A first course in noncommutative rings, t y lam 1

Graduate Texts in Mathematics 131 Editorial Board J.H Ewing F.w Gehring P.R Halmos Graduate Texts in Mathematics 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 TAKEUTI/ZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFFER Topological Vector Spaces HILTON/STAMM BACH A Course in Homological Algebra MAC LANE Categories for the Working Mathematician HUGHES/PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTI/ZARING Axiometic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FuLLER Rings and Categories of Modules GOLUBITSKY GUILEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nd ed., revised HUSEMOLLER Fibre Bundles 2nd ed HUMPHREYS Linear Algebraic Groups BARNES!MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKI/SAMUEL Commutative Algebra Vol I ZARISKI/SAMUEL Commutative Algebra Vol II JACOBSON Lectures in Abstract Algebra I Basic Concepts JACOBSON Lectures in Abstract Algebra II Linear Algebra JACOBSON Lectures in Abstract Algebra III Theory of Fields and Galois Theory HIRSCH Differential Topology SPITZER Principles of Random Walk 2nd ed WERMER Banach Algebras and Several Complex Variables 2nd ed KELLEY!NAMIOKA et al Linear Topological Spaces MONK Mathematical Logic GRAUERT/FRITZSCHE Several Complex Variables ARVESON An Invitation to C* -Algebras KEMENY/SNELL/KNAPP Denumerable Markov Chains 2nd ed ApOSTOL Modular Functions and Dirichlet Series in Number Theory SERRE Linear Representations of Finite Groups GILLMAN/JERISON Rings of Continuous Functions KENDIG Elementary Algebraic Geometry LOEVE PrObability Theory I 4th ed LOEVE Probability Theory II 4th ed MOISE Geometric Topology in Dimentions and continued after illdex T Y.Lam A First Course in Noncommutative Rings Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest T Y.Lam Department of Mathematics University of California Berkeley, CA 94720 USA Editorial Board H Ewing Department of Mathematics Indiana University Bloomington, IN 47405 USA F W Gehring Department of Mathematics University of Michigan Ann Arbor, MI 48109 USA P R Halmos Deparment of Mathematics Santa Clara University Santa Clara, CA 95053 USA Mathematics Subject Classification: 16-01, 16DlO, 16D30, 16D60, 16K20, 16K40, 16L30, 16N20, 16N60 Library of Congress Cataloging-in-Publication Data Lam, T Y (Tsit Yuen), 1942A first course in noncommutative rings / T Y Lam p cm -(Graduate texts in mathematics; 131) Includes bibliographical references and index ISBN-13: 978-1-4684-0408-1 e-ISBN-13: 978-1-4684-0406-7 DOl: 10.1007/978-1-4684-0406-7 Noncommutative rings I Title II Series QA251.4.L36 1991 512'.4-dc20 91-6893 Printed on acid-free paper ©1991 Springer-Verlag New York, Inc Softcover reprint of the hardcover I st edition 1991 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, NY 10010, USA), 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 if the 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 Camera-ready copy prepared using LaTEX 987654321 ISBN-13: 978-1-4684-0408-1 To Juwen, Fumei, Juleen and Dee-Dee who form a most delightful ring Preface One of my favorite graduate courses at Berkeley is Math 251, a one-semester course in ring theory offered to second-year level graduate students I taught this course in the Fall of 1983, and more recently in the Spring of 1990, both times 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 ring theory Ring theory is a subject of central importance in algebra Historically, some of the major discoveries in ring theory have helped shape the course of development of modern abstract algebra Today, ring theory is a fertile meeting 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, projective modules, 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 indispensable part of the education for any fledgling algebraist The purpose of my lectures was to give a general introduction to the theory of rings, building on what the students have learned from a standard first-year graduate course in abstract algebra We assume that, from such a course, the students would have been exposed to tensor products, chain conditions, some module theory, and a certain amount of commutative algebra