Graduate Texts in Mathematics 131 Editorial Board

S. Axler F.W. Gehring K.A. Ribet

Springer-Science+ Business Media, LLC

Graduate Texts in Mathematics

TAKEUTUZARING. lntroduction to 35 ALEXANDERlWERMER. Several Complex Axiomatic Set Theory. 2nd ed. Variables and Banach Algebras. 3rd ed.

2 OXTOBY. Measure and Category. 2nd ed. 36 KELLEy!NAMIOKA et al. Linear 3 SCHAEFER. Topological Vector Spaces. Topological Spaces.

2nded. 37 MONK. Mathematical Logic. 4 HILTON/STAMMBACH. A Course in 38 GRAUERTIFRITZSCHE. Several Complex

Homological Algebra. 2nd ed. Variables. 5 MAC LANE. Categories for the Working 39 ARVESON. An lnvitation to C*-Algebras.

Mathematician. 2nd ed. 40 KEMENY/SNELLlKNAPP. Denumerable 6 HUGHEslPIPER. Projective Planes. Markov Chains. 2nd ed. 7 SERRE. A Course in Arithmetic. 41 ApOSTOL. Modular Functions and Dirichlet 8 T AKEUTUZARING. Axiomatic Set Theory. Series in Number Theory. 9 HUMPHREYS. Introduction to Lie Algebras 2nded.

and Representation Theory. 42 SERRE. Linear Representations of Finite 10 COHEN. A Course in Simple Homotopy Groups.

Theory. 43 GILLMAN/JERISON. Rings ofContinuous Il CONWAY. Functions ofOne Complex Functions.

Variable 1. 2nd ed. 44 KENDIG. Elementary Algebraic Geometry. 12 BEALS. Advanced Mathematical Analysis. 45 LOEVE. Probability Theory 1. 4th ed. 13 ANDERSON/FuLLER. Rings and Categories 46 LoEVE. Probability Theory Il. 4th ed.

ofModules. 2nd ed. 47 MOISE. Geometric Topology in 14 GOLUBITSKy/GUILLEMIN. Stable Mappings Dimensions 2 and 3.

and Their Singularities. 48 SACHSlWu. General Relativity for 15 BERBERIAN. Lectures in Functional Mathematicians.

Analysis and Operator Theory. 49 GRUENBERGIWEIR. Linear Geometry. 16 WINTER. The Structure ofFields. 2nded. 17 ROSENBLATT. Random Processes. 2nd ed. 50 EDWARDS. Fermat's Last Theorem. 18 HALMOS. Measure Theory. 51 KLINGENBERG. A Course in Differential 19 HALMOS. A Hilbert Space Problem Book. Geometry.

2nded. 52 HARTSHORNE. Algebraic Geometry. 20 HUSEMOLLER. Fibre Bundles. 3rd ed. 53 MANIN. A Course in Mathematical Logic. 21 HUMPHREYS. Linear Algebraic Groups. 54 GRAVERIW ATKINS. Combinatorics with 22 BARNES/MACK. An Algebraic Introduction Emphasis on the Theory of Graphs.

to Mathematical Logic. 55 BROWN/PEARCY. Introduction to Operator 23 GREUB. Linear Algebra. 4th ed. Theory 1: Elements of Functional 24 HOLMES. Geometric Functional Analysis Analysis.

and Its Applications. 56 MASSEY. Algebraic Topology: An 25 HEWITT/STROMBERG. Real and Abstract Introduction.

Analysis. 57 CROWELUFox. lntroduction to Knot 26 MANES. Algebraic Theories. Theory. 27 KELLEY. General Topology. 58 KOBLlTZ. p-adic Numbers, p-adic Analysis, 28 ZARISKIlSAMUEL. Commutative Algebra. and Zeta-Functions. 2nd ed.

Vol.l. 59 LANG. Cyclotomic Fields. 29 ZARISKUSAMUEL. Commutative Algebra. 60 ARNOLD. Mathematical Methods in

VoI.II. Classical Mechanics. 2nd ed. 30 JACOBSON. Lectures in Abstract Algebra 1. 61 WHITEHEAD. Elements of Homotopy

Basic Concepts. Theory. 31 JACOBSON. Lectures in Abstract Algebra Il. 62 KARGAPOLOvIMERLZJAKOV. Fundamentals

Linear Algebra. ofthe Theory of Groups. 32 JACOBSON. Lectures in Abstract Algebra 63 BOLLOBAS. Graph Theory.

