)eilj,,,

~

I 9

where 6 is the Kronecker 6. Definition A.6. A set of matric units {ei,j,,..., eiJn}is a cycle if . . j l = i z , j 2 = i.3,.. . , J , = t l .

We shall now take m = n+ 1, and define p'(tl,. ..,5,) Lemma A.7.

Suppose p(Cl,. . .,5,,+

= p(tl,. .

.,t., 51).

has the following two properties:

(i) ( t i - t j ) l p f o r every i < j , except possibly for (i,j) = ( l , n + l ) ; (ii) p'(tl,..., 4,) is symmetric in t l , . . ,<,. 7henfp(eilj,,. . .,eidm,C;= ,tieii) = 0 unless {eiljl,...,einjm}is a cycle, in which casefp(eiljl,.. .,ei,,j,,,CT=itieii) = ~'(ti,...,tn)ei~i~. Proof.

By inspection. QED

APP. A1

Central Polynomials of Formanek

317

Call p admissible if p satisfies properties (i), (ii) of Lemma A.7. Now write 9 p ( x , , . . ., X"+1 ) = CnESym(m)fp(Xnl,. . ., x,,, x,,1) and g;(Xl,. . ., X , , , ) = fp(Xnl,.. . ,X,,, X,+l),summednowonlyoverallcycles n of Sym(n).

x

Proposition A.8.

Suppose p is admissible. Then

p ' ( t l , ..., 5,). 1

if, for some n in Syrp(n), {eixljnl~ ;..,eixgJ is cycle i= 1 ){o otherwise. Proof. (i) is immediate by Lemma A.7. To see (ii), note that g,,(eiljl,.. .,einjn,C1= tieii)= 0 unless some {einl j n , , . . . ,einnjrn} is a cycle. By symmetry, we may assume that 7~ = (1). But then for every v in Sym(n), ei,j,t,..., er,. J.. is a cycle iff v is a cycle in Sym(n), so we are done by (1) and Lemma A.7. QED

(ii) g p eiljl,. . . , ei,,j.,

(

1 Sieii

=

Theorem A.9. I f p is admissible, then g,(M,(C)) G C and gb(M,(C)) E C for every commutative ring C .

Proof. Since g p is n-linear, we apply Corollary A.4 to Proposition A.8. QED Of course, the real question is when p is M,(C)-central. The answer is simple : Theorem A.10. Suppose p is admissible and F is a jield. 7hen the following conditions are equivalent:

(i) g p is M,(F)-cenrral; (ii) gb is M,(F)-central; (iii) fJM,(F)) # 0; (iv) For some matrix in M,(F) having characteristic values ql,.. . ,q,, we have p'(ql,.. .,q,) # 0. Proof. Obviously (i) o (iii) and ( i i ) o (iii) by Corollary A.4, Lemma A.7, and Proposition A.8 ;similarly, (iii) 3 (iv) is clear. To prove (iv) 3 (iii),we aim to show that given a matrix r in M,(F), having characteristic values ql,.. .,q,, with p'(ql,. ..,q,) # 0, thatf,((M,(F))(") x (r)) # 0. Note, since p is admissible, that ql, . . . , q n must be distinct. Now by Lemma 1.1.31 it suffices to show f , ( ( M , ( ~ ) ) ' " ){r}) x # 0, where F is algebraically closed. By Remark 1.3.16 we may assume r is diagonal; hence, we are done by Remark A.5. QED

318

LAPP. A

CENTRAL POLYNOMIALS OF FORMANEK

Corollary A . l l . Suppose p is admissible; g, and gb are M,(C)-central for every commutative ring C i r f o r ei:eryJield F there is a suitable matrix in M , ( F ) having characteristic values q , , .. .,q, with p ’ ( q l , .. .,q,) # 0.

Immediate. QED

Proof.

The simplest example of an admissible p is pO(tl,...,
cn

(tl-~i)(tn+l-ti)

1n

(ti-<j)’-

2Ci<jQn

i=2

2

Write go. = goo and gb,, = gb,. Clearly pb(51, ..., tn)= r I i ~ i < j s n ( ( i - < j ) so p is admissible; note also that in a field p6(ql, ..., q.) # 0 ifT q l , ..., q,, are distinct. 7

Theorem A.12. central.

For every commutative ring C go, and gb, are M,(C)-

Proof. The theorem follows from Corollary A.11 and the well-known fact that for every field F M,(F) has a matrix with distinct characteristic values. [If F is finite, take the companion matrix of an irreducible monic polynomial (in one indeterminate) of degree n.] QED

We call gb, Formanek’s central polynomial. Corollary A.13. Let gin(Xl,Xz)=fio(X1,. . ., X I , X2). Then g;,, is M,(C)-central for every commutative ring C. Proof. One sees easily that applying ( n - 1 ) times the linearization process A, to g;,, yields go,, so g;,,(M,(C)) # 0. To show that g i n is M,(C)central for all C, it then suffices to assume C = Z[5]. But go, is M,(Z[<])centraland clearly go,,(X1,...,X,,X,) = n ! g i , ( X , , X , ) , implyingg’d,(X,,X,) is M,(Z[[])-central. QED

For n = 2, po = (51-t2)(53-5~) = 5 1 5 3 - 5 2 5 3 - 5 1 5 ~ + 5 : ‘ ~ ~ ’ ~ f,”

=

x3x,x,x, -x,x3x,x3 -X3X,X,X, +x,x:x,,

and g;, =

x2x:x2--x,x2x,x,-x,x,x,x,+x,x:x, = -[X1,X,I2.

the well-known central polynomial. Note that go,, gb, and g:, all have degree nz, and they have the lowest degree among all known M,(F)-central polynomials. We return now to other examples of admissible p. Theorem A.14. Suppose a(tl,. and ler p = pea.

tl,. . .,t,,

(i) p is admissihle.

. .,(,) is an element ofZ[t], symmetric in

APP A1

Exercises

319

,,

‘ , are the characteristic values (ii) Suppose a,, . . .,a,+ E M,(F), and q .. . ,1 ofan + 1. Thengp(a19 ,.., a n + 1 ) = ~ (1 , .4 . Vnfgon(a19* * * , a n + 1 L &(a,, .-.,a n + 1 ) (viewed in some field extension containing = a(? 1 , . . . , V,)go,(al, . . . ,a,+ - 9

. .,V n ) .

~ 1 , .

(iii) g , ( M , ( C ) ) E C and g ; ( M , ( C ) )G C f o r all cornmutative C . Proof. (i) Immediate. (ii) We may work in M,(Z([)) and assume a,+ is a generic matrix. Let c1 = a(ql, ..., q , , ) , f ( X ,,..., X , , , ) = g p ( X 1 ..., , Xn+l)c1gOn(X1,. . ., X , + , ) ; we claim that f ( r l , .. . ,I , , a , , , ) = 0 for all I , , . . ., r,, in M , ( Z ( [ ) ) .By Remark 1.3.16we may assume a,+ is diagonal. Then, by Lemma 1.1.31 we may assume the ri are matric units; we are done by Proposition A.8. (iii) Immediate from (ii). QED

,

Corollary A.15 (Amitsur). Let g i n= gpo0,.,where ginis the ith elementaryfunction in n variables. Then g i n ( M , ( C ) G ) C and

is an identity of M,,(C)forall C.

This very interesting observation gives a polynomial identity that captures the essence of the Cayley-Hamilton formula.

EX ERCl SES Note.

Exercises 1.4.12 and 3.1.9 should be much easier after reading Appendix A.

*l. What is the M,(Q)-central polynomial having smallest positive degree? (Maybe it is Formanek’s polynomial.) 2. Write down gOZ,d,2rg:3,g:4.Show (for all i, n) that gynis invariant under “reversal” (the involution of L { X } induced by the homomorphism L{X} + Z{X}””which takes all Xi + X i ) . *3. Is the multilinearkation of go. an M,(C)-central polynomial for each commutative ring C? (Perhaps it is an identity.) 4. Let C be acommutative ring and let (s)be the canonical symplectic involution of MZn(C). Then, for g = gin(X1,..., X,,X,+, + X : + l ) ~ Z { X ; * } show , that g + g * is either an identity of ( M , ” ( C ) , s or ) is (M,,(C),s)-central. [Hint:Suppose X i H r i . We may assume C is the algebraic closure ofZ(5); then we may assume that r , + , is diagonal, so r , + , = Zr=lci(eii+ei+n.i+n) for suitable ci in C. Now finish “a la Formanek”.] This exercise is the key step in constructing central polynomials for all reduced Jordan algebras (cf. Racine [76]). 5. With the notation as in Exercise 4,let gin = gin(Xl+ X:, . . ., X I+ X:, X, + X;).Then Go. is (M,,(C),s)-central for all C, and Z.l=o(-l)igi,,(Xl,X,)(X,+X?)”-’ is an identity of (M2m.S).

6. Construct a central (*)-polynomial g of ( M , J F ) , t), t the transpose involution, such that [X,,g] is not special. [Hint:apply Formanek‘s method to (XI -X:)*.] 7. Can we use Formanek’s method to construct a square-central (*)-polynomial for matrix

rings with involution? (This would havevery important consequences ;cf. Rowen [75a, Proposition 311.)

APPENDIX B

THETHEORY OFV3 ELEMENTARY CONDITIONS O N RINGS We have contended throughout that the PI-theory is the “correct” setting for finite-dimensional algebras and, in particular, central simple algebras. However, several important theorems (such as the Skolem-Noether theorem) apply to rings finite dimensional over (noncommutative) division rings, in particular M,(D), and one might be interested in a polynomial-like theory that would include M , ( D ) for arbitrary division rings D. This is not an easy matter, because every condition we have applied so far (cf. Chapters 7 and 8) to a division ring implies it has finite degree. In this appendix we briefly describe one theory, .initiated by Drazin [57], which, although unsatisfying in some respects, does encompass several major structural results from the PI-theory. Definition 6.1. Let ./({Xl,...,XJk = (monomials h of the free monoid./t{X, ,..., X , j I h # X , X , . . . X , a n d t <deg(h)ik},afiniteset.Ris (t, &)-elementaryif, given r l , . . . ,rl in R , one can find a right regular element x in R, as well as elements x,, for each h in J V { X ~ , . . . , X , ) such ~ , that xrl - - - I , = E,,.Y,,h(r,,.. . ,r,),thesum taken over all h i n . /{Xl,. . . ,XrJk.Theelement Y is very important; we shall call .Y the pivotal coeficient (corresponding to r l , . . .,r,).If for all r , , . ..,r, in R we can take x = , l ,then we say R is absolutely ( t ,&)-elementary.

If R satisfies a multilinear polynomial identity.f(X,, . . . ,X,), then, assuming has , coefficient 1 inj; we see immediately that R is absolutely (t, t)X, - - - X elementary. Thus, the theory to be described below generalizes PI-theory. Proposition 6.2. I f R contains no strictly increasing chain of minimal left ideals, then R is absolutelj ( t ,2t)-elementary.

x>=i

(I

+ 1)

Proof. Given r in R, set Li= Rd. Then L, E L,_ c_ ... c_ Lo, so by hypothesis Li- = lAifor some i, implying r i - E L i ; so f E C;Lg+ Rd. The proposition is concluded by a modification of the multilinearization technique. QED Corollary 6.3.

,

M , ( D ) is absolutely (t,2t)-elementaryfor all t 2 n.

Corollary B.4. If R is n-dimensional over a division subring D,then R is ( t ,2t)-elementary for all t 2 n. 320

APP. BI

The Theory of V3 Conditions

321

Corollary B.3 has the following important converse: Theorem 6.5 (Drazin [57]). !f R is primitioe and absolutely (t,k)elementaryfor some k, then R : M,(D),for some n < t and some dioision ring D. Proof. Otherwise, write R as a dense subring of End M , , choose Dindependent elements y l , . . .,y, of M , and for 1 < i < t pick ri in R such that riyi = y i + l and r i y j = 0 for all j # i . We have built a “staircase” that contradicts Definition B.l. QED .+

,

Thus “absolutely elementary” is a necessary and sufficient condition for a primitive ring to be simple artinian. Note that “ R is absolutely (t,k)elementary” can be written logically as (Vrl,. . .,r , E R ) ( 3 x , , . . . ,.x,,,): ( r l r 2... Y,

= x x , , h ( r l , .. . ,r,)),

where m is the order of b { X l , , . . ,X , } k .This kind of sentence, an atomic V3 sentence, is logically the simplest kind of sentence that has not yet been investigated (identities are the atomic V sentences, and generalized identities are the atomic 3V sentences). Thus, the absolutely elementary condition is very natural to examine, both by its intrinsic nature and its application to primitive rings. If R is absolutely ( t , k)-elementary, then every homomorRemark B.6. phic image of R is absolutely (t,k)-elementary. If { R ; l y e T }are each ( t , k ) elementary, then n { R , l y E r>is (t,k)-elementary. However, a subring of a ( t ,k)-elementary ring need not be (r, k)-elementary. One of the main structural techniques of PI-theory and of GI-theory was to pass to the polynomial ring R[A].However, a pretty theorem of Amitsur [60] says that a division ring D has finite degree < t iff every primitive homomorphic image of D[A] is a k x k matrix ring over a division ring for suitable k < t . [His proof is actually based on Wedderburn’s factorization is Kaplansky’s theorem. Conversely, suppose d E D is method, as follows: (a) arbitrary. For any x in D, define L , = D[A] (A- .ud.u- I ) , a maximal left ideal of D[A], and let B = n { L , J s t z D ) . Then B is a right D-module, implying B D[A]. Since D[A] is a PLID, we can write B = D[A]p(A) for some polynomial p ( A ) in D[A]. By Lemma 3.2.15 p ( A ) is a multiple of the minimal polynomial of d [over Z ( D ) ] .Since D [ A ] / B is primitive, D[A]/B :M,(D’) for some division ring D‘, so B is maximal ;hence p ( A ) is the minimal polynomial of d, and this polynomial has degree d t . Thus every element of D is algebraic of degree d t over Z(D), implying D has degree < t . ] Let us now restate this theorem in terms of elementary conditions.

a

Theorem 6.7 (Amitsur). For a dicision ring D, D[A] is absolutely ( r , 2t)-elementary implies deg(D) = t .

322

THE THEORY OF

v3 CONDITIONS

LAPP. B

An open question is, “If R has a faithful, irreducible right module and is absolutely (t, k)-elementary, then does R have the form M J D ) for suitable n and D?’ The best we can do so far, under these circumstances, is to note that Jac(R) = 0, so by the ultraproduct construction of $7.5 R can be injected into a primitive ring P that is absolutely (t, k)-elementary, implying P = M,(D) for suitable n < t and a suitable division ring D. When examining prime rings, we should like to obtain Posner’s original theorem, i.e., that the classical ring of left quotients of an absolutely ( t , k)elementary prime ring has the form M J D ) for some n < t. Actually, a slightly better result is obtained in Rowen [76b]. Call an element r of R Ore regular if r is left and right regular and if Rr is a large left ideal of R. Definition 6.8. R is strongly (t,k)-elementary if R is (r, k)-elementary and, if lor all r,, . . . .r, in R we can choose the pivotal coefficient to be Ore regular.

Note that absolutely (t, k)-elementary implies strongly (t, k)-elementary, which implies (t, k)-elementary. Theorem 6.9.

7he.followingconditionsare equivalentfor a prime ring R :

(i) R is strongly (t, 2t)-elementary; (ii) R is srronglv (t, k)-elementary,for some k; (iii) for suitable n < t and a suitable division ring D,M J D ) is the classical left quotient ring of R. Proof. (i) * (ii) is trivial; (ii) (iii) is Rowen [76b, Theorem 61 and is omitted here; (iii) (i) follows easily from Proposition B.2 and the Faith-Utumi theorem (cf. Jacobson [64B, p. 2721). QED Example 6.10. Let B be the ideal of Q{X,,X,} generated by (X,X,-X,X, - 1) and let R = Q(X,, X,}/B. We write the elements of R in the form xiqiXyX;L,where qi E Q, and define the degree to be the exponent ( i l , i2) of XyX:: such that (i,, iz) is maximal according to the lexicographic ordering. We list several facts about R.