Starting with these prerequisites, I designed a course dealing almost exclusively with the theory of noncommutative rings In accordance with the historical development of the subject, the course begins with the Wedderburn-Artin theory of semisimple rings, then goes on to Jacobson's general theory of the radical for rings possibly not satisfying any chain con- viii Preface ditions After an excursion into representation theory in the style of Emmy Noether, the course continues with the study of prime and semiprime rings, primitive and semiprimitive rings, division rings, ordered rings, local and semilocal rings, and finally, perfect and semi perfect rings This material, which was as much as I managed to cover in a one-semester course, appears here in a somewhat expanded form as the eight chapters of this book Of course, the topics described above correspond only to part of the foundations of ring theory After my course in Fall, 1983, a self-selected group 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 in the 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 This intended second volume will cover, among other things, the theory of modules, rings of quotients and Goldie's Theorem, noetherian rings, rings with polynomial identities, Brauer groups and the structure theory of finitedimensional central simple algebras The reasons for publishing the present volume first are two-fold: first it will give me the opportunity to class-test the second volume some more before it goes to press, and secondly, since the present volume is entirely self-contained and technically independent of what comes 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 inevitable challenge to things differently from other authors who have written earlier on the same subject But no doubt the number of available proofs for any given theorem is finite, and by definition the best approach to any specific body of mathematical knowledge is unique Thus, no matter how hard an author 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 the more painfully aware of this since the path to basic ring theory is so well-trodden, and so many good books have been written on the subject If, of necessity, I have to borrow so heavily from these earlier books, what are the new features of this one to justify its existence? In answer to this, I might offer the following comments Although a good number of books have been written on ring theory, many of them are monographs devoted to specialized topics (e.g., group rings, division rings, noetherian rings, von Neumann regular rings, or module theory, PI-theory, radical theory, localization theory) A few others offer general surveys of the subject, and are encyclopedic in nature If an instructor tries to look for an introductory graduate text for a one-semester (or two-semester) course in ring theory, the choices are still surprisingly few It is hoped, therefore, that the present book (and its sequel) will add to this choice By aiming the Preface ix level of 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 also for self-study in the subject by interested graduate students Since this book is a by-product of my lectures, it certainly reflects much more on my teaching style and my personal taste in ring theory than on ring theory itself In a graduate course one has only a limited number of lectures at one's disposal, so there is the need to "get to the point" as quickly as possible in the presentation of any material This perhaps explains the often business-like style in the resulting lecture notes appearing here Nevertheless, we are fully cognizant of the importance of motivation and examples, and we have tried hard to ensure that they don't play second fiddle to theorems and 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 open problems in ring theory, for instance, the Kothe Conjecture for rings with zero upper nilradical (§10), the semiprimitivity problem and the zero-divisor problem for group rings (§6), etc The fact that these natural and very easily stated problems have remained unsolved for so long seemed to have captured the students' imagination A few other possibly "unusual" topics are included