III. Theory of Fields and Galois Theory. 64 EDWARDS. Fourier Series. VoI. 1. 2nd ed. 33 HIRSCH. Differential Topology. 65 WELLS. Differential Analysis on Complex 34 SPITZER. Principles of Random Walk. Manifolds. 2nd ed.

2nd ed. (continued afler index)

T.Y. Lam

A First Course in N oncommutative Rings

Second Edition

, Springer

T.Y. Lam Department of Mathematics University of California, Berkeley Berkeley, CA 94720-0001

Editorial Board S. Axler Mathematics Department San Francisco State

University San Francisco, CA 94132 USA

F. W. Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA

K.A. Ribet Mathematics Department University of California

at Berkeley Berkeley, CA 94720-3840 USA

Mathematics Subject Classification (2000): 16-01, 16DlO, 16D30, 16D60

Library of Congress Cataloging-in-Publication Data Lam, T.Y. (Tsit-Yuen), 1942-

A first course in noncommutative rings / T.Y. Lam. - 2nd ed. p. cm. - (Graduate texts in mathematics; 131)

lncludes bibliographical references and index. ISBN 978-0-387-95325-0 ISBN 978-1-4419-8616-0 (eBook) DOI 10.1007/978-1-4419-8616-0 1. Noncornrnutative rings. I. Title. II. Series.

QA251.4 .L36 2001 512'.4-dc21 00-052277

© 2001 Springer Science+Business Media New York Originally published by Springer-Verlag New York in 2001 AII rights reserved. This work may not be translated or copied in whole or in part without the written permis sion ofthe publisher (Springer-Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even ifthe former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production managed by Terry Komak; manufacturing supervised by Jerome Basma. Typeset by Asco Typesetters, North Point, Hong Kong.

987 6 5 4 3 2 1

To Juwen, Fumei, Juleen , and Dee-Dee

my most delightful ring

Preface to the Second Edition

The wonderful reception given to the first edition of this book by the mathe­matical community was encouraging. It gives me much pleasure to bring outnow a new edition , exactly ten years after the book first appeared.

In the 1990s, two related projects have been completed. The first is theproblem book for "First Course" (Lam [95]), which contains the solutions of(and commentaries on) the original 329 exercises and 71 additional ones.The second is the intended "sequel" to this book (once called " SecondCourse"), which has now appeared under the different title " Lectures onModules and Rings" (Lam [98]) . These two other books will be useful com­panion volumes for this one. In the present book , occasional references aremade to " Lectures" , but the former has no logical dependence on the latter.In fact, all three books can be used essentially independently.

In this new edition of "First Course" , the entire text has been retyped,some proofs were rewritten, and numerous improvements in the expositionhave been included. The original chapters and sections have remained un­changed, with the exception of the addition of an Appendix (on uniserialmodules) to §20. All known typographical errors were corrected (althoughno doubt a few new ones have been introduced in the process!) . The originalexercises in the first edition have been replaced by the 400 exercises in theproblem book (Lam [95]), and I have added at least 30 more in this editionfor the convenience of the reader. As before, the book should be suitable as atext for a one-semester or a full-year graduate course in noncommutativering theory.

I take this opportunity to thank heartily all of my students, colleagues,and other users of "First Course" all over the world for sending in correc­tions on the first edition, and for communicating to me their thoughtson possible improvements in the text. Most of their suggestions have been

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viii Preface to the Second Edition

followed in this new edition . Needless to say, I will continue to welcome suchfeedback from my readers, which can be sent to me by email at the address"Iam @math.berkeley.edu".

T.y.L.Berkeley, California01/01/01

Preface to the First Edition

One of my favorite graduate courses at Berkeley is Math 251, a one-semestercourse in ring theory offered to second-year level graduate students . I taughtthis course in the Fall of 1983, and more recently in the Spring of 1990, bothtimes focusing on the theory of noncommutative rings. This book is an out­growth of my lectures in these two courses, and is intended for use by in­structors and graduate students in a similar one-semester course in basic ringtheory.