(1) R is simple. [Indeed, if A # 0 is an ideal of R, take a nonzero element r of A of minimal degree (il,i2). If i, > 0, then [r,X,] has leading term i,Xt-’Xk # O,contrarytotheminimalityofdeg(r);ifi,= Oand i, > 0,then [X,, r] has leading term i2X$-’ # 0, contrary to minimality of deg(r).Hence r E Q, implying A = R.] (2) R is a domain. [Indeed, deg(r,r,) = deg(r,)+deg(r,).] (3) R is a (left) Ore domain. (This is seen in two stages. Viewing Q { X , ) s R canonically, check that every nonzero element of Q{X,}is Ore regular, so one formally inverts Q{ X , ) - { 0) to form a ring R,, which can be seen to be a PLID, and is thus Ore. It follows that R is Ore.)

APP. BI (4)

The Theory of V3 Conditions

323

R is right Ore. [An argument analogous to (3).]

Thus, R is strongly (1,2)-elementary by Theorem B.9, but is not absolutely ( t ,k)-elementary for any t and k. Of course, this theory has no place for anything resembling central polynomials, and is a repository for old (i.e.,precentral polynomial), standard proofs in PI-theory; the omitted proof of Theorem B.9 was in fact modeled closely after Posner's original proof of his theorem. Along a similar vein, we shall now determine much of the structure of semiprime rings satisfying an V3 elementary condition. V

Proposition 8.11 . Suppose R is semiprime and (t,k)-elementary, and L R . (i) Ann V j = Ann V'Jbr all j 2 t . (ii) Ann' V j = Ann' V ' j b r all j 3 t .

Proof. (i) Let A' = Ann V ' for all i. It suffices to prove that A , , , c A,, because obviously ,4,cr A ' + , , . Now, for all k 2 i A k V k - ' G A i , implying A , V k - ' G AiVi-'.Pick ri in V ' A , for each i. Since rirk= 0 for all i < k, the only possible nonzero product, of length > t , of r , , . . ., r,, is rrrr-I " ' r , . Hence, by definitionof(t, k)-elementary,rr,r,- " ' r , = Oforsomerightregularelements of R, implying r r r r - ," - r l = 0. Thus 0 = ( V f A , ) ~ ~ ~ (=V VA'l()A r V ' - ' ) ~ ~ ~ ( A I3 V )VA'I( A r + , V r ) ' ,

so ( A r + ,V'R)'+' = 0, implying A , , , V' = 0, so A , , , (ii) Proof is analogous to (i). QED

c A,.

Under the assumptions qf Proposition B.11, if V is Corollary 8.12. nilpotent, then V' = 0. Theorem 8.13.

Suppose R is semiprime and ( t ,k)-elementary.

(i) Every nil subring of R is nilpotent ofdegree d t . (ii) R is left and right nonsingular. Proof. (i) Suppose A is a nil subring of R . Since each element a of A is nilpotent, Proposition B . l l says a' = 0 for all a in A. Thus, by Proposition 1.6.34 A is locally nilpotent. Hence, for any a , , . . . , a , in A , { a l , ..., a,} is nilpotent, implying { u l , .. . ,ar}' = 0, so a , ' " a , = 0; thus A' = 0. (ii) Let A be the (left)singular ideal of R . We claim that A is nil. (Indeed, if a E A and a' # 0, then 0 # Ra' n Ann, u, so we have some nonzero element rat such that ra"' = 0, which is impossible by Proposition B.ll, letting V = { a } . ) But then A' = 0 by part (i), implying A = 0. (The proof that R is right nonsingular is symmetrical.) QED

Thus any semiprime ring which is (t, k)-elementary has a "maximal ring of left (resp. right) quotients," but these need not be the same, as evidenced by the

324

THE THEORY OF

v3 CONDITIONS

LAPP. B

example in $8.2 of a left Goldie domain that is not right Goldie. We see that many important structural properties of semiprime, prime, and primitive rings are characterized by these elementary conditions, and we are led to examine the socle of a primitive ring. Recall that a dense subring R of End M , has an R-properGI itTsoc(R) # Oanddeg(D)isfinite;a naturalquestion is whetherone can characterize merely when soc(R) # 0 by an elementary condition. First we observe that soc(R) # 0 precisely when R satisfies the sentence, “R has a rank 1 idempotent,” i.e., (3e E R)(Vr, F R)(3r2E R):((er,e = 0 ) v (er,er2e = e)) A (e2 = e ) . (Onecan even make this sentence atomic, with a little manipulation.) Thus our program has some chance of working, if we “generalize” our notion of “absolutely elementary.” Accordingly, fix a finite subset W of R, and let .M{W,X,, . . . ,XI), = {generalized monomials of W ( W; X . . ,X,} having multilinear label In //{XI, . . . , X , } , } , a finite set. I

Definition 8.14.

Let g be a generalized monomial having label

X,...X,,withcoefficientsinW S a y g i s (R;U?-pivotalif,givenr,, ..., r , i n R , we have.u,,foreach hin //(W;X,, ..., X,},suchthat g ( r l , .. ., r,) = ~ h . ~ , , h ( r. l.,, r. , ) . Theorem B.15. Suppose R is a dense subring ofEnd M , andg is (R ; W)pivotalfor somefinite subset Wof R. 7here is a number n, dependent only on the number of elements of W, the number of monomials of g. und t , such thqt rank(g(r,, . . . ,r,)) < nfor all r,,. . .,r, in R.

Instead ofwriting down the proof ofthis theorem, we shall talk about it. The obvious approach is to try t o mimic the statement and proof of Theorem 7.2.2, which is fine, except that g need not be (RF; W)-pivotal when we pass to R F. So instead of passing to RF, we take a D-base {y,y,, . . .} of M, and for each d in D we obtain d in End M , by defining d’(Cyidi)= xyiddi.The map d w d yields an antihomomorphism $: D + End M,, so we see that [R, $(D)]= 0, and R+(D) is a dense subring of EndM,. Now we really are interested in the following situation. Let D’ = I(/(D). Given finite-dimensional D-subspaces V,, . . ., I: of M , say a generalized monomial g (having label X , , . . .,XI and coefficients in a finite subset W‘ of WD‘)is (I/;.)-ualuedif. given r l , . . . ,r, in R such that r, = 0,l < i 6 t, and given y in M, we can find an element r = xx,,h(rl,.. .,r,), summed over all h in //{ W’; X,, ..., X,},suchthat(g(r,, ..., r , ) - r ) y ~V.Nowwearereadytoprove the following statement, using essentially the same proof of Theorem 7.2.2. Ifg is (F)-valued,t hen for some finite subset W“ of WD‘ and for some number n depending only on [ WD’ : 0’1,the number of monomials of g, and t, we can rewrite g as a sum 01 monomials with coefficients in W,each monomial having

APP. BI

The Theory of V3 Conditions

325

a coefficient of rank
Corollary B.18. Suppose R is primitive. R has an R-strong, ( R ;W)pivotal generalized monomial i f R 2 M,(D) for suitable n,D. In .fact, if R 2 M,(D), then X , X , is ( R ;{eijl1 6 i,j d n})-pivotal.

(In some ways this result is more useful than Theorem B.5, as we shall see in Exercise 3.) We have a very satisfactory theory for primitive rings, and, using the techniques of $7.2and $7.4, one should be able to extend these results to the involutory case. (This has not been done in print; partial results are in DesMarais-Martindale [751.) Now we would like to extend Martindale’s theorem. This seems to be quite difficult, because all the proofs of Martindale’s theorem involve passing to a central extension, preserving all multilinear GIs, but not pivotal generalized monomials. The best known result is in Rowen [77d] (for which it is essential to read the correction). Theorem B.19. Suppose R satisfies an R-proper, ( R ; W)-pivotal generalized monomial and R satisfies the.fd1owiiv.j property:

(a) There is a primitive ring P intersects R nontrivially.

2

R such that every nonzero lefi ideal OfP

Then soc(P) # 0 and so the maximal ring of left quotients has nonzero socle. (The converse is also true.) This raises the question as to which prime rings R satisfy property (a). R satisfies (a) if R is nonsingular by a deep theorem of Goodearl [73], but attempts to extend the result to arbitrary prime rings have failed so far. The theory of elementary V3 conditions described above can be put into a slightly more general setting, enlarging . &’{XI,. . .,X,},, but one loses the logically elementary property of the conditions, and thus certain standard tools such as ultraproduct constructions. The theory as set above is very nice, because it naturally encounters many familiar and important concepts in noncommutative ring theory, such as “M,(D),” “Goldie rings,” “singular ideal,” “socle,” and “maximal quotient ring.” Its main defect is the lack of

326

THE THEORY OF v3 CONDITIONS

[APP. BI

nontrivial constructions. How can we check directly that a ring satisfies one of these elementary conditions?

EXERCISES 1. (Jain [66]) IfJac(R) = Oand R[L] is absolutely (t,k)-elementary. then S,, is an identity of R. ‘2. If R is prime and (t,k)-elementary, then is R[A] (t,k)-elementary? 3. Suppose R is primitive and has an element r that is a root of a polynomial go.) with g’(r) invertible (where g’ is the derivative of 9). If C,lr) is simple artinian. then R is simple artinian. 4. Example B.10 is left and right Noetherian. 5. IfR is prime and semiprimitive,satisfyingan R-proper (R ; W)-generalizedrnonomkal, then the maximal left quotient ring of R is primitive with nonzero socle.

APPENDIX C

NONASSOCIATIVE PI-THEORY This book has been written about associative algebras. However, there is an analogous “PI-theory’’ that can be carried out for nonassociative rings, which is outlined here. (A detailed discussion is given in Rowen [78a, b].) The theory can be done for arbitrary R-rings, but, for simplicity, we stick to &algebras, where 4 is an associative, commutative ring. “Algebra” will now mean, “Nonassociative &algebra with 1.” (Actually, we can do without 1 in places, for those interested in Lie algebras.) Our treatment is highly PItheoretic; a completely different (and more classical) point of view is taken in Osborn [72]. The free nonassociative monoid . f l ’ ( X )is defined as follows, inductively: 1 E .//’(X) and each ( X i )E . / / ‘ ( X ) ; if k,, 11, E N ‘ ( X ) , then (h,h,) E . N ‘ ( X ) . The product of two elements of / / ‘ ( X ) is defined in the obvious way: h , . 1 = 1 - h , = h , and h, ‘ h , = ( l i l h 2 ) for all h , , h, in . N ’ ( X ) .For example, , is different the product of ( X , X , ) and ( X 3 X 4 )is ( ( X l X 2 ) ( X 3 X 4 ) )which from (( x2)x3)x4)* The free nonassociative algebra 4’{X } is the nonassociative algebra freely generated by A ’ ’ ( X ) ; an identity of an algebra R is an element of @ ’ { X }in the kernel of every homomorphism from c$’{X} to R . We often leave off the outermost pair of parentheses in a monomial, for notational convenience. For example, in any algebra R write [ r , , r,, r 3 ] to denote the “associator” (rlr2)r3-r1(r2r3). Obviously [ X , , X , , X,] is an identity of R iff R is associative. R is called alternative if [ X , ,X , ,X , ] and [ X , ,X , , X J are identities of R ; R is power-associatiue if [ X i , X { , X i ] are identities of R for all i, j, k in Z f . R is Jordan if [ X I ,X , ] and [ X : , X , , X , ] are identities of R . Much of the literature is spent on examining identities which are “weaker” than associativity, in the sense that they lie in the T-ideal of $ ’ ( X } generated by [ X , , X , , X , ] . However, this is obviously an entirely different subject from a PI-theory, which would examine the structure of an algebra R satisfying an identityfthat is strong in the following sense: The canonical homomorphic image of ,f in the free associative ring 4 { X } (taken by 1. “erasing all parentheses”) has a monomial whose coefficient is [Actually the situation is more complicated for Jordan algebras because we want to exclude [ X , , X,], as well as all “s-identities” (cf. Jacobson [68B]) from being counted as strong identities; one workable definition of “Jordan PI-algebra’’ is a Jordan algebra satisfying a strong identity which is not an s-identity.] There are nonassociative analogues of the associative ring-theoretic concepts. An ideal A of R is a &submodule of R such that A R 5 A and 327

328

NONASSOCIATIVE PI-THEORY

[APP. c

RA G A ; left and right ideals are defined analogousy. R is prime if A , A , # 0 for all nonzero ideals A,, A , of R. Unfortunately, if A d R, it is not necessarily true that Ann, A Q R . If B E R , write ( B ) for the intersection of all ideals (of R) containing B. The center of R, written Z(R), is {z ERI[z, r] = 0 = [z, r, r’] = [r, z, r‘] = [ r , r’, z ] for all r , r‘ in R]. Note that Z(R) is a commutative, associative ring, and R is a Z(R)-algebra. From our vantage point, we can proceed in two possible directions. O n the one hand, we can examine very general classes of rings satisfying certain strong identities; on the other hand, we can examine specific classes of PIalgebras (e.g., associative PI-rings or alternative PI-algebras or Jordan PIalgebras). As we shall see, the theory of alternative PI-algebras is just about as rich as associative PI-rings. N PI- Rings Let us start with the first direction, which concerns the Capelli polynomial C,, Actually, there are several nonassociative versions of C 2 , + , , arising from the possible placement of parentheses in X 1 X r t 2 X 2 X 1 + 3 . X. , X z , + l X , + I ; for example, for t = 1, we have the nonassociative Capelli polynomials ( ( X l X 3 ) X z ) -( ( X , X , ) X , ) and ( X , ( X , X , ) ) - ( X , C X , X , ) ) . Weshall call these polynomials the worrassociatit~e Capelli t + 1-normal polynomials (“t-normal” is analogous to the definition for associative polynomials). The number of nonassociative Capelli t + 1-normal polynomials (for t fixed) is precisely the number of ways v(2t + 1) we can write parentheses in Xi X , + , X , X , + ... X , X , , X , to form an element i n . k ’ ( X ) . [This number is not hard to compute, using the recursive formula v ( n ) = \,(i)v(tz--i) ] Just as in the associative theory, one gets:

, ,

Remark C.l. If every nonassociative Capelli t-normal polynomial is an identity of R, then every t-normal polynomial is an identity of R . Definition C.2.

t

R is NPI of degree t if every nonassociative Capelli

+ 1-normal polynomial is an identity of R.

A guideline to follow when attempting to generalize results from the associative theory is that, whereas results intrinsically concerned with the center tend to generalize with straightforward extensions of the proof, facts further from the center require more elaborate arguments. A good example of this rule is found when attempting to generalize Martindale’s “central closure” of a prime algebra R with center Z , treated in $7.6. The construction given in 57.6 is often described as Q , ( R ) = lim,{Hom,(A, K ) I O # /1 4 R } , where each ideal is considered as a left module; R and its extended centroid are subrings of Q o ( R ) and generate the

APP. CI

329

Nonassociative PI-Theory

central closure. But in the nonassociative case there is no injection R + Q o ( R )[because Q o ( R )is associative]. Nevertheless, we can still view the extended centroid 2 as follows: An admissible pair ( $ , A ) is an ideal .4 of R and an additive group homomorphism t,b: A + R such that t,b(ra)= r$(a) and $(ar) = $(a)r for all r in R . Say ($,, A , ) ($,, A , ) iff $, and $, have the same restriction to some nonzero ideal contained in A , n A,, and define 2 to be the natural ring structure on the equivalenceclasses [($, A)], i.e., [($,,A,)] [(t,b2,A2)] = [($1++2r A1 n.4211 and [($l, A,)”,, A211 = [ ( $ 1 0 $ 2 7 (AIA2))l. Paralleling $7.6, we see easily that 2 is a field containing Z . The problem is, “How do we form the central extension RZ?” We follow the approach of Erickson-Martindale-Osborn [75] in the next three results. First observe that for any commutative, associative Z-algebra H , R OZH has a (nonassociative) multiplication induced from that of R , and thus is an algebra which is a central extension of R 0 1. In case H = Z , we set I = { x i ( a i0aipi-$i(ai) 0pi)laiE A i , ($,, A , ) is admissible, a, = [($i, A,)] E 2, and piE Z } .

-

+

Lemma C.3.

Proof.