in the text: for instance, noncommutative ordered rings are treated in §17, and a detailed exposition of the Mal'cevNeumann construction of general Laurent series rings is given in §14 Such material is not easily available in standard textbooks on ring theory, so we hope its inclusion here will 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 be labeled 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-reliant readers 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 material from them freely Thus lowe a great literary debt to many earlier authors in the field My graduate classes in Fall 1983 and Spring 1990 at Berkeley were attended by many excellent students; their enthusiasm for ring theory made the class a joy to teach, and their vigilance has helped save me from many slips I take this opportunity to express my appreciation for the role they played in making these notes possible A number of friends and colleagues have given their time generously to help me with the manuscript It is my great pleasure to thank especially Detlev Hoffmann, Andre Leroy, Ka-Hin Leung, Mike May, Dan Shapiro, Tara Smith x Preface and Jean-Pierre Tignol for their 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-Hirzebruch has been most understanding of an author's plight, and deserves a word of special thanks for bringing this long overdue project to fruition Keyboarder Kate MacDougall did an excellent job in transforming my handwritten manuscript into LaTex, and the Production Department's efficient handling of the entire project has been exemplary Last, first, and always, lowe the greatest debt to members of my family My wife Chee-King graciously endured yet another book project, and our four children bring cheers and joy into my life Whatever inner strength I can muster in my various endeavors is in large measure a result of their love, devotion, and unstinting support T.Y.L Berkeley, California November, 1990 References o 383 Zariski and P Samuel [58]: Commutative Algebra, Vols I, II, Van Nostrand, 1958 (Reprinted by Springer-Verlag as Graduate Texts in Math., Vols 28, 29.) Name Index Adjan, S.I., 150 Akizuki, Y., 350, 351 Albert, A.A., 220, 259, 281, 290 Amitsur, S.A., xiii, 45, 63, 75, 77, 85,86,92,104,171,181, 216, 226, 381 Archimedes, 105, 245, 282, 284, 292 Artin, E., vii, 1,21, 26, 31, 35, 51, 58, 201, 248, 255, 275, 277, 279, 284 Atiyah, M.F., 2, 302 Auslander, M., 123 Azumaya, G., 64, 301, 302 Baer, R., 168, 270 Banach, S., 85 Bass, H., 313, 335, 358, 369 Bergman, G.M., 75, 82, 182, 335 Berman, S.D., 144 Birkhoff, G.D., 215 Birkhoff, G., 13, 102, 205 Bjork, J.E., 355, 357 Blackburn, N., 103 Boole, G., 212, 343 Brauer, R., vii, 123, 126, 132, 172, 220, 222, 224, 226, 236, 259, 375, 377 Braun, A., 176 Bray, U., 268 Brown, B., 68 Brungs, H.H., 309 Burnside, W., 108, 109, 150, 151, 152 Cartan, E., 51, 346, 376 Cartan, H., 222 Cassels, J.W.S., 290 Cauchy, A., 84, 92, 330 Chevalley, C., 182, 191 Clifford, A.H., 130 Clifford, W.K., 13 Cohen, I.S., 306, 350 Cohn, P.M., 247, 248, 274, 318, 381 Coleman, D.B., 147 Connell, I.G., 92, 170 Cozzens, J.H., 42, 381 Curtis, C.W., 126, 142, 149, 381 Dade, E.C., 342 Dauns, J., 247, 381 Dedekind, R., 4, 283, 302 Deuring, M., 304 Dickson, L.E., 83, 125, 228, 229, 233, 238, 248, 265, 308, 381 Dietzmann, A.P., 105, 170 Dieudonne, J., 23 Dirac, P.A.M., Divinsky, N.J., 176,206 Eagon, J.A., 73, 381 Emmanuel, J., 124 Euclid, 23, 187, 262 Faith, C., 42, 223, 224, 381 Feit, W., 142 Fermat, P de, 115 Fitting, H., 299 Formanek, E., 94, 198 Frobenius, F.G., 82, 136, 219 386 Fuchs, L., 275, 281, 381 Gabriel, P., 302 Galois, E., 80, 229, 231, 240 Gerstenhaber, M., 256, 382 Goldie, A.W., viii, 22 Golod, E.S., 150 Gordon, B., 263, 266 Gratzer, G., 282 Green, J.A., 146 Grothendieck, A., vii, 377 Haar, A., 90 Hahn, H., 228, 241 Hall, Jr., M., 150 Hamilton, W.R., 5, 220 Hasse, H., 220 Heisenberg, W., 14 Herstein, I.N., 85, 104, 180, 209, 210, 213, 217, 222, 225, 257,382 Higman, G., 142, 144 Hilbert, D., 10, 11,23,60,71,174, 228, 229, 241, 279, 282, 286,288 Hirsch, K.A., 50 Hoffmann, D., ix Holder, 0., 34, 105,304 Hopkins, C., 21, 32, 59 Hua, L.K., 222, 223, 260 Humphreys, J.E., 161 Huppert, B., 