Ring theory is a subject of central importance in algebra. Historically ,some of the major discoveries in ring theory have helped shape the course ofdevelopment of modem abstract algebra. Today, ring theory is a fertilemeeting ground for group theory (group rings), representation theory (mod­ules), functional analysis (operator algebras) , Lie theory (enveloping alge­bras) , algebraic geometry (finitely generated algebras , differential operators,invariant theory) , arithmetic (orders , Brauer groups) , universal algebra (va­rieties of rings), and homological algebra (cohomology of rings, projectivemodules , Grothendieck and higher K-groups) . In view of these basic con­nections between ring theory and other branches of mathematics, it is per­haps no exaggeration to say that a course in ring theory is an indispensablepart of the education for any fledgling algebraist.

The purpose of my lectures was to give a general introduction to thetheory of rings, building on what the students have learned from a standardfirst-year graduate course in abstract algebra. We assume that, from sucha course, the students would have been exposed to tensor products, chainconditions, some module theory , and a certain amount of commutativealgebra . Starting with these prerequisites , I designed a course dealing al­most exclusively with the theory of noncommutative rings. In accordancewith the historical development of the subject, the course begins with theWedderburn-Actin theory of semisimple rings, then goes on to Jacobson's

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x Preface to the First Edition

general theory of the radical for rings possibly not satisfying any chain con­ditions . After an excursion into representation theory in the style of EmmyNoether, the course continues with the study of prime and semiprime rings,primitive and semiprimitive rings, division rings, ordered rings, local andsemilocal rings, and finally, perfect and semiperfect rings. This material,which was as much as I managed to cover in a one-semester course, appearshere in a somewhat expanded form as the eight chapters of this book .

Of course, the topics described above correspond only to part of thefoundations of ring theory. After my course in Fall, 1983, a self-selectedgroup of students from this course went on to take with me a second course(Math 274, Topics in Algebra) , in which I taught some further basic topics inthe subject. The notes for this second course, at present only partly written ,will hopefully also appear in the future , as a sequel to the present work. Thisintended second volume will cover, among other things, the theory of mod­ules, rings of quotients and Goldie's Theorem, noetherian rings, rings withpolynomial identities, Brauer groups and the structure theory of finite­dimensional central simple algebras . The reasons for publishing the presentvolume first are two-fold: first it will give me the opportunity to class-test thesecond volume some more before it goes to press, and secondly, since thepresent volume is entirely self-contained and technically indepedent of whatcomes after , I believe it is of sufficient interest and merit to stand on its own.

Every author of a textbook in mathematics is faced with the inevitablechallenge to do things differently from other authors who have written earlieron the same subject. But no doubt the number of available proofs for anygiven theorem is finite, and by definition the best approach to any specificbody of mathematical knowledge is unique . Thus , no matter how hard anauthor strives to appear original , it is difficult for him to avoid a certain de­gree of "plagiarism" in the writing of a text. In the present case I am all themore painfully aware of this since the path to basic ring theory is so well­trodden, and so many good books have been written on the subject. If, ofnecessity, I have to borrow so heavily from these earlier books , what are thenew features of this one to justify its existence?

In answer to this, I might offer the following comments. Although a goodnumber of books have been written on ring theory , many of them aremonographs devoted to specialized topics (e.g., group rings, division rings,noetherian rings, von Neumann regular rings, or module theory, PI-theory,radical theory, loalization theory). A few others offer general surveys of thesubject, and are encyclopedic in nature. If an instructor tries to look for anintroductory graduate text for a one-semester (or two-semester) course inring theory , the choices are still surprisingly few. It is hoped , therefore, thatthe present book (and its sequel) will add to this choice. By aiming the levelof writing at the novice rather than the connoisseur, we have sought to pro­duce a text which is suitable not only for use in a graduate course, but alsofor self-study in the subject by interested graduate students.