R

2

R

01

G

R O Z2. I

a R OZZ , und Z n ( R 0 1 ) = 0.

Straightforward verifications. Q E D

Now view R c R @,2 and let P a R @,2 be maximal such that I and P n R = 0. We form R = ( R @,Z)/P. QED

G

P

Theorem C.4 (Erickson-Martindale-Osborn). (i) R is a central extension of R, and every nonzero ideal of R intersects R nontrivially. Consequently R is prime. (ii) The extended centroid o j R is Z . Proof. (i) is clear. To see (ii), suppose B 4R and $ :B + R is admissible. Now B n R # 0, so, taking some nonzero h in B n R , write $ ( b ) = rimi for suitable r, in R , ai in 2. Writing a, = [(t+hi, Ai)J for suitable A i R and admissible ($,, Ai),let A = ( ( b ) ( A , n ... n A k ) ) ,and let $ be the restriction of $ to A . Since I has been sent to 0 in 8, we see that $ ( A ) G R , and (3, A ) is admissible; thus, writing a = [($, A)] E Z , we see easily that the map [ ( $ , A ) ] + CL yields an identification of the extended centroid of R with 2. QED

x!=

a

(Erickson-Martindale-Osborn). I f x , , . . .,x, are eleTheorem C.5 ments of R that are %independent, then there exist polynomials f ( X I , . . ,Xu!, linear in X I and of degree < 1 in X , , . . . ,Xuas well as elements r , , . . .,r, in R such t h a t f ( x , , r 2 , .. .,r,) # 0 a n d f ( x j ,r,, . . .,r,) = 0,for all j # 1. Proof. Induction on t . Let us assume the theorem is true for ( t - l), and false for t . Thus, letting A = {f(.x2, r 2 , .. ., r,)ll < u < 00, r 2 , . . . ,r,ER, ,fis

330

LAPP. c

NONASSOCIATIVE PI-THEORY

l-linear and deg’l’ 6 1 for all j, and f ( x , r2,. . . ,r,) = 0 for all i > 2) (taking A = R if t = 2 ) , we see easily that 0 # A Q ; moreover, by assumption, if , f ( x z r, 2 , . . . ,r,) = 0. then .f‘(x,. r 2 , .. ., r,) = 0. Hence, defining II/: A -,R by II/(,f(xZ,r 2 , . . . r u ) )= . f ( x , ,r2.. . . ,r,), we see that ($, A ) is admissible; letting a = [($, A ) ] E Z , we have,f’(sl-ax2, r2,. . ., r,) = 0 for all u, all rz,. . .,r, in R , and all 1-linear f ’ with degjj’d 1 for all j such that f ( x i ,r 2 , .. ., r,) = 0 for all i > 2. But this means, by the induction hypothesis, that .yl - a.yz. x 3 , .. .,s,are Z-dependent, contrary to assumption that Y , , xZ,. . . .x,

.

are 2-independent. QED (This proof inspired the proof of Theorem 7.6.10.) Corollary C.6. Suppose xl,. ..,.\;,are given, Z-independent dements of R . There is a polynomial g ( X ,, . . . , xk) (for suitable k ) that is t-linear and has degree 6 1 in each indeterminate, us well us elements r,+ . .. , rk in 8, such that,for all jibetweell 1 and t , g ( x j , ,. . .,y j , , rr+ . . .,r k )= 0 unless j , = ijor all i, in which case g ( . y l . . . . ,s,,r, + . . . ,rk)# 0.

,,

,,

The case t = 1 is Theorem (2.5; the corollary follows easily by induction, using the fact R is prime. QED Theorem C.7. Assume R is prime. I f [ R : Z ] 3 t , then there is a t-normal polynomial that is not an identity of R. Proof.

Take g as in Corollary C.6 and define . f ( X I . . . ’ , d Y k )=

1

( s g n ) y ( X , ,,..., X R I ’ X L +. .L. . . X J.

n ESyrnW

Then f’is t-normal and is not an identity o f R . But deg’fd 1 for all j , sof‘is not an identity of R (for otherwise f would be the sum of multilinear identities of R, and would thus be an identity of I?). QED This is just what we need to get the following major structure theorem for NPI-rings, which generalizes Posner’s original theorem: Theorem C.8.

If R is prime and NPI

of degree t. then

[a : 21 6 t .

This reduces the prime NPI-ring theory to finite-dimensional algebras over fields, which ace well known in most instances. Here is another theorem, modeled after $1.4, with similar proof: Theorem C.9. Suppose R is an arbitrary NPI-ring of degree t and has a t-normal ceritruf polynornial g such that 1 € g ( R ) R . Then there is a 1 : 1 correspondence.frotn {ideals q f R ) to {ideals o f Z ( R ) ) given by A -+ A n Z ( R ) , and A = ( A n Z ( R )IR: also R is ajnitely generated Z(R)-module. Moreover, if1 Eg(R),then R is u.free Z(R)-module ofdimension t .

APP- CI

Nonassociative PI-Theory

33 1

One can define central localization (with respect to a submonoid of Z(R)). in the obvious way, i.e., by localizing R as a Z(R)-module and defining multiplication (rls; ‘ ) ( r 2 s ;I ) = (rl r2)(s1sJ1. Then one can combine the above results to prove (cf. Rowen [78b, Theorem 4.51): Corollary C.10. If R is a pririze NPI-algebra with I-normal central polynomial, then the algebra of central quotients of R is simple (i.e., without proper nonzero ideals) andjinite dimensional over its center.

Every semiprime PI-algebra is NPI and, if recent work of E. Zelmanov (dueto appear in Algebra i Logika) is correct, every semiprime Jordan PI-algebra is a subdirect product of a semiprime N P I Jordan algebra and prime Jordan algebras whose algebras of central quotients are generically algebraic of degree 2. Nevertheless, the NPI condition is sometimes difficult to verify, and we shall sketch a different structure theory motivated by 4$1.5, 1.6. Kaplansky Classes of Algebras

For simplicity, we restrict our attention to power-associative algebras. Thus there is no ambiguity in writing rk to denote r multiplied by itself k times. An ideal of nilpotent elements is called nil; there is a unique maximal nil ideal, called the nilradical NiI(R). An ideal A is quasi-regular if for each element a in A , R has no proper left ideals containing (1 -a) ; Jac(R) is the sum of all quasi-regular ideals. Likewise, A is quasi-invertible if for each element a in A there exists a’ in R such that a’(1 - a ) = 1. If R is power associative and Nil(R) = 0, then one can modify the proof of Amitsur’s theorem to show that R[A] has no nonzero quasi-invertible ideal. Unfortunately, for nonassociative rings one cannot conclude that the Jacobson radical is 0. Thus, one introduces the following conditions for a PI-theory built on central polynomials to work : Definition C . l l . A class of algebras % is Kaplansky if it satisfies the following conditions for R 6 % :

(i) if R is prime and Nil(R) = 0, then R has an R-stable central polynomial ; (ii) R/Nil(R)E%,and R/NiI(R) is a subdirect product of prime algebras in % that have nilradical 0; (iii) R[A] E % ; (iv) if Nil(R) = 0, then Jac(R[A]) = 0. This class was called “Kaplansky” because Kaplansky’s theorem is at the heart of associative PI-theory, but the class could just as well have been called “Formanek” [for (i)] or “Amitsur” [for (iv)]. Conditions (ii) and (iii)

332

NONASSOCIATIVE PI-THEORY

LAPP. c

are technical conditions introduced to assure % has enough algebras to work with, and are usually very easy to verify. In the presence of (i), (ii), and (iii), condition (iv) is equivalent to each of the following two conditions for all R in $9 (cf. Rowen [78b, Theorem 3.171): (iv’) if R is prime and Nil@) = 0, then the algebra of central .quotients of R is simple; (iv”) if Nil(R) = 0, then R[A] is a subdirect product of simple algebras. One can prove formally, without difficulty, that in a Kaplansky class, if Nil@) = 0, then every ideal of R intersects Z ( R ) nontrivially (cf. Rowen [78b, Theorem 3.31). Application: Alternative Rings Let us see how these considerations relate to alternative rings. The best way to study alternative rings is to verify additional identities; in fact, we shall accumulate many identities over the next few pages. Remark C.12. [X,,X,,X,]-(sgn)[X,,,X,,,X,,] is an identity of every alternative ring, for every n E Sym(3). (Indeed, just multilinearize [ X , , X , , X , ] and [ X I ,X , , X , ] . ) Thus [ X I ,X , , X , ] is an identity of every alternative ring. Proposition C.13 (Moufang identities). Every alternative ring satisfies the Jollowrng three identities: (i) ( ( X , X , ) X , ) X ,- X l ( X 2 ( X l X 3 ) ) ; (4x 3 ~ x 1 ~ x ~ x 1 ~ (iii) ~ ~ ( X l~ ( X~2 X~ 3 )3) X-~,( X~, X~L ) ~( X 23 X ~l ) . ~

Proof.

(i)

For all a, b, .K in an alternative ring we have

((ab)a).u-a(b(a.\-I)= [ a b , ~ , x ] + [ ~ , b , U . = K ]- [ Q , ~ ~ , . Y ] - [ u , u . Y , ~ ] = - (a(ab)).y+a((ab).u)- (a(a.\-))b +a((a.y)b) = - (a2b).u- (a2s)b+a((ab).y + (a.y)b) = -a2(b.u)-[a2, b, - Y ] -a2(sb)-[a2,.u, b] +a((ab).u (ax)b) = a( - a(bx)- a(xb) (ab)x (ax)b) = ~ ( [ ab,, X ] + [ a , X, b ] ) = 0.

+

+

+

(ii) Symmetric proof to (i). (iii) For all a, b. .K we have (u(ab)).u-(xa)(b.u)= -[.Y, a, b].u+[.ua, b, x ] = - [.u, a, b ] .~ (b(W))X + ((b.u)a).u[using (ii) and the fact [.Y, a, x ] = 01 = (- [.u, a, b] [ b , X, ~ J ] ) . K = 0. QED

+

It is a bit neater to prove identities in thefiee alterriarive ritig Z ’ { X } / P , where .f is the (nonassociative) T-ideal generated by [ X I , X I ,X , ] and [XI, X , , X,].We shall let H’{R}denote the free alternative ring.

1

~

APP. CI

Nonassociative PI-Theory

333

Corollary C.14. [X3,X1X2,X1]+[X3,X1,X2]X1 is an identity of every alternative algebra. Proof.

In the free alternative ring, [x3

9

X 1 X 2 , X I ] = ( x 3 ( X , X 2 ) ) x ,-

=

-[x39

=

-[Lxl,XZ]xI

XZ]xl

X~((X~XZ)X~) + ( ( x 3 X , ) x , ) X ,-x3(x,(X2x,))

by the second Moufang identity. QED

Proof.

Apply the multilinearization step A 1 4 to Corollary C.14. QED

We are ready for a key result on alternative rings, called “Artin’s Theorem.” Any alternative ring generated (over its center) by two Theorem C.16. elements is associative. Proof. It suffices to prove Z’{X1,X2} is associative; since the associative identity is multilinear, it suffices to prove that [f1,f2,f3] = 0 for all monomialsf,,f,,f, of Z ’ { X , , X2}.This is obvious unless deg(f;-) 2 1 for 1 < i < 3. We proceed by induction on deg(fi). By induction, we may ji€{ 1,2}. Thus, write fi = ( X j i h i )for suitable monomials hi in Z’{X1, two ji are the same and by Remark C.12, we may assume j , = j , = j . So, using Corollary C.15, we have

x:=

[,/;3./;7,/3]

= [ X j k ~ Xjil2,,/3] ,

x;]

x2>,

xj,

-[Xjhl,./3h2, - [ x j / l l , h2]./3 -[Rjhl,,/3, / l Z ] X j (by induction) = - [ X j h , , . J 3 h , , Xj] -0-0 (by Corollary C.14) = [,/;/i2, Xjhl, Rj] = -[,fib,, X j , =0 (by induction). QED =

hl]xj

Thus we can apply associative PI-t heory directly to alternative algebras in some striking situations. Theorem C.17 (Kaplansky). I f R is an alternative ring satisfying the identity [ X I , X2Im,then {nilpotent elements o f R ) is a nil ideal. Proof. Case I . R is associative. Passing to R/Nil(R), we may assume Nil(R) = 0. Assume s # 0 is nilpotent in R. Then for some k, x k = 0 and x k - ’ = 0. Replacing .Y by . y k - l , we may assume x 2 = 0 and Y # 0. Now, for

334

NONASSOCIATIVE PI-THEORY

LAPP.

c

any r in R , (r.Y)"'+' = r s [ r , .v]" = 0, so R.Y is a nil left ideal of R. But for associative PI-rings, the nilradical contains all nil left ideals, contrary to NiI(R) = 0. Case 11. R is arbitrary. Pick s,y E R. Then by Artin's theorem s and y generate an associative subring R x y . If s is nilpotent then srNil(R,,) by Case I, so xy is nilpotent and yx is nilpotent; moreover, if y is nilpotent then (x+y)ENil(R,,) is nilpotent. Thus, { x ~ R l xis nilpotent)aR, as desired. QED Corollary C.18. Any alternative ring with nilradical 0, satisfying the identity [ X , , XZ]", is cornmutarice. Remark C.19. algebra :

The following identity holds in the free nonassociative

xi [x2,x,, X4] + [ X I , x2,

x.2,x,, x4]+ [xi,xz x,,x4] &x4].

[xi

-[XI

%

x2,

Definition C.20. N ( R ) = :.\

E Rl[.\,

Remark C.21.

N ( R ) is a subalgebra of R. If R is alternative and

R, R ]

=

[ R , S, R ]

=

[R, R, .Y] = 0;.

[x, R , R] = 0, then YEN(R).

Remark C.22. For any algebra R and any a, 6, c in R and x E N(R), we have the following four equations: x [ a , 6, c] = [xu, 6, c ] , [ax, 6, c] = [a,x6,c], [a, hx,c] = [a, h, xc], and [a, 6, CX] = [a,b,c]x. (Use Remark C 19.)

Here is yet another identity, whose proof follows easily by applying the multilinearizing step A , 4 to the third Moufang identity (cf, Jacobson [68B, Equation (34) and p. 201): Proposition C.23.

CX4,[X,,

Eaery alterriative ririg satisjes the ideritity

xz, x,II+cx,x29x,, X,I+[X,X,,

XI, X,]+[X,X,,

x2, X,].

Corollary C.24. I f R is alternative and x e N( R ) , then [x, [ R , R , R]] are now ready to develop more properties of N(R).

= 0. We

Proposition C.25. Proof.

[ N ( R ) ,R]

G

N ( R ) f o r every alternatiite ring R .

For all . Y E N ( R ) and r, a, 6 in R, we have

[a, 6, [s,r]]

6, r . ~ = ] [ s r , a, 61 - [ a , 6, r . ~ ] = ~ [ ra,, 61 - [a, 6, r ] s (using Remark C.77) = [.Y, [ a , 6, r]] = 0 by Corollary C.24. QED = [a, b, .xr] - [ a ,

APP. CI

If R

Corollary C.26. Ann",, R] R.

a

is alternative and N ,

Proof. Let A = Ann",, N , , r , , r2 in R , we have

,

335

Nonassociative PI-Theory

( a r )[ s,r2]

= a(r

,[

.Y,

c N ( R ) , then itfollows that

clearly a $-module. For all a in A, x in

R],

,

,

r,] ) = a(r (.vz ) - r (r2 s))= a(r ( s r 2 ) )- a( (r r , 1s) a ( [ r l ,s]r2) = (a[rl, s])r2 = 0,

= a ( r l ( r r 2 ) )- a ( . y ( r ,r 2 ) ) =

and ( r , a ) [ x ,r2]

=

= 0. Thus

r , ( a [ x ,r,])

A d R. QED

We can use Corollary C.26 to gain decisive information about prime alternative algebras. Lemma C.27. all nonzero r in R.