103 Hurwitz, A., Isaacs, I.M., 142, 382 Ito, N., 139 Iwasawa, K, 102 Jacobson, N., vii, 32, 41, 52, 53, 64,85,164,176,182,191, 195, 201, 209, 213, 219, 226, 240, 255, 257, 259, 261, 269, 273, 356, 382 Jategaonkar, A.V., 182 Johnson, R.E., 275, 279 Jonah, D., 355 Name Index Jordan, C., 34, 120, 304 Jordan, D.A., 46, 382 Jordan, P., Kaplansky, I., 2, 25, 147, 153, 195, 210, 213, 258, 260, 307 Karrass, A., 103 Killing, W.KJ., 51 Kolchin, E.R., 108, 157, 160 Kothe, G., ix, 77, 174, 181,247 Krempa, J., 181 Kronecker, L., 135, 227 Krull, W., 64, 249, 291, 293, 296, 301, 302, 310 Kummer, E., 249 Lam, T.Y., 273, 289, 290, 382 Laurent, H., 9, 11, 228, 241 Leroy, A., ix Leung, KH., ix, 292 Levi, F.W., 102 Levitzki, J., 21, 32, 52, 59, 163, 173, 176, 177, 180 Lie, S., vii, 13, 51, 108, 156, 157, 216 Littlewood, D.E., 7, 149 Macdonald, I G., Magnus, W., 103 Mal'cev, A.I., ix, 228, 241, 247, 286 Maschke, H., 83, 85 May, M.K, ix McCoy, N.H., 68, 165, 171, 180, 206 Molien, T., 41, 51, 125 Moore, E.H., 214 Morita, K, 41, 323, 375 Motzkin, T.S., 263, 266 Moufang, R., 247 Nagata, M., 64, 382 Nakayama, T., 64, 377 Nathanson, M.B., 179 Nazarova, L.A., 123 Nesbitt, C.J., 377 Name Index Neumann, B.H., ix, 97, 102, 105, 228, 241, 247, 281, 286, 382 Neumann, J von, viii, 4, 65, 211 Newton, I., Niven, I., 269, 271, 272, 382 Noether, E., viii, 1, 21, 32, 82, 107, 124, 220, 247, 257, 304,382 Novikov, P.S., 150 Ore, 0., 12, 247 Osima, M., 375 Osofsky, B.L., 30 Palais, R.S., 220 Passman, D.S., 75, 86, 90, 92, 93, 98, 103, 105, 170,382 Peirce, C.S., 318 Pfister, A., 289 Pickert, G., 275, 286 Pierce, R.S., 123, 382 Plancherel, M., 147 Poincare, H., 13 Quebbermann, H.-G., 50 Ram, J., 178 Reiner, I., 126, 142, 149, 381 Remak, R., 302 Rentschler, R., 370 Rickart, C.E., 85, 86, 90, 382 Rieffel, M.A., 39 Riemann, G.F.B., 297 Ringel, C.M., 123 Roiter, A.V., 123 Rowen, L.H., 30, 370, 381, 382 Samuel, P., 2, 195, 196, 305, 383 Sanov, LN., 150 Scharlau, W., 289 Schmidt, E., 282 Schmidt, O.Ju., 293, 301, 302 Schofield, A., 248 Schreier, 0., 255, 275, 277 387 Schur, I., 35, 82, 108, 138, 153, 154 Scott, W.R., 225 Seidenberg, A., 306 Serre, J.-P., 275, 277 Shapiro, D.B., ix Small, L.W., 381, 382 Smith, T.L., ix Snapper, E., 70 Solitar, D., 103 Steinitz, E., 302 Stonehewer, S.E., 146 Suprunenko, D.A., 157 Swan, R.G., 145, 305, 314 Sylow, L., 158 Szele, T., 275, 286 Takahashi, S., 145 Thrall, R.M., 123 Tignol, J.-P., ix, 233, 236 Trazom, A.W., 398 Tschimmel, A., 289 Utumi, Y., 175 Vandermonde, A.T., 273 Vandiver, H.S., 215 Villamayor, O.E., 146 Wadsworth, A.R., 291 Waerden, van der, B.L., 179 Wallace, D.A.R., 94, 105, 131 Wedderburn, J.H.M., vii, 1, 26, 31, 35, 41, 51, 104, 209, 213, 214, 233, 265, 302 Weyl, H., 7, 32 Whaples, G., 201, 268 Wigner, E.P., Willems, W., 146 Witt, E., 13, 103, 215 Yang, C.T., 256, 382 Zariski, 0., 2, 72, 305, 383 Zorn, M., 27 Subject Index A , 6,140 As, 140-141, 148 abelian Lie algebra, 13 absolutely irreducible module, 111 absolutely simple module, 111 additive category, 15 additive commutator, 115,214,216, 261 adjoint of an operator, 174 affine algebra, 72, 73 Akizuki-Cohen Theorem, 350 Albert's Theorem, 290 Albert-Neumann-Fuchs Theorem, 281 algebra of quantum mechanics, algebraic algebra, 63, 219, 226, 257 algebraic element, 25, 62 algebraic K-theory, 313, 314, 316, 377 algebraically closed division ring, 269 271 Amitsur-McCoy Theorem, 171 Amitsur's Theorems, 45, 63, 75, 77,92,216 annihilator (left, right), 49,53,175, 179,202 archimedean class, 245, 292 archimedean ordered group, 105 archimedean ordered ring, 282-283 artinian module, 20 artinian ring, 20 Artin-8chreier Theorems, 255, 256, 277 Artin-Whaples Theorem, 201 ascending chain condition (ACC), 1,20 ACC on annihilators, 175 ACC on ideals, 180,337 ACC on principal right ideals, 355 augmentation ideal, 85, 131,211 augmentation map, 84 Baer's lower nilradical, 168 Baer-McCoy radical, 168 Baer's Theorem, 270 Banach algebra, 85-86, 90 basic idempotent, 372 basic ring (of a ring), 372 basic ring, 372, §25 basic semiperfect ring, 375 Bass' Theorems, 313, 354, 369 Bergman's examples, 182, 335 Bergman's Theorem (on graded rings),82 Berman's Theorem, 144 bimodule, 17 binary tetrahedral