Since this book is a by-product of my lectures, it certainly reflects much

Preface to the First Edition xi

more on my teaching style and my personal taste in ring theory than on ringtheory itself. In a graduate course one has only a limited number of lecturesat one's disposal, so there is the need to "get to the point" as quickly aspossible in the presentation of any material. This perhaps explains the oftenbusiness-like style in the resulting lecture notes appearing here. Nevertheless,we are fully cognizant of the importance of motivation and examples, andwe have tried hard to ensure that they don 't play second fiddle to theoremsand proofs. As far as the choice of the material is concerned, we have per­haps given more than the usual emphasis to a few of the famous openproblems in ring theory, for instance, the Kothe Conjecture for rings withzero upper nilradical (§IO) , the semiprimitivity problem and the zero-divisorproblem for group rings (§6), etc. The fact that these natural and very easilystated problems have remained unsolved for so long seemed to have cap­tured the students' imagination. A few other possibly "unusual" topics areincluded in the text: for instance, noncommutative ordered rings are treatedin §17, and a detailed exposition of the Mal'cev-Neumann construction ofgeneral Laurent series rings is given in §14. Such material is not easilyavailable in standard textbooks on ring theory, so we hope its inclusion herewill be a useful addition to the literature.

There are altogether twenty five sections in this book , which are consec­utively numbered independently of the chapters. Results in Section x will belabeled in the form (x.y). Each section is equipped with a collection of ex­ercises at the end. In almost all cases, the exercises are perfectly "doable"problems which build on the text material in the same section. Some ex­ercises are accompanied by copious hints; however, the more self-reliantreaders should not feel obliged to use these.

As I have mentioned before, in writing up these lecture notes I have con­sulted extensively the existing books on ring theory, and drawn materialfrom them freely. Thus lowe a great literary debt to many earlier authors inthe field. My graduate classes in Fall 1983 and Spring 1990at Berkeley wereattended by many excellent students; their enthusiasm for ring theory madethe class a joy to teach, and their vigilance has helped save me from manyslips. I take this opportunity to express my appreciation for the role theyplayed in making these notes possible. A number of friends and colleagueshave given their time generously to help me with the manuscript. It is mygreat pleasure to thank especially Detlev Hoffmann , Andre Leroy, Ka-HinLeung, Mike May, Dan Shapiro, Tara Smith and Jean-Pierre Tignol fortheir valuable comments, suggestions, and corrections. Of course, the re­sponsibility for any flaws or inaccuracies in the exposition remains my own.As mathematics editor at Springer-Verlag, Ulrike Schmickler-Hirzebruchhas been most understanding of an author's plight, and deserves a word ofspecial thanks for bringing this long overdue project to fruition. KeyboarderKate MacDougall did an excellent job in transforming my handwrittenmanuscript into LaTex, and the Production Department's efficient handlingof the entire project has been exemplary.

xii Preface to the First Edition

Last, first, and always, lowe the greatest debt to members of my family.My wife Chee-King graciously endured yet another book project, and ourfour children bring cheers and joy into my life. Whatever inner strength I canmuster in my various endeavors is in large measure a result of their love,devotion , and unstinting support.

T .Y.L.Berkeley, CaliforniaNovember, 1990

Contents

Preface to the Second Edition

Preface to the First Edition

Notes to the Reader

CHAPTER 1Wedderburn-Artin Theory

§1. Basic Terminology and ExamplesExercises for §I

§2. SemisimplicityExercises for §2

§3. Structure of Semisimple RingsExercises for §3

CHAPTER 2Jacobson Radical Theory

§4. The Jacobson RadicalExercises for §4

§5. Jacobson Radical Under Change of RingsExercises for §5

§6. Group Rings and the J-Semisimplicity ProblemExercises for §6

CHAPTER 3Introduction to Representation Theory

§7. Modules over Finite-Dimensional AlgebrasExercises for §7

vii

ix

xvii

22225293045

48

506367777898

101

102116

Xlli

xiv

§8. Representations of GroupsExercises for §8

§9. Linear GroupsExercises for §9

CHAPTER 4Prime and Primitive Rings

§1O. The Prime Radical; Prime and Semiprime RingsExercises for §10

§11. Structure of Primitive Rings; the Density TheoremExercises for §ll

§12. Subdirect Products and Commutativity TheoremsExercises for §12

CHAPTER 5Introduction to Division Rings

§13. Division RingsExercises for §13

§14. Some Classical ConstructionsExercises for §14

§15. Tensor Products and Maximal SubfieldsExercises for §15

§16. Polynomials over Division RingsExercises for §16

CHAPTER 6Ordered Structures in Rings

§17. Orderings and Preorderings in RingsExercises for §17

§18. Ordered Division RingsExercises for §18

CHAPTER 7Local Rings , Semilocal Rings, and Idempotents

§19. Local RingsExercises for §19

§20. Semilocal RingsAppendix : Endomorphism Rings of Uniserial ModulesExercises for §20