I f R is prime alternative and 0 # A

R, then Ar # 0 for

Proof. We shall show in fact that Ann', A 4 R, and thus is 0 since R is prime. Clearly it is enough to show that if A r , = 0 then A ( r , r ) = 0 and A ( r r , ) = 0 for all r in R . But for all a in A , a(rl r ) = ( a r l ) r - [a, r l , r] = [ r , a, r,] = (ra)rl- r ( a r l ) = 0 (since ra E ,4); likewise a ( r r l )= ( a r k l [a,r,r,] = O+[r,a,r,] = 0. QED

Lemma C.28.

For all ri in R and x in N(R), [ [ I - , , r 2 , r 3 ] , r4, r5][r6, x]

= 0.

Proof. Using the fact Corollary C.24, we have "'19

r29

r3]7r49 r5][r6? .y]

[r6,.Y]EN(R),

as well as Remark C.22 and

[[rl,r2?r3][r6>.y],

r4, r5]

= - [([I1

~

~

r3]-y)r6, 2 9 r 4 r Ys]

= [ [ r l r r 2 , r 3 ] r 6 , . y ~ 4 , r s ] -[(.y[r1,r2,r3])r6,r4,r5] - [-yr47 [ r i , r 2 , r3]rs, r5]

-[.y([rl,r2,r3]r6)?

r49r5]

= .~(-"Y4,[rl,r2,r3]~6rrs] -[[ri,r2,r3]rhrr4,rS])

= 0.

QED

We shall soon see that nonassociative prime alternative rings are better behaved (vis a vis the center) than associative rings. The reason stems from the following very important theorem. Theorem C.29 (Kleinfeld). If'R is prime alternative, then either R is associative or N ( R ) = Z(R). Proof.

Let N

= N ( R ) .It

suffices to prove N

E Z ( R ) , i.e.,

[ R ,N ]

= 0. By

336

LAPP. c

NONASSOCIATIVE PI-THEORY

Lemma C.28 [[R, R, R],R,R] c Ann[R, N].Now if Ann[R, N] # 0 then by Corollary C.26 and Lemma C.27 [R,N] = 0, and so we are done. Thus X4,X,] is an identity of R , i.e., we may assume [[X,,X,,X,], [R, R,R] G N. We claim now that the product of any two associators having a common element is 0; indeed, for all ri in R,by Remark C.22 and Corollary C.14 -

r2, r3I[ri3

1 4 7

151 = [ r 2 ,

=

rir r3][ri,

r4r r5]

=

[ [ r , , ri

9

r3]ri7 r4,r 5 ]

- [ [ r , , rl r3,r,], r4, r 5 ] = 0.

Now take r l , r,, r3 arbitrary in R, and let u = [ r l , r,, r3] EN.For all r in R, using Remark C.19, we have (ur)u = -r1[r2, r3, r]u+[rlr2,r 3 ,r]u-[r,,

r2r3,

r]u+[r,, r 2 . r3r]u = 0

by the above claim. Thus uEAnn[u, R]. But by Corollary C.26 Ann[u, R]a R,so by Lemma C.27 [u, R] = 0, i.e., u E Z(R).Since uRu = 0, we conclude u = 0. Therefore [R, R,R] = 0,so R is associative. QED

,,

Lemma C.30. [X:,X ,, X 3 ] - X [X X X 3] - [X ideritity of'every ulternative ring. Proof.

1,

X , , X 3] X is an

In an alternative ring,

r1[rl,r2,r3] +['1,r?,r3]r1

= r1[r1,r2,r3]-[r3,r1r2,rl]

(by Corollary C.14)

= ri[ri9r2,r3] +[ri,rirz,r3]

= rl((rlr2)r3-r:(12r3)+ (rtr2)r3-rl((rlr2)r3) = [r:, 12, r3].

Proposition C.31. Suppose R with nilradical 0. Then R is associative.

Proof. 2[ri,

QED is commutatitre and

utterriatitle

First note that for all ri in R,

129r31 = [ ~ I r29 V 1 3 1 - [ r 2 , r1, r3] = -ri(r2r3)

+r2(rir3) = -[I29

r3,rllr

implying 3[r1, r,, r 3 ] = 0. In particular, by Lemma C.30, for all ri, [r:, r,, r3] = 2r1[r1,r2,r3] = -rl[rl,r2,r3]. Let u = [r1,r2,r3]. We shall use Corollary C.14 repeatedly, with the above identities. Now (riU)r2 = -[r?.r2,r3]r2

=

= -ri[ri7r2r3qr2]

= ri(Ur2)7

implying0 = - [ r l . u, r,] = [u, r l , r,], i.e., [[X,, X,,X3], X,,X,]isan identity of R. Next, [ur,,r,,r31 = -[[r:J2J3I7r2J3] = 0, so ~[x,,x2,x31x,.x2~x33

APP CI

Nonassociative PI-Theory

337

is an identity. Thus, by Remark C.19, - [ u , r l , r 2 r 3 ]= +[r1,u,r2r3] = r1[u,r2,r3] + [ ~ 1 , ~ , r z ] ~ , - [ r , u , r 2 , r ~ l + [ ~ l , ~ r 2=, ~03. ] Putting everything together and using Corollary (2.15, we get u2 = [ r l , r 2 , r 3 ] u= - [ r l r r 2 ~ . 3 r ~ ] - [ r l , ~ r 2 , r 3 ] - [ r l , ~ , r 2 ] r 3= 0. Thus, by Theorem C.17, U E NiI(R) = 0. Thus [ X , , X , , X , ] is an identity of R. QED Kleinfeld [53]-Bruck, using an intricate but not difficult argument, verified the identities [ X I ,X , ] [ [ x , , X 2 l 2 ,x , , x41and " x , , x,]', x,, X , ] [ x , , x21, so using Lemma C.30 we have Proposition C.32 every alternative ririg.

(Kleinfeld). [ [ X , , X , ] , , X 3 , X , ] i s an identity of

Putting together the pieces, we have the following crucial result. Theorem C.33. Every nonassociative prime alternative ring with nilradical 0 satisjies the central polynonzial [ X , , X , I 4 .

Thus % = {alternative PI-rings} satisfies conditions (i)-(iii) of Definition C . l l . T o show '% is Kaplansky, it suffices to verify (iv'), which is equivalent to the next theorem. Remark C.34.

Every nonassociative ring satisfies the identity

[ X I ,X 2 , X 3 ] - [ X I , X 3 , X 2 ] + [ X 3 , X I ,X2I + X i [ X 2 9 X 3 ] + [ X i , x3]x2 -[X,Xz, X3l. Theorem C.35. I f R is a prime alternative PI-algebra and NiI(R) = 0, then every nonzero ideal A O f R intersects Z(R) nontrivially. Proof. If R is associative, then this is Theorem 1.6.27, so we may assume R is nonassociative. Thus we are done by Theorem C.33, unless [ a , rI4 = 0 for all elements a in A, r in R. We shall show this implies R is commutative and associative. Fix r in R, and let R' be the subring of R generated by .4 and r. Now [ X 1 , X 2 ] ' is a n identity of R', so by Theorem C.17, [nilpotent elements of R'J R'. Thus, if a , and a, are nilpotent elements of A, a , + a , , r a , , and a , r are nilpotent; since r was arbitrary, we see that {nilpotent elements of A ] is a nil ideal of R, which is thus 0. In particular [ A , R] = 0. We use Remark C.34 twice. First, specializing X 3 to A yields [3A, R, R ] = 0 ; then, specializing X I to A (and X,, X 3 to R) yields A[R, R1 = 0. By Lemma C.27 [ R , R] = 0, so by Proposition C.31 R is associative; Theorem C.35 follows at once. QED

a

Thus {alternative PI-rings) is Kaplansky, and yields a very nice structure

338

NON ASSOCIATIVE PI-THEORY

CAPP. CI

theory. In view of Theorem C.33, the nonassociatioe alternative rings are always PI, and thus are often better behaved than associative (non-PI) rings. This observation was made by Slater in a number of papers, and stimulated my interest in fitting Slater’s results into the PI framework; cf. Exercises 2-9. There is also a very pretty Jordan PI-theory that is too involved to go into here. Let us conclude by remarking that the papers of Shirshov [57a, 57b, 57c] included proofs of his theorems for alternative algebras and special Jordan algebras (assuming the invertibility of 2). EXERClS ES If 3 has an inverse in a commutative, alternative ring R, then R is associative. (Cayley-Dickson-Graves) Suppose Q is an arbitrary quaternion F-algebra with symplectic involution (*) of the first kind. Then the F-vector space R = Q 0 Q can be given multiplication ( q 1 , q 2 ) & , q 4 ) = (qlqo+vq:q,,q,q, + q 2 q 3 , where v is any fixed element of F. R is then an alternative algebra, of dimension 8 over Z ( R ) = F, and is called a C-D algebra. 3. Every C-D algebra is simple. 4. Forevery infinitedomain 4 there isa C-D algebra without zerodivisors.Conclude that ifR isaC-Dalgebra with infinitecenter, then I ( R )has no zero divisors,so, in particular, [XIXJ2is Rcentral. 1.

2.

We shall now use the theorem of Kleinfeld [53] that every simple nonassociatioe, alternative algebra is C-D. 6. If R is prime, alternative, and nonassociative with nilradical 0, then R and M z ( 4 ) satisfy the same ?-variable identities. Every alternative PI-algebra with nilradical 0 satisfies a 2-variable Formanek central polynomial. 7. If A d R and A ‘Z N ( R ) , then A[R, R, R ] = [ R , R , R I A = 0. 8. Suppose R is alternative with nilradical 0, and N ( R ) contains no nonzero ideals of R. Then R is a subdirect product of nonassociative, prime alternative rings having nilradical 0. [X,,X,]’ is Rcentral, and every ideal of R intersects Z(R) nontrivially. 9. Every alternative PI-algebra satisfies a power of a Capelli identity. 10. If R is an alternative ring with nilradical 0 and if, for some r in R, [r, [r. x]] = 0 for all x in R, then 2 r e Z ( R ) .

POSTSCRIPT

SOME ASPECTS OF THE HISTORY Just as world history sometimes is sketched in terms of major events, I will try to give an idea of the trends of PI-history, through its major theorems. An excellent history before the advent of central polynomials is given in Amitsur [74b] (cf. also Jacobson [74]); the reader probably could round this out with the historical remarks made in this book. 1. Kurosch’s problem. The problem of Kurosch [41] was posed as a ringtheoretical analog of Burnside’s problem for groups, and inspired the outstanding progress of Jacobson [45] in describing the structure ofrings. The first positive answer to Kurosch’s question came for algebraic algebras of bounded degree, independently by Kaplansky [46] and Levitzki [46]. (Kaplansky required a sufficiently large base field.) Shortly thereafter, Jacobson observed that every algebraic algebra of bounded degree is PI, and then Kaplansky [SO] found a positive answer for PI-algebras. In 1952, using the Levitzki radical, Amitsur gave an algebraic proof (in Jacobson [64B]), and Levitzki [53] also had an algebraic proof. More generally, one can ask, “If R is PI and finitely generated and integral over Z ( R ) , then is [R:Z(R)] finite?” Shirshov [57a,b,c] answered this affirmatively, but in the West nobody knew Shirshov’s theorem, and the same question inspired much activity in the early 1970s.Upon learning of Shirshov’s work, Procesi [73B] gave a noncombinatorial proof of the theorem, which Schelter [75] generalized to the form used in this book. We used Shirshov’s proof.

2. Amitsur-Levitzki theorem. The famous paper of Amitsur-Levitzki [SO] used an ingenious proof based on reverse induction on the number of idempotents in the arguments. Subsequent proofs went in two diametrically opposed directions. Whereas Schutzenberger and Kakutani (and probably others) viewed the theorem as essentially graph theoretic, using “Eulerian paths” of matric units (leading to proofs by Swan [63] and Passman [77B]), Kostant [58] found a totally different proof by relating PI-theory to better known branches of algebra; Kostant’s proof contained seeds of a short proof by Formanek [75P] using properties of traces. Razmyslov [74a] gave the trace argument used here, in compact form. (Formanek explained the details to me.) Rosset [76] found another easy proof. 3 39

340

SOME ASPECTS OF THE HISTORY

[Postscript]

3. Posner’s theorem. Posner [60] proved directly that a prime PI-ring is Goldie, and then appealed to Goldie’s theorem ;parts of the paper have rough spots. Using ultraproducts (cf. 47.5),Amitsur [67] reworked Posner’s theorem rigorously ; later, Amitsur [72b] presented a short, elegant proof. The additional fact, that the ring of central quotients is simple, became apparent from the application of Formanek’s central polynomials to the Artin-Procesi theorem; a direct proof was given by Rowen [73]. The best proof uses t-normalcentralpolynomials,andisimplicitin Amitsur [7S] and Rowen [75P]. 4. Artin-Procesi theorem. The original paper by Artin [69] was for algebras over a field, and is a landmark in PI-theory, remarkable all the more since the only PI-theory then available was essentially Kaplansky’s theorem and Posner’s theorem. Procesi [72a] gave a purely algebraic proof, removing the restriction of algebra over a field; simplifications using central polynomials were found by Amitsur [73]. Rowen [74a], and Goldie [76]. The breakthrough came independently and simultaneously by Amitsur [75] and Rowen [75P] ; there, the basic properties of Azumaya algebras were verified from the A,-ring conditions. A further refinement was found by Schelter. All these proofs were that A,-rings are Azumaya. The other direction has a very recent, easy proof, worked out by Amitsur with a few details supplied by Rowen. 5. 7he noncrossed product. The crowning achievement so far in PI-theory (which I doubt is surpassable) is Amitsur [72a]. The idea of confronting two incompatible crossed products to show UD(4,n) is not a crossed product underlies Amitsur’s result, as well as subsequent refinements (obtained through sharper arithmetical results) by Schacher-Small[73], Amitsur [74a], Jacobson [7SB], Fein-Schacher [76], and Risman [77b]. Saltman [78a] has recently shown that division algebras of index n and exponent < n can also be noncrossed products.

The first two digits appearing after an author's name indicate the year (+1900) of publication; for example, Amitsur [74b] appeared in 1974. The symbol " B denotes " b o o k and "P' denotes "preprint"; for preprints, the indicated year is the year of distribution. For easier reference, the books are listed first, and then a number of "prehistoric" articles (before 1948, the year of Kaplansky's theorem). Then follows a lengthy list of all post-1948 articles which, t o my knowledge, pushed forward the subject matter of this book. Occasionally an article has been deleted if it was superseded later by the same author.

Books [hlB] Albert, A. A., Structure o/ ,4lgebras. Amer. Math. SOC.Colloq. Publ. 24, Providence, Rhode Island. [57B] Artin, E., Geometric Algebra. Wiley (Interscience), New York. [72B] Bourbaki, N., Commutariue Algebra (in English). Addison-Wesley, Reading, Massachusetts. [hSB] Cohn, P. M., Universal Algebra. Harper, New York. [77B7 Cohn, P. M., Skew-Field Consrructions, London Math. SOC.Lecture Notes 27. Cambridge Univ. Press, London and New York. [71B] Demeyer, F. and Ingraham, E., Seperable Algebras over Commutative Rings, Lecture Notes in Math. 181. Springer-Verlag, Berlin and New York. [70B] Dieudonne, J. A,, and Carrell, J. B., Inmriant Theory 0ld.and New, Academic Press, New York. [75B] Goldie, A. W., Rings with Polytmniul Identities. Carleton University Lecture Notes, Ottowa. [67B] Hall, M., Cornbinatorial Theory. Ginn (Blaisdell), Waltham, Massachusetts. [64B] Herstein, I. N., Topics in Algebru. Xerox, Lexington, Massachusetts. [68B] Herstein, 1. N., Noncornmuratiiv Rings, Carus Monograph 15 (Math. Assoc. Amer.). Wiley, New York. [76B] Herstein, I. N., Rings with Iiiuolution. Univ. of Chicago Press, Chicago, Illinois. [43B] Jacobson, N., Theory of Rings, Math. Surveys 2. Amer. Math. SOC.,Providence, Rhode Island. [64B] Jacobson, N., Structure of Rings. Amer. Math. SOC.Colloq. Pub. 37, Providence, Rhode Island. [68B] Jacobson, N., Structure and Representof ions of Jordan Algebras. Amer. Math. SOC. Colloq. Publ. 39, Providence, Rhode Island. [75B] Jacobson, N., PI-Algebras: An Introduction, Springer Lecture Notes in Math. 441. Springer-Verlag, Berlin and New York. [70B] Kaplansky, I., Commutariue Rings. Allyn and Bacon, Boston, Massachusetts. [72B] Kaplansky, I., Fields and Rings. Univ. of Chicago Press, Chicago, Illinois. [55B] Kelley, J., General Topology. Van Nostrand-Reinhold, Princeton, New Jersey.