group, 6, 215 Birkhoff's Theorem, 205 Birkhoff-Iwasawa-Neumann Theorem, 102 Birkhoff-Vandiver Theorem, 215 Bjork's Theorem, 355, 357 block decomposition, 337, §22 Boolean ring, 212, 343 Brauer-Albert Theorem, 259 Brauer group, vii, viii Brauer's example, 236 Brauer's Lemma, 172 Subject Index 390 Brauer's Theorems, 126, 132, 226 Brauer-Nesbitt-Nakayama Theorem, 377 Brauer-Thrall Conjecture, 123 Bray-Whaples Theorem, 268 Brown-McCoy radical, 68 Burnside's Lemma, 109 Burnside's Problems, 150 Burnside's Theorems, 151, 152 C* -algebra, 90 Cancellation Theorem (for modules),315 Cartan-Brauer-Hua Theorem, 222, 226 Cartan matrix, 376-380 Cassels' Theorem, 290 Cauchy's Theorem, 84, 92 center, 48, 49, 124, 228-229 center of a group ring, 134 center of a left primitive ring, 195, 201 central idempotents, 24, 135,319, 336, 343, §22 central multiplicative set, 77 central simple algebra, viii, 214 central units (in group rings), 142 centralizer of a set, 214 centrally finite division ring, 213, 228-229 centrally infinite division ring, 213, 228-229, 247 centrally primitive idempotent, 336 change of rings, §5 character, 117 character table, 139 class function, 136 Clifford algebra, 13 Clifford's Theorem, 130, 160 Cohn's examples, 248 cohomology of rings, vii column rank (of a matrix), 227 commutator subgroup, 96 complementary idempotent, 318 completely primary ring, 301 completely (or strongly) prime ideal, 206 completely reducible linear group, 154 completely reducible (or semisimpie) module, 26 composition series, 20 convolution, 90 countably generated algebra, 64 crossed product, 247 cyclic algebra, 220, 229 cyclic division algebra, 232, 233, 238-239, 241 cyclic module, 29, 53 cyclotomic polynomial, 127, 215 8-conjugate, 187 8-conjugacy class, 187 8-ideal, 44 8-simple ring, 44 Ll(G),97 Ll+(G), 105 Dade's Lemma, 342 Dedekind cut, 283 Dedekind domain, 302, 305 Dedekind-finite ring, 4, 24, 25, 49, 67, 295, 313, 314, 316, 328 dense ring, 190 dense set of linear operators, 192 Density Theorem, 191-193, §11, 252 derivation, 11 descending chain condition (DCC), 1, 20, 241 DCC on ideals, 337 DCC on principal left ideals, 173, 354, 355 DCC on finitely generated submodules, 354 diagonalizable operator, 42 Dickson's example, 238 Dickson's Theorems, 265, 308 Dietzmann's Lemma, 105 Dieudonne's example, 23 Subject Index dihedral group (finite), 148 dihedral group (infinite), 94, 97, 105 differential operator, vii, differential polynomial ring, 11, 12, 44-45, 62, 187 direct product of rings, 24 division-closed preordering, 280 division closure of a preordering, 279 division rings, 1, 4, 9, Chapter domain, double centralizer property, 39, 50 Double Centralizer Theorem, 254, 261 entire function, 89 enveloping algebra, vii, 13 equivalent representations, 83 Euclidean algorithm, 23, 187,262 extendibility of an ordering, 280 exterior algebra, 13, 60, 297 faithful semisimple module, 182 faithful simple module, 182, 188189, 201 Fermat's Little Theorem, 115 finite conjugate (f.c.) group, 97 finite-dimensional algebras, 108, §7 finite representation type, 123 finitely generated algebra, vii, 7172, 176,332 finitely presented module, 124 Fitting Decomposition Theorem, 299 flat module, 365 formal power series ring, formal reality, 91, 277 formally real ring, 91, 279, 290291 Formanek's Theorem, 198 Fourier exansion, 146 free group, 102 free k-ring, 6, 196-197,247,283 free product of semigroups, 198 391 Frobenius' Theorem, 219, 256 full idempotent, 322 generators and relations, Gerstenhaber-Yang Theorem, 256 Goldie's Theorem, viii Gordon-Motzkin Theorems, 263, 266 graded ideal, 82 graded ring, 82, 238 Grothendieck groups, vii, 377 group character, 117, 125 group representations, 83, Chapter group ring, 8, §6 Heisenberg Lie algebra, 14 Herstein's Conjecture, 222 Herstein's Lemma, 217 Herstein's Theorem, 225 Herstein-Kaplansky Theorem, 210 Higman's Theorem, 142, 144 Hilbert Basis Theorem, 23 Hilbert domain, 71 Hilbert ring, 71-72 Hilbert space, 174 Hilbert's 17th Problem, 279 Hilbert's example, 288 Hilbert's Nullstellensatz, 60, 72, 73 Hilbert's twist, 10 Holder's Theorem, 105 Hopkins-Levitzki Theorem, 21, 32, 59 homological algebra, vii, §24 Hurwitz' ring