§21. Th Theory ofIdempotentsExercises for §21

§22. Central Idempotents and Block DecompositionsExercises for §22

Contents

117137141152

153

154168171188191198

202

203214216235238247248258

261

262269270276

279

279293296302306308322326333

Contents

CHAPTER 8Perfect and Semiperfect Rings

xv

335

§23. Perfect and Semiperfect Rings 336Exercisesfor §23 346

§24. Homological Characterizations of Perfect and Semiperfect Rings 347Exercisesfor §24 358

§25 . Principal Indecomposables and Basic Rings 359Exercisesfor §25 368

References

Name Index

Subject Index

370

373

377

Notes to the Reader

As we have explained in the Preface, the twenty five sections in this book arenumbered independently of the eight chapters. A cross-reference such as(12.7) refers to the result so labeled in §12. On the other hand, Exercise 12.7will refer to Exercise 7 appearing at the end of §12. In referring to an exerciseappearing (or to appear) in the same section, we shall sometimes drop thesection number from the reference. Thus, when we refer to "Exercise 7"anywhere within §12, we shall mean Exercise 12.7.

Since this is an exposition and not a treatise, the writing is by no meansencyclopedic. In particular, in most places, no systematic attempt is made togive attributions, or to trace the results discussed to their original sources.References to a book or a paper are given only sporadically where they seemmore essential to the material under consideration . A reference in bracketssuch as Amitsur [56] (or [Amitsur: 56]) shall refer to the 1956 paper ofAmitsur listed in the reference section at the end of the book.

Occasionally, references will be made to the intended sequel of this book,which will be briefly called Lectures. Such references will always be periph­eral in nature ; their only purpose is to point to material which lies ahead . Inparticular, no result in this book will depend logically on any result to ap­pear later in Lectures.

Throughout the text, we use the standard notations of modern mathe­matics. For the reader's convenience, a partial list of the notations com­monly used in basic algebra and ring theory is given on the following pages.

xvii

xviii

Some Frequently Used Notations

Notes to the Reader

c ,~

s;IAI, Card AA\BA-B6ijEijtr<x)Z(G)CG(A)[G :H][K:F][K : D]t , [K : D],KGMR,RNM®RNHomR(M ,N)EndR(M)nM (or Mn)I1Ri

charRU(R) ,R*U(D) , D*, iJGLn(R)GL(V)radRNil*(R)Nil*(R)Nil Rannt(S), ann,(S)kG,k[G]k[Xi: iEI]

kcx, : iEI)

ring of integersfield of rational numbersfield of real numbersfield of complex numbersfinite field with q elementsset of n x n matrices with entries from Sused interchangeably for inclusionstrict inclusionused interchangeably for the cardinality of the set Aset-theoretic differencesurjective mapping from A onto BKronecker deltasmatrix unitstrace (of a matrix or a field element)cyclic group generated by xcenter of the group (or the ring) Gcentralizer of A in Gindex of subgroup H in a group Gfield extension degreeleft, right dimensions of K 2 D as D-vectorspaceG-fixed points on Kright R-module M, left R-module Ntensor product of M R and RN

group of R-homomorphisms from M to Nring of R-endomorphisms of MM EB . . . EB M (n times)direct product of the rings {Ri }

characteristic of a ring Rgroup of units of the ring Rmultiplicative group of the division ring Dgroup of invertible n x n matrices over Rgroup of linear automorphisms of a vector space VJacobson radical of Rupper nilradical of Rlower nilradical (or prime radical) of Rideal of nilpotent elements in a commutative ring Rleft , right annihilators of the set S(semi)group ring of the (semi)group G over the ring kpolynomial ring over k with (commuting) variables{Xi : iEI}free ring over k generated by {Xi : i EI}

Notes to the Reader

ACCDCCLHSRHS

ascending chain conditiondescending chain conditionleft-hand sideright-hand side

xix