34 1

342

BIBLIOGRAPHY

[74B] Knus, M., and Ojanguran, M., Theoriu de la Descente er Algebrcis J.4:urnapa. Lecture Notes in Math. 189. Springer-Verlag, Berlin and New York. [66B] Lambek. J., Lecrures on R i ~ i g sand Modules. Ginn (Blaisdell), Waltham, Massachusetts. [65B] Lang, S.. 4lgrbrti. Addison-Wesley, Reading, Massachusetts. [74B] McDonald, H. K.. Finite Rings wirh Identity, Pure and Appl. math. Vol. 28. Dekker, New York. [73B] Maltsev, A. I., AIgebruic Svsrems. Springer-Verlag, Berlin and New York. [72B] Miller, W. Jr.. S,~rnmetryGroups and their Applications. Academic Press, New York. ~ Wiley (Interscience), New York. [62B] Nagata, M., L O CRings. [67B] Neumann, H., krtrieries ?/Groups. Springer-Verlag, Berlin and New York. [77B] Passman, D. S., 4lgebruic Structure qJ’Group Rings. Wiley. New York. [73B] Procesi, t.,Rings wirh Polynomial Identities. Dekker, New York. [75B] Reiner, I., Moximal Orders. Academic Press, London and New York. [63B] Weiss, E., Algebraic Number Theory. McCraw-Hill, New York. [46B] Weyl, H., The ( ’lossical Groups. their Intiariants arid Representations. Princeton Llniv. Press, Princeton. New Jersey. and Samuel, P., Conimutatirie Algebra. Van Nostrand-Reinhold, Princeton, [58B] Zariski, 0.. New Jersey and Springer-Verlag. Berlin and New York. Early Papers (before 1948) Dehn, M., ”Uber die Grundlagen der projektiven Geometrie und allgemeine Zahlsysteme,” Math. Ann. 85, 184-193. Eilenberg. S., and MacLane, S., “Cohomology theory in abstract groups.” .4nn.Math. 48, 51-78. Hall, M.. ”Projective planes,” Trans. Amer. Math. Soc. 54, 229-277. Jacobson. N.. “Siructure theory for algebraic algebras of bounded degree.” .h. Math. 46, 695-707. Jennings,S. A.,‘Y )n rings whoseassociated Lie rings areniIpotent,”Bull. Amer. Math. So1,.53, 593-598. Kaplansk y. I..“On a problem of Kurosch and Jacobson,”Bull. Amer. Marh. SOC.52,496-500, Kurosch. M.. “Kingtheoretische Probleme, die mit dem Burnsideschen Probleme uber periodiche Gruppen in Zusammenhang stehen,” Bull. Acad. Sci. U R S S Ser. Math. 5, 223-240. Levi, F. W.. ”On skewfields of a given degree,” J. Indian Math. Soc. 11, 85- 86. Levitzki. J., “On a problem of Kurosch,“ Bull. Amer. Math. Soc. 52, 1033- 1035. Littlewood, D. F., “Identical relations in algebra,” Proc. London Math. Soc. 32, 311-320. Richardson, A. K., “Equations over a division algebra,” Messeng. Math. 57, 1-6. Wagner. W., * Uber die Grundlagen der projektiven Geometrie und allgemeine Zahlsysteme.” I 113, 518-567. Wedderburn. J. H. M., ”On division algebras.” Trans. .4mer. Math. Soc. 22, 129-135. Main Bibliographi (selected articles)+ Albert, A. A [%!I “On simple .Ilternattve rings,” Caiiad. J. Muth. 4, 129- 135.

t Works of each author ordered by year of pubfieatinn. Page numbers o f the Russian journals Algebra i Logiku (Alyebra and Logic), Math. USSR-lsi>.,Math. USSR-Sb., and Siberim Math. J. are given in the English translation whenever possible.

Bibliography

343

Amitsur. S. A. . Soc. 2, 538-540. “Nil PI-rings,” Proc. A t ~ e r .2ltr/h. “An embedding of PI-rings,” Proc. An7rr. Muth. Soc. 3, 3-9. “Identities of PI-rings,” Proc. .4nlrr. M a t h . Soc. 4, 27-34. “Applications to a polynomial identity,” Riceon Lrtnoteniur. 7, 30-32. “On rings with identities,” J . h i d o r 7 Mtrth. Soc. 30, 464-470. “The T-ideals of the free ring,” J . Lontloii Moth. See. 30, 470-475. “Identities and generators of matrix rings,” Bull. Rrs. Council lsruel, Sect. A, 5, 5- 10. “Some results on central simple algebras,” Ann. M a t h . 63, 285-293. “A generalization of Hilbert’s Nullstellensatz,” Proc. Anrru. M u t h . Soc. 8, 649~-656. “Rings with a pivotal monomial,” Proc. h e r . Murh. Soc. 9, 635-642. “Finite dimensional central division algebras,” Proc. Amer. Math. Soc. 11, 28-31. “Groups with representations of bounded degree,” Illinois J . M a t h . 5, 198-205. “Generalized polynomial identities,” Trtrrrs. Amrr. M u t h . Soc. 114, 210-226. “Associative rings with identities.” S o w Aspeer., of Ri)ig Theory, C.I.M.E. Conference, I1 Circlo, 1-48. “Rational identities and applications to algebra and geometry,” J . Algebru 3, 304-359. “Nil semigroups of rings with a polynomial identity,“ Nagoyu Murk. J . 27, 103-1 I I . “Prime rings having polynomial identities with arbitrary coefficients,” Proc. L o d o n M a t h . Soe. 17, 470-486. “Identities in rings with involution,” Isruel J . Math. 7, 63-68. “A noncommutative Hilbert basis theorem and subrings of matrices,” Trurrs. Amer. MtJl/7.Soc. 149, 133- 142. “A note on PI-rings,” Isrtrel J . Mtrrh. 10, 210-21 1. “Embeddings in matrix rings.” Pttcjfic J . Muth. 36,21-29. “On central division algebras,” I.srtrel J . Muth. 12,408-410. ”On rings of quotients,” Symp. Mu/. .rh. Inst. Naz. Alt. Matematica, Bolgna. “Polynomial identities and Azumaya algebras,” J . Algebra 27, 1 17-1 25. “The generic division rings,” Isrtrrl J . Murh. 17, 241-247. “Polynomial identities,” Isruel J . Mtrth. 19. 183- 199. “Identities and linear independence.” Isruel J . M a t h . 22, 127-1 37. “On a central identity for matrix rings,” J . London M o t h . Soc. 14, 1-6. “Alternating identities,” Proc. Ohio Unit Ring Theurq. Con/: pp. 15-61. Dekker, New York. Amitsur, S. A., and Levitzki, J. [50] “Minimal identities for algebras,” Pro(-. ,4nlrr. Math. Soc. I. 449-463. “Remarks on minimal identities for algebras,” Proc. Amer. M a t h . Soc. 2,320-327. [SI] Amitsur, S. A,, and Procesi, C. “Jacobson rings and Hilbert algebras with polynomial identities,” ,4m. Mut. Purrr [66] Appl. 71,(4) 67-72. Amitsur, S. A., Rowen, L. H., and Tignol. J. P. “Division algebras of degree 4 and 8 with involution,” Israel J . Mtrth. [79] Amitsur, S. A,, and Saltman, D. “Generic abelian crossed products and p-algebras,” J . Algebra 51, 76-87. [78] Amitsur, S. A,, and Small, L. 1771 “Prime PI-rings,” Bull. Artreu. M u r k Soc. 83, 249-251. [78P] ”Prime ideals in PI-rings.” Ananin, A. 2. [77] “Locally finite approximable and locally representable varieties of algebras.” Algrbru i Logiku 16, I 16. -

344

BIBLIOGRAPHY

Armendariz, E. P. [76a] “Regular rings with a polynomial identity,” 4nn. Math. Con/: vth, Unic. Southwestern Imuisianu, Res. Ser. 41,33-39. [76b] “On semiprime PI-algebras over commutative regular rings,” Pacific J . Math. 66, 23-27. Armendariz, E. P.. and Fisher, J. W. “Regular PI-rings,” Proc. .Imer. Math. So<. 39,247-151. [73] Armendariz, E. P., Fisher, J. W., and Snider. R. “On injective and surjective endomorphisms of finitely generated modules,” Conrm. [78] in Algebra 6,059-672. Armendariz, E. P.. Fisher, J. W., and Steinberg, S. A. “Central localizations of regular rings,” Proc. Amer. Math. Soc. 46, 315-331. [74] Armendariz, E. P.. and Steinberg, S. A. “Regular self-injective rings with a polynomial identity.” Trans. Amer. Math. Soc. [74] 190,417-425. Artin, M. “On Azumaya algebras and finite dimensional representation of rings,” J. Algebra [69] I I . 532-563. Auslander, M., and Goldman, 0. “The Brauer group of a commutative ring.” Trans. Amer. Math. Soc. 97, 367-409 [60] Azumaya, G. [S 11 “On maximally central algebras,” Nagoya M&h. J. 2, 1 17--1 SO. Bang, C. M., and Mandelberg, K. “Finite basis theorem for rings and algebras satisfying a central chain condition,” J . [75] Algebra 34. 105- I 13. Barbaumov, V. E. “Automorphisms and PI-algebras,” Upsehi Mat. Nauk. 28,231-232. [73] [75a] “Algebraic automorphisms and PI-algebras,” Math. USSR-Sb. 26, 55-69. [75b] “Groups of aulomorphisms and PI-algebras,“ Siberian Math. J. 16,353- 359. Bahturin, Ju. A,, and Ol’sanskii, A. Ju “Identical relations in finite Lie rings,” Math. USSR-Sb. 25, 507-523. [7S] “Solvable just-non-Cross varieties of Lie rings,” Math. USSR-Sb. 29, 345-358. [76] Baxter, W. E., and Martindale, W. S. “Rings with involution and polynomial identities,” Canad. J . Math. 20.465 -473. [68] Beidar, L. I., and Ten, V. D. “On the local finiteness of some PI-algebras,’’ Siberian Math. J. 18,663-666. [77] Belluce, L. P., and Jain, S K. “Prime rings with a one-sided ideal satisfying a polynomial identity,” Pact& J. [68] Math. 24,421- 424. Bergman, G. M. [68P] “Skew fields of rational functions, alter Amitsur.” [70] “Skew fields of noncommutative rational functions, after Amitsur,” Seminaire Schutzenberger-Lentin-Nivat Annee 1967/70, No. 16, Paris. [74] “Some examples in PI ring theory,” Israel J. Math. 18, 257-277. [74aP] “Lifting prime ideals t o extensions by centralizing elements.” [74bP] “More on centralizing elements.” [75P] “A lemma of Razmyslov, and central polynomials.” [76] “Rational relations and rational identities in division rings I, 11,” .1. Algebra 43, 252-297. Bergman, G., and Small. I,.

Bibliography

345

“PI degrees and prime ideals.” J . .4lgehra 33,435462. [75] Berman, E. “Tensor products of polynomial identity algebras,” Trans. Amer. Math. Soc. 156, [71] 259-271. “Group rings, matrix rings, and polynomial identities,” Trans. Amer. Math. Soc. 172, [72] 241-248. “Matrix rings over polynomial identity rings 11,” Pacijc J . Math. 58, 37-41. [75] Blair, W. D. “Right Noetherian rings integral over their centers,” J . Algebra 27, 187-198. [73] Bloch, S. “Torsion algebraic cycles, K,, and the Brauer groups of function fields,” Bull. Amer. [74] Math. SOC.80,941-945. Braun, A. [77] Doctoral dissertation, Univ. of Chicago, Chicago, Illinois. [77P] “Affine PI-rings and their generalizations.’‘ [78P] “A characterization of prime Noetherian PI-rings and a theorem of Mori-Nagata.” Burgin, M. S. [74] “Schreier varieties of linear Q-algebras,” Math. USSR-Sb. 22, 561-579. Cauchon, G . [74] “Sur L’intersection des puissances d u radical d’un T-anneau noetherien,” C . R . Acad. Sci. Paris (Ser A ) 279,91 93. [76] Anneaux premiers, noetheriens, a identites polynomiales, Bull. Soc. Math. Frarice 104.99-1 11. [77] Doctoral dissertation, Univ. of Paris, Centre d’Orsay. Chamarie, M. [77] “Maximal orders and R-maximal orders,” C . R . Acad. Sci. Paris Ser. A 285, A989-A992. Chamarie, M. and Maury, G . “Un theoreme d e I’ideal a gauche principal dans les anneaux premiers, noetheriens. a [78] identite polynomiale dont le centre est un domaine de Krull,” C . R . Acad. Sci. Paris Ser. A 286, A609-A611. Chatters, A. W. “PI-rings with zero singular ideal,” J . LonJon Math. Soc. (2),6,629-632. [73] Cohn, P. M. [72] “Generalized rational identities,” Ririg Theory (R. Gordon, ed.), pp. 107-1 17. Academic Press, New York. DesMarais, P., and Martindale, W. S. 111 “Primitive rings with involution and pivotal monomials,” Israel J . Math. 22, [75] 118-126. Dilworth, R. P. “A decomposition theorem for partially ordered sets,” A m . Math. 51, 161-166. [SO] “Abstract commutative ideal theory,” Pacific J . Math. 12,481-498. [62] Dorofeev, G. V. “Properties of the join of varieties of algebras, Algebra i Loyika 16, 17-26. [77] Drazin, M. P. “A generalization of polynomial identities in rings,” P r w . , h e r . Math. Soc. 8, [57] 352-361. Eisenbud, D. “Subrings of Artinian and Noetherban rings,” Math. Ann. 185, 247-249. [70] Erickson, T. E., Martindale, W. S. 111, and Osborn, J . M.