of quaternions, I-adically complete, 330 I -adic completion, 330 icosahedral representations, 142 icosahedron, 141 ideal, 3, 19 ideals in matrix rings, 32 392 idempotent, 24, 145, §21 idempotent ideal, 181 idempotents in integral group rings, 145, 335 indecomposable module, 119, 120-123, 298 indecomposable ring, 336 infinite representation type, 123 infinitely large element, 282 infinitely small element, 282 injective module, 30 inner derivation, 12, 216 inner order, 46 inseparable element, 81, 274 integral domain, integral group ring, integral quaternions, invariant theory, vii invariant valuation ring, 310 inverse, invertible, involution, 87, 91 irreducible linear group, 161 irreducible module, 26 irreducible representation, 83 isomorphic idempotents, 326 Jacobson-Azumaya,64 Jacobson-Chevalley Density Theorem, 191 Jacobson-Herstein Theorem, 209 Jacobson-semisimple ring, 52, 55, 57,62,68,71,86,92-94, 101, 102, 105, 182 J-semisimple ring (or semiprimitive ring), ibid J-semisimplicity Problem, ix, §6 Jacobson radical, 21, 52, 53, §4 Jacobson's example, 328 Jacobson's Theorems, 209, 219, 226, 261 Jategaonkar's examples, 182 Johnson's Theorem, 279 Jonah's Theorem, 355 Jordan canonical form, 120 Subject Index Jordan-Holder Theorem, 34, 36, 38 Kaplansky's Theorem, 307 Killing form, 51 Klein 4-group, 146 Kolchin's Theorem, 160 Kothe's Conjecture, ix, 77, 174, 176, 181 Kronecker's Rank Theorem, 227 Krull-Azumaya,64 Krull-Schmidt decomposition, 294,301 Krull-Schmidt Theorem, 303, 332 Krull-Schmidt-Azumaya Theorem, 302 Krull valuation, 249, 291, 309 Krull's Theorem, 310 £1(G),90 A-potent element, 156, 157 A-potent group, 156 Laurent polynomial ring, 11, 95 Laurent series ring, 9, 243 left algebraically closed, 269 left artinian ring, 20 left dimension, 248 left Goldie ring, 22 left ideal, 3, 18 left identity, 25 left inverse, 4, 24, 25 left-invertible, 4, 24, 53, 54, 55 left irreducible idempotent, 323 left noetherian ring, 20 left operator, 33 left perfect ring, 353 left polynomial, 10 left primitive ideal, 183 left primitive ring, 41, 163, 182, 198, §11 left quasi-regular, 67 left stable range 1, 314 left T-nilpotent set, 351 left zero-divisor, 3, 10 Leung's example, 292 Subject Index Leung's Theorem, 292 level of a division ring, 289 Levitzki radical, 52, 163, 177 Levitzki-Herstein Theorem, 180 Levitzki's Theorems, 102, 176 lexicographic ordering, 102 Lie algebra, 13, 51 Lie bracket (or Lie product), 13, 115 Lie ideal, 216 Lie-Kolchin-Suprunenko Theorem, 157 lifting idempotents, 326, 329, 330 lifting pairwise orthogonal idempotents, 328 linear group, 149, §9 linkage of primitive idempotents, 337, 372 little ideal, 204 Littlewood's formula, 149 local group ring, 309 local idempotent, 312 local ring, 13, 295, §19 localization, 81, 294, 297 locally compact group, 90 locally finite group, 104, 154 locally nilpotent (one-sided) ideal, 176-177 locally nilpotent set, 176 logarithmic derivative, 187 lower central series, 158 lower nilradical (or prime radical), 163, 164, 168 m-system, 166 m-transitive, 192, 194 Magnus-Witt Theorem, 103 Mal'cev-Neumann construction, ix,241-242 Mal'cev-Neumann ring, 242 Mal'cev-Neumann-Moufang Theorem, 247 Maschke's Theorem, 83, 91, 104, 146, 155 393 matrix ring, 32, 33, 49, 61, 172, 181, 323, 356 matrix units, 33, 61, 328, 335 maximal ideal, 68,165,179 maximal left or right ideal, 53, 294 maximal preordering, 278 maximal subfield, 230, 248-249, §15, 254-255 McCoy's Theorems, 180 metro equation, 274 minimal idempotent, 333 minimal left ideal, 25, 35, 38, 148, 172, 185, 202 minimal polynomial, 42, 264, 265, 272, 274 minimal prime ideal, 180, 207 modular left ideal, 68 modular representation, 125 monogenic algebra, 72 (multiplicative) commutator, 214, 221-222 Morita Theory, 323, 370, 375 n-abelian group, 96 n-system, 167 Nakayama's Lemma, 64, 352 Neumann's Lemma, 97 Neumann's Theorem, 105 Newton's law, nil ideal, 52, 56, 105, 181 nil (left or right) ideal, 91-92, 173-176 nilpotent (left or right) ideal, 52,56, 169 nilpotent group (of class r), 158,223 nilradical of a commutative ring, 60, 70, 164 Niven-Jacobson Theorem, 269 Niven's Theorems, 271, 272 Noether-Deuring Theorem, 304 Noether-Jacobson Theorem, 257 noetherian module, 20 394 noetherian ring, viii, 20 noetherian simple ring, 46 nonassociative ring, 51 nontrivial idempotent, 293 nontrivial unit (in group rings), 94, 147 norm, 231 Op(G), 130 octahedral representation, 148 operator algebras, vii, 174 opposite ring, 5, 252 ordered division ring, 285, §18 ordered group, 100, 103, 242, 249, 283,287 ordered ring, ix, 276 ordering of a ring, 276 orthogonal idempotents, 320 orthogonality relations, 136 p-adic integers, 297, 299 p-adic numbers, 297 p-group, 104, 122, 129, 157, 131, 138, 150, 298 p'_group, 92, 93, 101 p-regular conjugacy class, 132 p-regular element, 132 Passman's Theorems, 93, 105 Peirce decomposition, 318 perfect commutative ring, 357 perfect field, 113 perfect ring, 353 Pfister's Theorem, 289 PI-algebra, 176 Plancherel formula, 147 Poincare-Birkhoff-Witt Theorem, 13 positive cone, 100, 276 preordering of a ring, 277 prime ideal, 164, 165 prime group ring, 170 prime matrix ring, 172 prime polynomial ring, 171 prime radical (or lower nilradical), 163, 164, 168 Subject Index prime ring, 163, 168, §1O primitive idempotent, 320 principal indecomposable module, 308, 371, §25 principal left ideal domain, 23 projective cover, 361 projective module, 29, 145, 307, 353,360 projective space, 224, 267 pseudo-inverse, 66 quadratic form, 13 quasi-regular, 67 quaternion algebra (generalized), 232, 238, 256 quaternions, 5, 16, 25, 219, 230, 269-270 quaternion group, 5, 6, 127, 146, 148 quotient ring, 3, radical ideal, 164 radical of an ideal, 164, 166, 168 radical of a module, 358-360 radical ring extension, 255 Ram's example, 178-179 rank of a free module, 49 rank of a linear operator, 50, 202 rational quaternions, 5, 128 real-closed field, 91, 220, 255-256, 270-271 reduced ideal, 208 reduced norm, 236 reduced ring, 4, 71, 163 reductive group, 161 regular one-sided ideal, 68 relatively archimedean, 245 Remainder Theorem, 262 representations, 83, Chapter resolvent function, 90 resolvent set, 90 Rickart's Theorem, 85-86 Rieffel's proof (of the double centralizer property), 39 Subject Index Riemann surface, 297 right algebraic, 272 right algebraically closed, 269 right artinian ring, 20 right dimension, 248 right discrete valuation ring, 309 right Goldie ring, 22 right hereditary ring, 378 right ideal, 3, 18 right identity, 25 right inverse, 4, 24, 25 right-invertible, 4, 24 right irreducible idempotent, 323 right noetherian ring, 20 right operator, 33 right perfect ring, 58, 353 right polynomial, 10 right primitive ideal, 183 right primitive ring, 182, §11 right quasi-regular, 67 right root, 262 right T-nilpotent set, 351 right zero-divisor, 3, 10 ring, ring of differential operators, ring of quotients, viii, 247, 281 ring with involution, 87, 91 ring with polynomial identity, viii ring without identity, 24, 25, 6768, 180, 281-282 row rank (of a matrix), 227 83 , 129, 146, 149 , 139-140, 148 a-conjugate, 189 a-conjugacy class, 189 a-derivation, 12 a-ideal, 44 scalar extension, 78, 110 Scharlau-Tschimmel Theorem, 289 Schofield's examples, 248 Schur's Lemma, 35 Schur's Theorems, 138, 154 395 Scott's Theorem, 225 semigroup ring, 8, 198 semilocal ring, 311, §20 semiperfect commutative ring, 350 semiperfect endomorphism ring, 348 semiperfect ring, 346, §23 semiprimary ring, 55, 317 semiprime group ring, 170 semiprime ideal, 167 semiprime matrix ring, 172 semiprime polynomial ring, 171 semiprime ring, 163, 168, §1O semiprimitive ring (see Jacobson semisimple ring) Semiprimitivity Problem for group rings, ix, §6 semisimple Lie algebra, 51 semisimple module, 26, 27, 31 semisimple operator, 42 semisimple ring, 26, 28, 29, 32, §2, §3 semisimplicity, 1, 26, §2, §3 separable algebra, 113 separable algebraic extension, 79-80, 180 separable element, 257 separable maximal subfield, 257 sfield (or division ring), 213 sign character, 139 simple artinian ring, 38, 39, 40, 253 simple (or Wedderburn) component, 37 simple decomposition, 37 simple domain, 48 simple Lie algebra, 51 simple module, 26, 28 simple ring, 3, 32, 33, 43-48 skew field (or division ring), 213 skew group ring, 14, 15, 247, 250 skew Laurent polynomial ring, 11,46 skew Laurent series ring, 10, 243 skew polynomial ring, 