346

BIBLJOG RAPHY

[75] “Prime nonaslociative algebras,” Pacrfic J . Mtrilr. 60, 49-63. Fein, B., and Schacher. hl. ”Galois groups and division algebras,“ J. 4lyebru 38, 182--191. [76] Fields, K. L. “Examples of orders over discrete valuation rings,’’ Mrrrh. 2. 11 1. 126-130. [69] Fisher, J. W. “Structure of semiprime PI-rings.“ Prrr. Amrr. Murh. Soc. 39,465-467. [73] Fisher. J. W.. and Rowen, L. H. [74] “An embedding of semiprime PI-rings,” Pacific J . Math. 52, 369 -375. Formanek, E. [72] “Central polynomials for matrix rings.” J. Algebra 23, 129-132. [73] “Faithful Noelherian modules,” Proc. Amrr. Math. So<.41, 381-383. “Noetherian PI-rings,” Comm. Algebrrr l,, 79%86. [74] [75P] ”A short prool of the Amitsur- Levitzki theorem.” [76] Oral communication. [78P] “The center of the ring of 3 x 3 generic matrices.” Formanek, E., and Jategaonkar, A. [74] “Subrings of Noetherian rings,” Proc. AmtFr. Math. Soc. 46, 181-186. Gavrilov, M. B. [68] “Varieties o f avsociative algebras,” C.R. Acud. Bulyare. Sci. 21, 989-992 (in Russian) [69] “T-ideals with the element [xi, x2, x3. x4, [xg, x6]],” Bulgar. Akud. Naut. Otdel. Ma1 Fir. Nauk. Iru Mat. Inst. I I . 269--271 (in Russian). [69] “On certain Tideals in free associative algebras,” AIqebra i Loyikrr 8.97F99. [70] Doctoral dissertation, Univ. Sofia. Gavrilov, M. B.. Davidot, L. I., and Tonov, 1. K. “Some remarhs on PI-algebras,” Anrruaire Uriiu. Sofia Far. Math. 64, 277-291 (in [71] Russian). Glassmire, W. “A note on noncrossed products,” J . Algebra 31, 206-207. [74] Goldie, A. W. “The structure of prime rings under ascending chain conditions,” Proc. London [58] Math. SOC.(3) 8, 589-608. “Semiprime rings with maximum condition,” Proc. Lotidoti Murh. Sot*. (7) 10, [60] 20 1 220. “A note on prime rings with polynomial identities,” J. London Mark. Soc. 1. [69] 606-608. ”Azumaya algebras and rings with polynomial identity,” Moth. Prnc Camhridqe [76] Phil. Sos. 79. 303-398: 80, (1976), 383 -384. Golod, E. S. 1641 “On nil algebras and finitely approximable p-groups,” 1 : ~.4kad. Nauk. SSSR. Srr. M a t . 28.273-276. Golubchik, I. Z., and Mikhalev, A. ”A note on varieties of semiprime rings with semigroup identities,” J . Algebrir 54, [78] 42-45. Goodaire, E. G. “Algebras with a diagonable subspace whose centralizer satisfies a polynomial [77] identity. Cunutl. J. Math. 29, 277-33. Goodearl 1731 “Prime ideals i n regular self-injective rings.“ Cartad. J. Math. 25, 829-839. Guseva. I. L. ~

Bibliography

347

“On tensor products of PI-algebras,” Siberiuri Mutb. J . 11, 58-65, [70] Heinicke. A. G. “A remark about noncommutative integral extensions,” Cariud. M a t h . Bull. 13, [70] 359-361. Herstein, I . N. [65] ”A counter-example in Noetherian rings,” Proc. Nar. Acad. Sci. U.S.A. 54, 1036-1037. “Special simple rings with involution,” J . ,4lgebru 6, 369-375. [67] “Notes from a ring theory conference,’’ C B M S Regionul Car$ Ser. Mutb.. p. 9, [70] American Mathematical Society, Providence, Rhode Island. Herstein, I . N., and Small, L. “Regular elements in PI-rings.” Pticjfjc J . M a t h . 36,327-330. [71] “The intersection of the powers of the radical in Noetherian PI-rings,” Israel J . [73] M a t h . 16, 176- 180. Higman, G. “Ordering by divisibility in abstract algebras,” Proc. Loridon Math. Soc. 2, 326-336. [52] [56] “On a conjecture of Nagata,” Pntc. Curribridge Pbilos. Soc. 52, 1-4. Hoechsmann, K . ”Lifting ideals in noncommutative integral extensions,” Cnmd. M a t h . Bull. 13, [70] 129- 130. Hutchinson, J . ”Eulerian graphs and polynomial identities for skew-symmetric matrices,” Cunud. J . [75] M a t k . 27, 590-609. Ilamed, Y. [771 “A characterization of the standard polynomial of even degree,” Group Theoretic Methods iti Physics (Pruc. / ~ r r . Colloy., < t / i ) ,Academic Press, New York. Isaacs, I . M., and Passman, D. S. .‘G~oupswith representations of bounded degree,” Catzud. J . Math. 16,299-309. [64] Jacobson, N. ~ Magid, and Smith, eds), pp. 1-30, “PI-algebras,” iri Ring T b c o r ~(McDonald, [74] Dekker, New York. “Some recent developments in the theory of PI-algebras,” Uniiwrstrle Algebrieji urid [76] Theorie der Rudikale, (Studien zur Algebra und ihre Anwendungen, Band I, pp. 17-2 I . Akademie-Verlag, Berlin. Jain, S. K. “Polynomial rings with a pivotal monomial,” Proc. Anier. M a t h . Sue. 17,942-945. [66] “Prime rings having a one-sided ideal with polynomial identity coincide with special [71] Johnson rings,” J . Alyebrri 19. 125- 130. [72P] “Rings with a polynomial identity.” Jain, S. K., and Singh, S. 1691 “Rings having one-sided ideals satisfying a polynomial identity,” Arcb. Muth. 20, 17-23. James, G . “A note on the T-ideal generated by S ,[x,. x2, x.,].” Israel J . M u r k 29, 105-1 12. [78] Jategaonkar, A. V. “A counterexample in ring theory and homological algebra,” J . Algebra 12,418-440. [69] [74a] “Jacobson’s conjecture and modules over fully bounded Noetherian rings,” J . Algebra 30, 103-121. [74b] “Relative Krull dimension and prime ideals in right Noetherkan rings,” Cornni. Algebra 2,429-468.

348

BIBLIOGRAPHY

“Principal ideal theorem for Noetherian PI-rings,” J. Algebra 35, I7--22. “Certain injective are Artinian:’ Noncommutative Ring Theory (Kent State), Springer Verlag Lecture Notes 545, pp. 128-1 39. Springer Verlag, Berlin and New York. Jensen, C. and Jondrup. S. “Centers and fixed-point rings of Artinian rings,” Math. Z. 130, 189- 197. [73] Kaplansky, I. “Rings with a polynomial identity,” Bull. Amer. Math. SOC.54, 575-580. [48] “Groups with representations of bounded degree,” Canad. J. Moth. 1. 105-112. [49] “Topological representations of algebras 11,” Trans. Amer. Math. SOC.68, 62-75. [SO] “The structure ofcertain operator algebras,” Trans. Amer. Math. SOC.70, 219-25s. [Sl] “Problems in the theory of rings revisited,” M A A Monthly 77,445-454. [70] Kemer, A. R. “A remark on the standard identity,” Mat. Zam. 23,753-757. [78] Kezlan, T. P. “Rings in which certain subsets satisfy a polynomial identity,” Trans. Amer. Math. [66] SOC.125,414- 421. Kharchenko, V. K. “Galois extensions and quotient rings,” Algebra i Logika 13,265-281 [74] “Generalized identities with automorphisms,” Algebra i Logika 14, 132- 148. [75] “Ring identities with automorphisms.” Siberian Math. J. 17, 349-363. [76] Klein, A. A. “PI-algebras satisfying identities of degree 3,” Trans. Amer. Math. SOC.201.263-277. [75] Kleinfeld, E. “Simple alternative rings,” Ann. q f M a t h . 58, 544547. [53] “Primitive alternative rings and semisimplicity,” Amer. J . Math. 77, 725-730. [5S] “Alternative nil rings,” Ann. o f M a t h . 66,395-399. [57] Kochen, S. “Ultraproducts in the theory ofmodels,” Ann. of Math. 74,221-261. [61] Kostant, B. “A theorem of Frobenius. a theorem of Amitsur-Levitzki. and cohomology theory,” [SS] Indiana J . Math. (and Mech.) 7, 237-264. Kovacs, A. “The n-center of a ring,” J . Algebra 40,107-124. [76] “On the nonevistence of hypercommuting polynomials,” Proc. Amer. Marh. SOC.66, [77] 241-246. Krakowski, K., and Regev, A. “The polynomial identities of the Grassman algebra,” Trans. Amer. Math. SOC.181, [73] 429-438. Kruse, R. L. “Identities satisfied by a finite ring,” J . Algebra 26, 298-318. [73] Kuyk, W. “Generic construction of non-cyclic division algebras,” 1. Pure Appl. Algebra 2, [72] 121-130. Latyshev (also LatySevI, V. N. “Algebras with identical relations,” Dokl. Acad. Nauk. SSSR 146, 1003-1006 (in [62] Russian). [63a] “Two remarks on PI-algebras,” Sibirsk. Mat. Z. 4, 1120-1121 (in Russian). [63b] “On the choice of basis in a T-ideal,” Sibirsk. Mat. Z. 4, 1122-1 127 (in Russian). “On the finiteness of the number of generators of a T-ideal with an element [65] [x,, x2, xg, x,],” Sibirsk. Mat. Z. 6, 1432-1434 (in Russian). [75a] [75b]

Bibliography

349

“Generalization of Hilbert’s theorem on the finiteness of bases,” Sibirsk. Mat. Z . 7 , 1422-1424 (in Russian). “On the Specht property of certain varieties of associative algebras,” Algebra i Logika 8,374-382. “On Regev’s theorem on identities in a tensor product of PI-algebras,” Ups. Mat. Nauk. 27, 213-214 (in Russian). “On some varieties of nonassociative algebras,” Math. USSR Izu. 7, 1011-1037. “Partially ordered sets and nonmatrix identities of associative algebras,” Algebra i Logika 15,34-44. “On the complexity of non-matrix varieties of associative algebras,” Algebra i Loaika 16.98-132. Latyshev, V. and’Shmelkin, A. L. (also LatySev and Smelkin) “A certain problem of Kapalansky,” Algebra i Logika 8,257. [69] Leron, U. Doctoral dissertation, Hebrew Univ. Jerusalem, Israel. [72] “Nil and powercentral polynomials in rings,” Trans. Amer. Math. SOC.202,97-103. [75] “Trace identities and polynomial identities of n x n matrices,” J . Algebra 42, [76] 369-377. “Young subgroups and polynomial identities,” J . Algebra 49,304-3 14. [77] Leron, U., and Vapne A. (now Regev, A.) “Polynomial identities of related rings,” Israel J . Math. 8, 127-137. [70] Lestman, P. “Simple going down in PI-rings,” Proc. Amer. Math. SOC.63,41-45. [77] “Going down and the Spec map in PI rings,” Comm. Alg. 6,1667-1691. [78] Levi, F. W. “Uber der Kommutativitatsrang in einem Ring,” Math. Ann. 121, 184-190. [49] Levitzki, J. “A theorem on polynomial identities,” Proc. Amer. Math. Soc. I, 334-341. [50] “On the structure of algebraic algebras and related rings,’’ Trans. Amer. Math. SOC. [53] 74,384-409. “On minimal central identities,” Riireon Lematemat. 8.41-48. [54] Lewin, J. “On some infinitely presented associative algebras,” J . Austral. Math. Soc. 16, [73] 290-293. “A matrix representation for associative algebras I, 11,” Trans. Amer. Marh. Soc. 188, [74] 293-317. Lo&,J. [551 “Quelques remarques, theortmes, et problemes sur les classes definissables d’algebres,” in Mathematical Interpretation of Formal Systems, pp. 98-1 13, North-Holland, Amsterdam.

h.,

L’vov, I. v. “Maximality conditions in algebras with identity relations,” Algebra i Logika 8, [69] 258-263. “On varieties of associative rings I, 11,” Algebra i Logika 12, 150-167.381-392. [73] “Finitely based identities of nonassociative rings,” Algebra i Logika 14, 7-15. [75] “Finite dimensional algebras with infinite bases of identities,” Siberian Math. J . 19, [78] 63-68. Malliavin-Brameret, M. “Dimension de Gelfand-Kirillov des algebra a identites polynomiales,” C.R. Acad. [76] Sci. Paris Ser. A-B. 282, Ai, A679-A681.

350

BIBLIOGRAPHY

Maltsev, A. 1. (also MaEev, A. I.) “On algebras satisfying defining identical relations,” Mat. Sb. 68, 11-33 (in Russian). [50] Maltsev, Ju. N. (also Mali-ev, Ju. N.) “Basis for identities of the algebra of upper triangular matrices,” Algebru i Loyika [71] 10,242-247 ”On the identities of full matrix algebras,” Siberian Math. J . 14,952-953. [73] [76a] “Almost commutative varieties of associative rings,” Siberian Math. 17, 803- 810. [76b] “Varieties of asqociative algebras,” Algcbra i Logika 15. 361-364. Maltsev, Yu. N., and Kuz’min, E. 2. “A basis for the identities of the algebra of second-order matrices over a finite field.” [78] Algebra i Loyiko 17, 18-20, Markov, V. T. “On the dimension of noncommutative affine algebras,” Math. U S S R I N 7, [73] 281 -285. Martindale, W. S. 111. “Primitive algehras with involution,” Pacific J . Math. 11, 1431-1441. “Rings with involution and polynomial identities,” J . Algebra 11, 186194. “Prime rings satisfying a generalized polynomial identity,” J . Algebra 12, 576584. “Primitive rings with involution whose symmetric elements satisfy a generalized polynomial identity,” Proc. Amer. Math. Soc. 24, 508-51 I. “Prime rings with involution and generalized polynomial identities,” J . .4lgebra 22, 502-516. “A note on generalized polynomial identities,” Canad. Math. Bull. 15, 605-606. “On semiprime PI-rings,’’ Proc. Amer. Matti. Soc. 40, 364-369. McCrimmon, K. “Alternative algebras satisfying polynomial identities,” J . Algebra 24,283- 292. [73] Montgomery, S . “Polynomial identity algebras with involution,” Proc. Amer. Math. Soc. 27, 53-56. [71] “Centralizers satisfying polynomial identities,” Israel J . Math. 18, 207-219. [74] “A structure theorem and a positive-definiteness condition in rings with involution,” [76] J . Algebra 43, IS 1- 192. Montgomery, S., and Smith, M. “Algebras with a separable subalgebra whose centralizer satisfies a polynomial [75] identity,” Coinm. Algebru 3, 151- 168. Mueller, B. J. [76a] “Localization in fully bounded noetherian rings,” Pac$c J . Math. 67, 233- 245. [76b] “Rings with poljnomial identities,” Monograjas del Insrituto de Mutematicas, No. 4. Universidad Nacional Autonoma d e Mexico, Mexico City. [77] “An example in non-commutative localization,” Ann. Inst. M a t . L’iiiu. N a c . Auroilomu Me.xit.0. Nagarajan, K. “Groups acting on Noetherian rings,” Nieuw. Arch. Wisk. 16, 25-29, [68] Nagata, M. “On the nilpotency of nil algebras,” J . Math. Soc. Japan 4,296-301. [53] Olsson, J., and Regev, A. [76a] “An application of representation theory to PI-algebras,” Proc. Amer. Math. Soc. 55. 253-257. [76bl “The colength of some T-ideals,” J. Algebra 38, 100-1 1 1 . ”On the T-ideal generated by a standard identity,” Israef J. Math. 26,97-104. [77] Osborn, J. M.

Bibliography [68]

35 1

“Commutative nonassociative algebras and identities of degree 4,” Canad. J . Math. 20, 769-794. “Varieties of algebras,” Adrartees iri M u t h . 8. 163-369.

[72] Page. A. [73a] ”Generalization aux anneaux a identite polynomiale d’un theorem de 1. Kaplansky,” C.R. Acad. Sci. Paris Ser. A 276. A233-A236. [73b] ”Resultats nouveaux sur les anneaux i identite polynomiale,” Set77. d‘dqrbrr m)/t commutative, Pub. mathematiques d’Orsay, 6#8,9. “Centre des anneaux i identite polynomiale,” Sem. d‘algebre tior1 commutatire, Pub. [74] mathematiques dOrsay, 596.7. Pare, R., and Schelter, W. “Finite extensions are integral.” .I. Algebra 53,477-479. [78] Pascaud, J. L. ”Anneaux d’endomorphismes a identites polynomiales,” C . R . Acad. Sci. Paris, Ser. A [75] 280, Ai, A 1093-A 1096. Passman, D. S. [7la] “Linear identities in group rings,” Pacific J . Math. 36,457-484. [71b] “Group rings satisfying a polynomial identity 11,” Pacific J . Murk. 39, 425-438. [72a] “Group rings satisfying a polynomial identity,” J . Algebra 20, 103-1 17. [72b] “Group rings satisfying a polynomial identity Ill,” Proc. Amer. Math. Soc. 31. 87-90. Patil, K. B., and Racine, M. L. “Central polynomials for Jordan algebras 11,” J . Algebra 41, 238-241. [76] Posner, E. 1601 “Prime rings satisfying a polynomial identity,” Proc. Amer. Math. Soc. 11, 180-184. Procesi, C. “The Burnside problem,” J . A l p b r a 4,421-425. [66] [67a] “Noncommutative affine rings.” Air; Accad. Na:. Liricei 8, 239-255. [67bJ ”Noncommutative Jacobson rings,” Atm. Scuolu Norm. Sup. Pisu 21,281-289. [68a] “Sulla anelli noncommutative zero dimensionale con identite polinomiale,” Rend Circ. Mat. Palermo Ser. 2 17. 5- 12. [68b] “Sulla identita delle algebre semplici,” R e d . Circ. M a r . Palermo Ser. 2 17, 13-18. [72a] “On a theorem of M. Artin,” J . Algebru 22, 309-315. [72b] “On the identities of Azumaya algebras,” in Ritig Theory (R. Gordon, ed.). Academic Press, New York. [72c] “Central polynomials and finite dimensional representations of rings.” Atus da Escola de Algebra, IMPA. “Finite-dimensional representations of algebras,” Israel J . Math. 19, 169-182. [74] 1761 “The invariant theory of n x n matrices,” Adoancrs ill Math. 19, 306-381. Procesi. C.. and Schacher, M. “A non-commutative real Nullstellensatz and Hilbert’s 17th problem,” A m . ( ? / M a t h . [76] 104,395-406. Procesi, C., and Small, L. “Endomorphisms of modules over PI-rings.” Math. Z . 106, 178-180. [68] Racine, M. L. ”Central polynomials for Jordan algebras I.” J . Algebra 41,224-237. [76] Razmyslov, Yu. P. [73a] “On a problem of Kaplansky,” Math. USSR 1st.. 7,479-496. [73b] “Finite basing of the identities of a matrix algebra of second order over a field of characteristic 0,” Alyebru i Logiku 12, 47-63. [74a] “Trace identities of full matrix algebras over a field of characteristic zero,” Murh.

352

[74b] [74c] Regev, A. [72] [74] [77a] [77b] [78a]

BIBLIOGRAPHY USSR-lsl). 8. 727-760. “The Jacobson radical in PI-algebras,” Algebra i Logika 13, 192-204. “The finite basis property for certain varieties of algebras.” Algehrtr i Logiku 13, 394- 299.

“Existence oridentities in A @ B,” IsraelJ. Math. 11, 131-152. “Nonassociative PI-algebras.” J . Algebru 28,247-252. “The T-ideal generated by the standard identity,” Israel J . Math. 26. 105--125. “Finitely generated T-ideals of PIdegree 3.” J. London Marh. Soc. 10, 261-266. “The representations of S,, and explicit identities for PI-algebras,” J . Algrhrci 51, 25-40. “On algebras satisfying a Capelli identity,” J. Pure Appl. Algebra.

[78b] Risman, L. [77a] “Noncyclic division algebras,” J . Pure A p p l . Algebra 11, 199--216. [77b] “Cyclic algebras, complete fields, and crossed products,” Israel J . Math. 28, 113- 128. Robson, J. C., and Small, L. W. [74] ”Hereditary prime PI-rings are classical hereditary orders,” J . London Marh. Soc. 8, 499-503. [76] “Idempotent ideals in Pl-rings,”J. London Math. Soc. 14, 120-121. Rosenberg, A,, and Zelinsky, D. [59] “Finiteness of the injective hull,” Moth. Z . 70, 372-380. Rosset. S. [75] “A note on symmetric functions in Formanek polynomials,” Proc. Amer. Math. Soc. 50, 127- 1 30. [76] “A short proof of the Amitsur-Levitzki theorem,” Isroel J . Marh. 23, 1x7- 188. [77aP] “Abelian splitting of division algebras of prime degrees.” [77bP] ”Group extensions and division algebras.” Rowen, L. H. Some results on the center of a ring with polynomial identity,” Bull. Amer. Math. SOC.79,2 19- 223. “On classical quotients of polynomial identity rings with involution.” Pro(.. Amer. Math. SOL..40, 23-29. “On rings with central polynomials,” J . .4igebra 31, 393-426. “Standard polynomials in matrix algebras,” Trans. Amer. Marh. Soc. 190, 253-284. “Universal PI-Algebras and algebras of generic matrices,” Israel J . Math. 18,65- 74. “Maximal quotients of semiprime PI-algebras,” Trans. Amer. M o t h S o r . 196, 127- 135. “Prime ideal, of Noetherian PI-rings.’’ “Identities in algebras with involution,” Israel J . Math. 20,70-95. “Generalized polynomial identities,” J . Algebra 34, 458-480. “The Artin-Procesi theorem revisited.” “Generalized polynomial identities 11,” J . Algebra 38, 380-392. “Monomial conditions on rings,’’ Israel J . Math. 23, 19-30. “Classes bf rings torsion lree over the center,” Pacific J . Math. 69, 527- 534. “The theory of generalized polynomial identities,” Proc. Ohio U n h . Ring Theory Cor~t.pp. 15- 61. Dekker. New York. “Nonassociative rings satisfying a normal polynomial identity,” J . Algebra 49, 104- I 11. “Monomial ionditions on prime rings,“ Israel J. Mur/t. 27, 131 -14V (Correction. ibid. 30, 192)

Bibliography

353

”Generalized polynomial identities 111.” J . Algebra 46, 305-314. “Central simple algebras,” Isruel J . Math. 29, 285-301. “Polynomial identities of nonassociative rings I, 11, 111,” Illinois J . Murh. 22, 341-378, 521-540. “Central polynomials for special Jordan rings without nilpotent elements, and an [79] analogue of Kaplansky’s theorem,” J . Algebru. Rowen, L. H., and Schild, U. “A scalar expression for matrices with symplictic involution,” Math. Comp. 32, 607. [78] Saltman, D. [74] Oral communication. “Splittings of cyclic p-algebras,” Proc. Amer. Math. Soc. 62, 223-228. [77] [78a] “Noncrossed products of small exponents,” Proc. Amer. Marh. Soc. 68, 165-168. [78b] “Noncrossed product p-algebras and Galois p-extensions,” J . Algebra 52,302-3 14. [78c] “Azumaya algebras with involution,” J . Algebra 52, 52C539. “Indecomposable division algebras,” Comrn. Algebra. [78] [78bP] “On p-power central polynomials.” Sandomierski, F. L. “Classical localization at prime ideals of fully bounded Noetherian rings,” in [77] Ring Theory: Proc. Ohio Uniu. Athens Ring Theory Conf:, pp. 169-181, Dekker, New York. Schacher, M. “The crossed product problem,” in Ring Theory 11, Proc. Oklahoma Con/:, ’nd [75] (McDonald-Morris, ed.),pp. 237-246, Dekker, New York. Schacher, M., and Small, L. [73] “Noncrossed products in characteristic p,” J . Algebra 24, 100-103. Scharlau, W. “Zur existence von involution auf einfachen algebren,” Moth. Z . 145,29-32. [75] Schelter, W. [75] “Essential extensions and intersection theorems,” Proc. Amer. Math. Soc. 53, 328-330. [75P] “Non-commutative affine PI-rings are catenary.” [76a] “Jntegral extensions of rings satisfying a polynomial identity,” J . Algebra 40, 245-257 [Correction, ibid. 44, 576 (1977)l. [76b] “On the Krull-Akizuki theorem,” J . Ltir7rk)n Math. Snc. 13, 263-264. [76c] “Intersection theorems for some noncommutative Noetherian rings,” J . Algebra 38, 244-250. [77a] “Afine PI-rings are catenary.” Bull. Amer. Math. Soc. 83, 1309-1310. [77b] “Azumaya algebras and Artin’s theorem,” J . Algebra 46,303-304. “Noncommutative affine PI-rings are catenary,” J . Algebra 51, 12- 18. [78] Schelter, W., and Small, L. “Some pathological rings of quotients,” J . Lor~doilMath. Soc. 14, 200-202. [76] Shirshov, A. I. (Also SirSov, A. 1.) [ V a l “On some nonassociative nil-rings and algebraic algebras,” M a t . Sb. 41, 381-394. [57b] “On rings with identity relations,” M a t . Sb. 43, 277-283. [57c] “On some nonassociative nil-rings and algebraic algebras,” Proc. All-Union Math. Cot$, 3rd Moscow 1.40 (1956). “On certain identity relationships in algebras,” Sibirsk. Mat. Z . 7, 963-966. [66] Slater, M. “Prime alternative rings I, 11,” J . Algebra 15,229-251. [70] “Prime alternative rings 111,” J . Algebru 21, 394-409. [72] [77e] [78a] [78b]

354

BIBLIOGRAPHY

Small, L. W. [66] “Orders i n artinian rings,“ J. .4/gebru 4, 13-41. [7la] “An example in PI-rings,” J. Algebra 17,434-436. [TI b] “Localizatioti in PI-rings.” J. Alyebrtr 18, 269-270. “Ideals in finitely generated PI-algebras,” Ring Theory (R. Gordon, ed.), pp. [72] 347-352. Academic Press, New York. [73] “Prime ideals in Noetherian PI-rings.’’ Bull. Arner. Math. Soc. 79,421-422. Smith, B. [75] Doctoral dissertation, Univ. of Virginia. [77] “A standard Jordan polynomial,“ t o m m . Algebra 5,207-218. [79] “An application of model theory to Jordan algebras,” J . Algebra. Smith, M. [71] “Group algebras,” J . Algebra 18.477 -499. [72] “Regular representations of direct groups,” J. Funcrioiml Analysis 11,401-406. [75] “Rings with .in integral element whose centralizer satisfies a polynomial identity,” Duke A4orh. J 42, 137-149. Snider, R. [78P] “Is the Brauei group generated by cyclic algebras.” Specht. W. [50] “Gesetze in Ringen I.” Math. Z. 52, 557-589. Streb, W. [75] ”Kommutatorbeziehungen in Ringen der Charakteristik 0 mit Einselement und ihren Einheitsgruppen,” Rend. Sem. Mar. C’nir. P U ~ W U 53, 105- 1 1 5. Swan, R. G . “An application of graph theory to algebra,” Proc. .4mer. Math. Soc. 14, 363-373. [63] [Correction. rhid. 21, 379-380 (1969)l. Tignol, J. P. “Sur les corps a involution de degre 8,” C . R . Acad. Sci. Puris Ser. A 284. 1349- 1352. [77] “On similarit) classes of involutorial division algebras of degree 8,” C . R . Acrid. Sci. [78] Puris Ser. 4 286. A876-AX77. Valette, J. “Anneaux d e groupes noetheriens, Comm. Alg. 5,365-374. [77] Vamos, P. “Semi-local hoetherian PI-rings.” Bull. Londori Math. Soc. 9, 3 1 - 3 6 , [77] Vaughan-Lee, M. R. “Varieties of I ie algebras,’’ Quart. J. .Math. O\-ford Ser. 21. ( 2 ) 297-308. [70] Watters, J. F. ”Polynomial extensions of Jacobson rings,” J . Algebra 36, 302-308. [75] Zelmanov, E. 1. “Semigroup algebras with identities,” Siberiati Math. J. 18, 557-562. [77]

MAJOR THEOREMS CONCERNING IDENTITIES

Theorem 1.3.11 1.4.1 1.4.12 1.4.14 1.4.21 1.4.26 t 1.4.34 1.5.16 1.6.27 *1.6.36 * 1.6.46 1.7.9 1.7.11 t1.7.34 1.8.48 1.8.50 *1.9.18 *1.9.21 1.10.10 1.10.18 1.10.23 t 1.10.28 1.10.69 1.10.71 1.11.12 2.1.8 2.2.24 *2.3.23 2.3.29 2.3.37 2.4.7 *2.4.9 t2.4.13 2.5.17

Content (in

brief) Algebra of generic matrices free in class of all algebras satisfying identities of n x n matrices. Szmis an identity of n X n matrices. Normal polynomials give characteristic values of transformations. Multilinear, ng-normal central polynomial of n X n matrices. PI-algebras as module over the center. Algebras having a central polynomial taking value 1. Polynomial criterion for linear independence of matrices. Primitive PI-rings are simple and finite dimensional. General structure of semiprime PI-rings. Nil multiplicative subsets of PI-rings. Every PI-algebra satisfies a power of a standard polynomial. Ring of central quotients of a prime PI-ring is simple. Localization at value taken by central polynomial. Semiprime PI-rings that have semisimple ring of quotients. Characterization of Azumaya algebras of well-defined rank. Characterization of proper maximally central algebras. Correspondence of certain prime ideals of R and Z ( R ) . Correspondence of certain maximal ideals of R and Z ( R ) . Extending prime PI-ring to algebra over valuation ring. Prime PI-algebras with center of finite transcendence degree have finite rank. Rank of afiine PI-algebra equals transcendence degree of center. Bound on rank of algebra of generic matrices. PI-degree of domain is multiple of degree of any subring. PI-degree of prime ring is spanned by degrees of its images. Center of semiprimitive PI-ring is semiprimitive. T-ideals correspond uniquely to relatively free algebras. Structure of semiprime PI-rings with involution. Identities of R[A] in terms of identities of R . R and R[A] have the same identities if center is nice. Identities pass to all central localizations. Jacobson radical of relatively free PI-ring is nil. Classification of prime T-ideals Identities of prime algebras with involution. Minimal identities of matrices with transpose involution.

t: Theorems preceded by * are sharpened further in the exercises; those preceded by t are improved later in the text. 355

MAJOR THEOREMS CONCERNING IDENTITIES Theorem

Content (in brief)

3.1.63 3.2.6 3.2.20 3.2.29 3.2.40 t3.3.12 3.3.29 3.3.31 *4.1.4 *4.1.8 4.1.9 4.2.8 4.3.1 4.3.5 *4.4.5

Classification of prime T-ideals with involution. The generic division algebra really is a division algebra. Cyclicity of division algebras in terms of central polynomials. Division algebras of degree 1, 2, 3, 4, 6, 12 are crossed products. Division algebras of degree 8 with involution are crossed products. Noncrossed products obtained without arithmetic. More noncrossed products. The bound of Theorem 1.10.28 is sharp. Integral PI-extensions satisfy “going up,” “lying over.” Integral PI-extensions satisfy “incomparability.” Jacobson radical under an algebraic PI-extension. Every integral PIextension is locally finite. The characteristic closure. A finitely generated subring of the characteristic closure. Every prime, finitely generated PI-extension of a semiprimitive, commutative ring is semiprimitive. Nilradical of a PI-extension of ring with ACC is often nilpotent. Jacobson radical of finitely generated PI-algebra is often nilpotent. “Going down” for PI-extensions of normal rings. For R affine. rank(R) = rank(P) + rank(R/P). Finitely generated PIextension of ring with ACC(ideals) satisfies ACC(semiprime ideals). Improvement of Theorem 4.4.17. Semiprime PI-ring with Noetherian Z(R)-subalgebra is Noetherian. Prime PI-rings with ACC(ideals) are Noetherian. Big commutative subrings of Noetherian rings are Noetherian. Principal ideal theorem for PI-rings with ACC(ideals). Prime ideals of Noetherian PI-rings have finite rank. Intersection of powers of the radical of Noetherian PI-ring is 0. Intersection of powers of certain prime ideals is 0. Tensor product of PI-rings is a PI-ring. Explicit version of Theorem 1.6.46. Certain T-ideals are finitely generated. Coefficients of generalized identities (written suitably) have finite rank. Characterizes primitive rings satisfying generalized identity. Ring with strong generalized identity is PI. (*)-version of Theorem 7.2.2. Ring with strong generalized (*)-identity is PI. Values of each GI of prime ring contained in socle of related primitive ring. The central closure is the primitive ring of Theorem 7.5.14. Generalization of Theorem 1.4.34 to prime rings. Infinite dimensional division algebras over infinite fields have no nontrivial rational identities. Generalization of the generic division algebras. Division PI-algebras are rationally equivalent to matrices. D, satisfies all rational identities of D2 if deg(D,) I deg(D,). Converse of Theorem 8.3.6. Multiplicative group of noncommutative division algebra has no nontrivial group identities.

t4.4.17 4.4.18 4.4.24 4.4.27 4.5.7 4.5.8 5.1.6 75.1.7 *5.1.12 5.2.12 5.2.19 ‘5.2.26 *5.2.29 6.1.1 6.2.22 6.3.8 *7.2.2 7.2.9 t7.2.15 7.4.8 7.4.12 7.5.14 7.6.15 7.6.16 8.2. I5 8.2.22 *8.3.5 8.3.6 8.3.7 8.4.2

MAJOR THEOREMS CONCERNING IDENTITIES

Theorem 8.4.21 8.4.26

A. 12 *c.33 *c.35

357

Conrenr (in brief) Rational identity yields intersection theorem in Desarguian plane. Intersection theorems of Desarguian plane correspond to rational identities of the underlying division ring. Formanek's central polynomial for matrices. Central polynomial for nonassociative prime alternative rings. Structure of prime alternative rings having nilradical 0.