10, 62, 178-179, 187, 230, 233 396 skew power series ring, 10, 297 skew symmetric matrix, 50 small (or superfluous) submodule, 358 Snapper's Theorem, 70 sode of a module, 69 so de (left or right) of a ring, 69, 186, 202-203, 205 solvable ideal, 51 solvable Lie algebra, 156 splitting field (of an algebra), 112 splitting field (of a group), 126 splitting field (of a polynomial), 113 square-product, 285 stably free module, 316 Steinitz Isomorphism Theorem, 302 strictly maximal subfield, 236 strongly indecomposable module, 298 strongly nilpotent element, 180 strongly von Neumann regular ring, 211 strongly (or completely) prime ideal, 206 sub direct product, 203 sub directly irreducible ring, 204 sub directly reducible ring, 204 subnormal subgroup, 156 subring,3 superfluous (or small) submodule, 358 support, 242 Swan's example, 305-306 symmetric algebra, 13 tensor algebra, 12 tensor product of algebras, 221, 251, §15 tetrahedral group, 6, 140 topologically nil element, 90 torsion group, 150, 154 torsion-free group, 82, 95, 97, 100 Subject Index totally positive element, 280, 286 trace, 86, 91, 92, 117 Trace Lemma, 151 transfer homomorphism, 96 triangular ring, 17, 18 trivial idempotent, 293 trivial unit (in group rings), 94, 95, 101, 142 twisted (or skew) polynomial, 7,10 two-sided ideal, unipotent group, 157 unipotent operator, 156 unipotent radical, 156, 160 uniserial module, 61 unit (or invertible element), 4, 24 unit of group rings, Unit Problem for group rings, 95 unitriangular group, 157 unitriangular matrix, 157 universal algebra, vii universal enveloping algebra, 13 universal property, 3, upper central series, 223 upper nilradical, 77,163,173-174 Utumi's argument, 175 valuation ring, 296 Vandermonde matrix, 273 varieties of rings, vii von Neumann-finite ring, von Neumann regular ring, 6567, 69, 105, 181, 334 Wadsworth's example, 291 Wallace's Theorem, 105 weak preordering, 279 weakly n-finite ring, 317 Wedderburn-Art in Theory, Chapter Wedderburn-Artin Theorem, 1,35 Wedderburn radical, 1, 31, 51-52 Wedderburn's Little Theorem, 104, 209, 214 Subject Index Wedderburn's Norm Condition, 233, 238 Wedderburn's Theorems, 41,104, 214, 233, 265 well-ordered (WO) set, 241 Weyl algebra, 7, 8, 12, 14, 45, 50, 202,283 397 Zariski's Lemma, 72 zero-divisor, Zero-Divisor Problem for group rings, ix, 95 Graduate Texts in Mathematics continued from page ii 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 SACHS/WU General Relativity for Mathematicians GRUENBERG/WEIR Linear Geometry 2nd ed EDWARDS Fermat's Last Theorem KLINGENBERG A Course in Differential Geometry HARTSHORNE Algebraic Geometry MANIN A Course in Mathematical Logic GRAVER/WATKINS Combinatorics with Emphasis on the Theory of Graphs BROWN/PEARCY Introduction to Operator Theory I: Elements of Functional Analysis MASSEY Algebraic Topology: An Introduction CROWELI./FOX Introduction to Knot Theory KOBLITZ p-adic Numbers p-adic Analysis, and Zeta-Functions 2nd ed LANG Cyclotomic 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Combined 2nd ed REMMERT Theory of Complex Functions Readings ill Mathematics EBBINGHAUS/HERMES et al Numbers Readillgs in Mathematics 124 125 126 127 128 129 DUBROVIN/FOMENKO!NOVIKOV Modern Geometry Methods and Applications Vol III BERENSTEIN/GAY Complex Variables: An Introduction BOREL Linear Algebraic Groups MASSEY A Basic Course in Algebraic Topology RAUCH Partial Differential Equations FULTON/HARRIS Representation Theory: A First Course Readings ill Mathematics 130 DODSON/POSTON Tensor Geometry 131 LAM A First Course in Noncommutative Rings ... rings This theorem, regarded by many as the first major result in the abstract structure theory of rings, has remained as important today as in the earlier part of the twentieth century when it was... right artinian, we say that R is artinian Again, we shall see that this is stronger than R being only one-sided artinian ? ?1 Basic Terminology and Examples 21 Needless to say, the nomenclature above... was to give a general introduction to the theory of rings, building on what the students have learned from a standard first- year graduate course in abstract algebra We assume that, from such a course,