MAJOR COUNTEREXAMPLES

Number * 1.7.13 E1.4.2 E1.7.2 E1.7.3 E1.9.5 E1.10.4 E l. 11.21 2.4.23 E2.5.9 E3.3.11 4.4.12

4.4.22 5.1.18 5.1.19 5.2.20 5.2.28 E5.1.1 E5.1.2 E5.1.3 E5. 1.4 E5.1.5 7.2.17 8.3.1

B.10

Example of Commutative, semiprime, nonsemiprimitive ring that is its own ring of central quotients. PI-ring without central polynomial. Central localization # Ore localization. Semiprimitive PI-ring not satisfying Ore condition. Prime PI-ring failing “lying over” from center. “Going up” failing from center, which is a valuation ring. PI-ring whose maximal left quotient ring is not PI. Ssnnor generating T-ideal of n x n matrices. Ssn-s not vanishing for antisymmetric n x n matrices. Generic matrix ring tensored with itself is not prime. Non-Noetherian PI-ring with ACC on right ideals. Affine PI-ring failing ACC on right annihilators. Prime Noetherian PI-ring not integral over the center. Prime, Noetherian, a f b e PI-algebra failing LO from center. Right Noetherian PI-ring with nonzero intersection of powers of radical. Prime Noetherian PI-ring with idempotent maximal ideals. Prime Noetherian PI-ring with non-Noetherian center. Prime Noetherian PI-ring infegral over non-Noetherian center. Further properties of Example 5.1.18. Prime Noetherian PI-ring failing “incomparability” from center. Principal ideal theorem failing for center of prime, atline Noetherian PI-*. Primitive, algebraic algebra over field, with nonzero socle, which is not locally finite. Rational identity of division algebras of degree 4, failing for division algebras of degree 3. Simple Noetherian Ore domain which is not Artinian.

* E means Exercise. 358

LIST OF PRINCIPAL NOTATION

Prerequisites: H, Q, Z(R), Hom,(M, MJ, End, M, ker 9,

page 2 4

5 6 8 10 12 13 14 I5 17 19 26 30 32 33 38 43

2 , II, 0, L, C,H+, S") Page 78 80 82 83 84 92 93 95 110 112 113 114 116 119

124

44 45 51 52 53 54 57 65 72 73 73 74 75

127 137 140

146 153 154 158 159 163 167 176

359

This Page Intentionally Left Blank

INDEX Burnside’s problem, u)7

A ACC, see Ascending chain condition Admissible algebra, 41f, 107, 215 A A n g , 70 Mine, 84f, 215f, see ako Extension Algebraic algebra, see Kurosch’s problem Algebraically independent, 17, 82f Algebraic system, 145f, 277f Algebraic, W-,202 Algebra of generic matrices, 16f, 176f, 196f, 300 Almost n-dimensional, 92f Alternating form, 2671 Alternative algebra, 327, 332f Amitsur filter, 276 Amitsur-Levitzki theorems, 2lf, 339 Amitsur’s method, 48f Amitsur’s theorem, 261 Annihilator (Ann, Ann?, 20 ideal, 56f Antiautomorphism, 114 Antisymmetric, 140 Artin-Procesi theorem, 65f, 69f, 340 Ascending chain condition (ACC), 137f, 212f on annihilator ideals, 56f on extended submodules, 226f on ideals, 212f, 220 on left. right annihilators, 215 on left, right ideals, 212f, 224f on semiprime ideals, 213f, 220, 223f on T-ideals, 138 Associated factor set, 160 Atomic sentence, 147, 321 Azumaya algebra, 67 of rant 1, 67f, 199, 220

B BergmanSmall theorems, 95f Brauer group, 72, 114, 157f, 173f, 199 361

C

Canonical symplectic involution, 140f. 169f, 184f Capelli, see PolynDmial Catenary, 217f Cayley-Hamilton theorem, 18,319 Center, xix of ring with involution, 121 Central closure, 2825 328f Central extension, 8, l25f Centralizer, 57 Central localization, 51f, 62.69, 75, 79, 105, 106, 123, 132,230 Central, n-, 180 Central polynomial (Rcentral), 4, 24f, 53f, 124, 3lSf, 337 Central quotients, 52f, 54f Central simple (simple PI), l5lf Characteristic closure, 208f, 219 Chinese remainder theorem, 55 Classical identity, 4 Class of systems, 147 Closed primitive ring, 35 Closure, 35f with respect to (*), 272f Codirnension, 24Of, 269 Coefficient, 5, 113, 116, 118 set, 118 Completion A-adic, 89f of valuation ring, 9Of, 192 Composition series, 229f Compositum, 194 Configuration, 306 Constructible configuration, 307 Countable, xx Crossed product, lS8f, 174f. 183, 186 abelian, 158, 183, 186f, 189 Cyclic algebra (cyclic crossed product), 158, 164f, 174,190,194

362

INDEX

D Degree of algebra, 151 Density theorem, 32f, 104.264, 266f Dararguian plane, 289, 304f Dilworth’s theorem, 241 Dimension, xix Direct power, 5 Direct product, 5 Direct sum. 54 Discrete valuation ring, 191f Division ring. 32f, 37f, 72, ISlf, 171f. 174f, 187,262.291f Domain (kind of ring), 96 Domain (of function), 145 Double centralizer theorem, 154

E Elementary conditions, 320f Elementary sentence, 146f Elementary symmetric function, 18 Endomorphism, xix algebra. 110 Epimorphism, 133 Equivalent involutions, 168f Equivalent rings, 124 Essential, 235 Exponent, 163f. 173 Extended centroid, 282f,329f Extension, 57, 74f, 202f. 221,238,284, 288 central, 8, 125 finitely generated. 84f, 203 finitely spanned, 203. 221 PI-, 203f, 238 Exterior algebra, 10, 104.243

F Factor set, 159f Faithful, 33 Filter. 276f Finite dimensional xix Finitely generated extension, 84f, 203 Finitely generated in&, 137 Formal power series, 192f Frame, 244 Free, 2 algebra, 3 commutative algebra, 3 group, 302f monoid, 2, 112

ring, 4 W-ring 113 Frobenius’ theorem, 200 C Gauss’ lemma, 79 Generalized identity (GI, GPI,W-identity), Illf, 254f Rpropcr, 119, 261 R-strong, 119,263 with (*), 114, 117,271f Generalized monomial 118,260f Generalized rational identity, 291 Generalized polynomial (W-polynomial), 113 Generic division rings (VD (tp,n)), 175f. 187f. 300 Generic matrix, I5f Going down, 93, 216 Going up (GU). 73f, 203, 204,209f. 221 Group algebra, 23,243, 313 Group identities, 302, 304

H Height (of polynomial), 127 Hermitian form, 267f Hilbert basis theorem, 212 Hilbert’s Nullstellensatz, 210,222 Hilbert’stheorem 90,161 Hook formula, 245,252

I Ideal, (*)-, I15 Idempotent, 88f. 255f id& U6f, 238 Idempotent-lifting, 88 Identity, see 4 h polynomial, 4. 100 W-,113 Incomparability (INC),73f. 93, 204.209f. 288

Index, 155f reduction factor, 156 Injection, xix problem, 42f, 107, 215 Inner automorphism, 152 Integral 9, 19,78f, 202 closure, 79 extension, 78f, 202f

363

INDEX

of bounded degree, 9,203

w-,202,221

Intersection theorem, M7f Involution, 114f, 119f, 123f. 136, 140f, 148f, 150, 167f, l99,265f, 271f, 2795 282f, 287,319 canonical (on W(X)), 116 canonical symplectic, 14Of, 169f, 184f exchange, 117 first kind, 121, 136, 167f orthogonal type, 169f reversal, 115 second kind, 121f, 150, 200 symplectic type, 169f transpose, 114, 14Of, 168f. 184f Irreducible, 33

Matrices, 14f, 21f. 315f under involution, 14Of, 150, 268,270 Maximal xx ideal, 74, 76, 106,222 subfield, 35, 37f, 157f Maximal, (A’,B)-, 73 Minimal left ideal, 34,USf, 266f Minimal prime, 78,231 Minimal prime, A-, 77 Monoid, xix algebra, 3 Monomial, 5 Monomial, W-,113 Mult-equivalent, 30, 124,279f Multilinearkation, 6f, 126f

N J Jacobson radical 38f, 43.77, 134, 204,21Of, 214f, 216,222,233,236,238 Jacobson ring, 211,222 Jordan-Holder theorem, 230

K Kaplansky class (of nonassociative algebras), 331f Kaplansky’s theorem, 35f Kurosch’s problem, 202,204f, 264, 287,339

L Label, 118 Large, 105 Laurent series, 193f Levitzki radical, 48 Level, 290 Lexicographic order, 3 Lie nilpotent, solvable, 10 Lies above, 81

Line of Desarguian plane, 304 of D“’, 291 Locally nilpotent, 47f Lying over (LO) 73f, 203,2@f, 221

M Martindale’s theorem, 282, 285 Matric unit, 14, 88

Nagata-Higman theorem, 100, 149 Newton’s formulas, 18, 150 Nice, (*)-, 167 Nil, 19f, 43f, 48, 105,323 of bounded degree, 45 Nilpotent, 19f Nilradical, 43f, 212,214,280 Noether-Jacobson theorem, I55 Noetherian, 224f Noncrossed products, 191, 196, 340 Nonsingular ring, 108,287, 323 Normal polynomial, 7f, I If, 24f, 328 Normal ring, 79,91f, 216f Nullstellensatz, 210f, 222 0

Opposite ring, 65f Ordered ring, 200, 313 Ore condition, 105 Ore domain, 292

P Partial order, 2 Pfaffmn, 142 PI-class, 30 PI-extension, 203f, 238 PI-ring, 5,50, 105 without 1, IOOf Point of D‘”,291 of Dcsarguian plane, 304

3 64

INDEX

Polynomial 5 alternating, r-, 7 blended, 127 central,4,24f,53f. 124.315f, 337

R

Rank of element, 256f of prime ideal 74f, 21 7f, 23 If central (R,*), 124 of ring, 75f,84f,234 Capelli, 12,23f,31, 216, 253, 299. 328f Rational identity, 289f, 291 completely homogeneous, 119 Rationally equivalent, 2%f correct, 104f Reduced algebra with I , 102 Formanek, 315f Reduced length, 23Of homogeneous, 5, I 1 9 Reduced trace, 166 identity, 5 Regular (element), 52 linear, 1-, 5, 8, 1 1, 316 Regular representation. 19 multilinear, 5, 1 I9 Relatively free, 1 IOf, 133f normal-r, 7f, I If, 24f, 328 Ring with involution, see also Involution primitive, 12, 103 fmt kind, 121, 167f proper, R-, 5 homorphism of, I I5 stable, 12Sf standard, 8, 13,21f, 103, 104, 138, 14f, ideal of, 1 I5 prime, 121f, 136,278f 150,246,248 primitive, 265f symmetric, 7 second kind, 121f,200 W-, 113 semiprime, 122f (W,*)-, 116 simple, IZOf, 185f. 200 Posner (-Formanek-Rowen) theorem, 53, special, 117, 132, 271 322, 330,340 Prime (ring), 34,47,53,83f,106, 135, 153,

ZOSf,210f,222, 225f. 2275 237, 278f, 282f, 322,325,335 Prime ideal 49,73f,96f. 106,209f,213f, 221, 222,231,237, 238,288 Prime spectrum, 73f Primitive (ring), 34f. 256f,264,321, 324f strongly primitive, 28 I Primitive ideal, 38 Principal ideal theorem, 229f Principal left ideal domain (PLID), 292f Projective geometry, 304f Proper (identity), 5, 119,261 Proper maximally central, 68,218f Pure monomial, 118 Purely transcendental, 87

Q Quasi-invertible (quasi-regular), 38f Quasi-local, 69 Quaternion, 185 Quotient rings, 59 central, 52,54 Ore (classical), 105. 292f,322f maximal, 59, 108

S

Schur’s lemma, 32 Semiprime (ring). 46f, 54f,105, 107,286,

287,323 Semiprime ideal, 49, 106, 137 Semiprimitive, 40,42,102, 105, 21 1, 222 Semisimple, 55f Separable splitting fields, 155 Sesquilinear form, 267 Shirshov’s theorem, 206 Simple, 36f,64f,ISlf,222, 286,299f,313 Singular ideal, I08 Skew polynomial ring, 188 Skolem-Noether theorem, 152,199 Socle, 255f, 28 1, 324f Specht’s conjecture (broblem), 138, 242f,

248f Special, 117, 132,271f Split, splitting field, 72f, 153f staircase, 22 Standard polynomial, see Polynomial Standard tableau, 244f Subdirect product, 39f Symmetric element, 140f System, 145f

INDEX

T T-ideal, 110, 133f, 137f, 170, 240f, 243f, 248 Tensor product, 59f of algebras, 61f of PI-algebras, 71, 239f Torsion-free, 52 Trace identity, 139f Transcendence basis, 82f Transcendence degree, 82f Transpose, see Involution

U Ultrafilter, 277f Ultraproduct, 278 Universal intersection theorem, 307f Universal PI-algebra (relatively free), I10 Upper nil radical, 43 Upper triangular matrices, 10.22, 252

Vandennonde argument, 130 Vector space, 32 Von Neumann regular, 108, 148

W Weakly multiplicative, 105 Wedderburn’s method, 178f Wedderburn’s theorem, 152 Word, 2f, 204f blank, 2 height of, 2 length of, 2 order on, 3

Y Young diagrams, 243f

V

Valuation ring, 80f. 90f of a discrete valuation, 191

3 65

6

Zorn, xx

Pure and Applied Mathematics A Series of Monographs and Textbooks

Editors

Samuel Elenberg and Hyman Ears Columbia University, New York

RECENT TITLES

I. MARTINISAACS. Character Theory of Finite Groups JAMESR. BROWN. Ergodic Theory and Topological Dynamics C. TRUESDELL. A First Course in Rational Continuum Mechanics: Volume 1, General Concepts GEORGE GBATZER. General Lattice Theory K. D. STROYAN A N D W. A. J. LUXEMBURG. Introduction to the Theory of Infinitesimals AND JOHN D. DIXON. Modular Representations of Finite B. M. PUITASWAMAIAH Groups M ELWN BERGER. Nonlinearity and Functional Analysis : Lectures on Nonlinear Problems in Mathematical Analysis CHARALAMBOS D. ALIPRANTIS A N D OWENBURKINSHAW. Locally Solid Riesz Spaces J A N MIKUSINSKI. The Bochner Integral THOMAS JECH. Set Theory CARLL. DEVITO.Functional Analysis MICHIELHAZEWINKEL. Formal Groups and Applications SIGURDUR HELGASON. Differential Geometry, Lie Groups, and Symmetric Spaces ROBERT B. BURCKEL. An Introduction to Classical Complex Analysis : Volume 1 JOSEPH J. ROTMAN. An Introduction to Homological Algebra A N D R. G. MUNCASTER. Fundamentals of Maxwell's Kinetic Theory of a C. TRUESDELL Simple Monatomic Gas : Treated as a Branch of Rational Mechanics BARRY SIMON. Functional Integration and Quantum Physics A N D ARTOSALOMAA. The Mathematical Theory of L Systems. GRZEGORZ ROZENBERG DAVIDKINDERLEHRER and GUIDOSTAMPACCHIA. An Introduction to Variational Inequalities and Their Applications. H. SEIFERT A N D W. THRELFALL. A Textbook of Topology; H. SEIFERT. Topology of 3-Dimensimal Fibered Spaces ROWEN.Polynominal Identities in Ring Theory LOUISHALLE

IN PREPARATION

ROBERT B. BURCKEL. An Introduction to Classical Complex Analysis : Volume 2 DRAWSM. CVETKOVIC, MICHAELDOOB,A N D HOST SACHS. Spectra of Graphs W. KAHN.Introduction to Global Analysis DONALD EDWARD PRUGOVECKI. Quantum Mechanics in Hilbert Space ROBERT M. YOUNG.An Introduction to Nonharmonic Fourier Series