Noncommutative Rings

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AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 15 15

Noncommutative Rings provides a cross-section of ideas, techniques, Noncommutative Rings I. N. Hersteein and results that give the reader an idea of that part of algebra which concerns itself with noncommutative rings. In the space of 200 pages, Herstein covers the Jacobson radical, semisimple rings, commuta- NONCOMMUTATIVE tivity theorems, simple algebras, representations of nite groups, polynomial identities, Goldie's theorem, and the Golod–Shafarevitch theorem. Almost every practicing ring theorist has studied portions of RINGS this classic monograph.

Herstein's book is a guided tour through a gallery of masterpieces. I. N. Herstein The author's style is always elegant and his proofs always enlight- ening… I had a lot of pleasure when I rst read this book while I was an undergraduate student attending to a course given by C. Procesi at the University of Rome. Today, I appreciate even more the author's mastery and real gift for exposition.

—Fabio Mainardi, MAA Reviews M A PRESS / MAA AMS

4-color Process 215 pages spine: 3/8" finish size: 5.5" X 8.5" 50 lb stock 10.1090/car/015

Noncommutative Rings

I. N. Herstein Fifth Printing 2005

Paperback ISBN 978-0-88385-039-8 eISBN 978-1-61444-015-4 Hardcover (out of print) ISBN 978-0-88385-015-2

Printed in the United States of America CARUS MATHEMATICAL MONOGRAPHS 15

Noncommutative Rings

I. N. Herstein

Published and Distributed by The Mathematical Association of America THE CARUS MATHEMATICAL MONOGRAPHS

Published by THE MATHEMATICAL ASSOCIATION OF AMERICA

Committee on Publications R. P. BOAS, Chairman

Subcommittee on Cants Monographs IVAN NFVEN, Chairman R. P. BOAS OLGA TAUSSKY The following Monographs have been published:

1. Calculus of Variations, by G A. Bliss (out of print) 2. Analytic Functions of a Complex Variable, by D. R. Curtiss (out of print) 3. Mathematical Statistics, by H. L. Rietz (out of print) 4. Projective Geometry, by J. W. Young (out of print) 5. A History of Mathematics in America before 1900, by D. E. Smith and Jekuthiel Ginsburg (out of print) 6. Fourier Series and Orthogonal Polynomials, by Dunham Jackson (out of print) 7. Vectors and Matrices, by C. C. MacDuffee (out of print) 8. Rings and Ideals, by Í. H. McCoy (out of print) 9. The Theory of Algebraic Numbers, second edition, by Harry Pollard and Harold G Diamond 10. The Arithmetic Theory of Quadratic Forms, by B. W. Jones (out of print) 11. Irrational Numbers, by Ivan Niven 12. Statistical Independence in Probability, Analysis and Number Theory, by Mark Kac 13. A Primer of Real Functions, fourth edition, by Ralph P. Boas, Jr. Revised and updated by Harold P. Boas 14. Combinatorial Mathematics, by Herbert J. Ryser 15. Noncommutative Rings, by I. N. Herstein 16. Dedekind Sums, by Hans Rademacher and Emil Grosswald 17. The Schwarz Function and its Applications, by Philip J. Davis 18. Celestial Mechanics, by Harry Pollard 19. Field Theory and its Classical Problems, by Charles Robert Hadlock 20. The Generalized Riemann Integral, by Robert M. McLeod 21. From Error-Correcting Codes through Sphere Packings to Simple Groups, by Thomas M. Thompson 22. Random Walks and Electric Networks, by Peter G. Doyle and J. Laurie Snell 23. Complex Analysis: The Geometric Viewpoint, second edition, by Steven G Krantz 24. Knot Theory, by Charles Livingston 25. Algebra and Tiling: Homomorphisms in the Service of Geometry, by Sherman Stein and Sandor Szabo 26. The Sensual (Quadratic) Form, by John H. Conway assisted by Francis Y. C. Fung 27. A Panorama of Harmonic Analysis, by Steven G. Krantz 28. Inequalities from Complex Analysis, by John P. D'Angelo 29. Ergodic Theory of Numbers, by Karma Dajani and Cor Kraaikamp 30. A Tour through Mathematical Logic, by Robert S. Wolf TO THE MEMORY OF MY FATHER

PREFACE

This book is not intended as a treatise on ring theory. Instead, the intent here is to present a certain cross- section of ideas, techniques and results that will give the reader some inkling of what is going on and what has gone on in that part of algebra which concerns itself with noncommutative rings. There are many portions of great importance in the theory which are not touched upon or which are merely mentioned in passing. On the other hand there is a rather detailed treatment given to some aspects of the subject. While the account given here is not completely self- contained, to follow it does not require a great deal beyond a good first course in algebra. Perhaps I should spell out what I would expect in such a course. To begin with one should have been introduced to some of the basic structures of algebra—groups, rings, fields, vector spaces—and to have seen some of the basic theorems about them. One would want a good familiarity with homomorphisms, the early homomorphism theorems, quotient structures and the like. One should have learned with some thoroughness linear algebra—the fundamental theorems about linear transformations on a vector space. This type of material can be found in many books, for instance, Birkhoff and MacLane A Survey of Modern Algebra or my book Topics in Algebra. Beyond these standard topics cited above I shall make frequent use of results from the theory of fields. All these can be found in the chapter on field theory in van der Waerden's Modern Algebra. My advice, to the reader not familiar with this material, is to read into a proof until such a result is cited and then to read about the notions arising in van der Waerden's book. Finally, I shall continually use Zorn's Lemma and the axiom of choice.

ix χ PREFACE

A great deal of what is done in this book is based on selected parts of two sets of my notes published in the University of Chicago lecture notes series. Part of this selection and weeding process, polishing and blending together was accomplished in a course I gave at Bow- doin College, under the auspices of the Mathematical Association of America, in the summer of 1965 to a group of mathematicians teaching at various colleges and smaller universities. I should like to thank the par- ticipants in that course for their patience and enthusi- asm. There are many others I should like to thank, Nathan Jacobson and Irving Kaplansky, for the part they and their work have played in my formation as a mathematician, Shimshon Amitsur for the many pleas- ant hours spent together working and discussing ring theory and my students, Claudio Procesi and Lance Small, for taking the notes at Bowdoin and for their stimulating comments, suggestions and improvements. CONTENTS 1. THE JACOBSON RADICAL 1 1. Modules 1 2. The radical of a ring 8 3. Artinian rings 18 4. Semisimple Artinian rings 25 References 37 2. SEMISIMPLE RINGS 39 1. The density theorem 39 2. Semisimple rings 52 3. Applications of Wedderburn's theorem 56 References 67 3. COMMUTATIVITY THEOREMS 69 1. Wedderburn's Theorem and some generalizations 69 2. Some special rings 76 References 87 4. SIMPLE ALGEBRAS 89 1. The Brauer group 89 2. Maximal subfields 94 3. Some classic theorems 96 4. Crossed products 107 References 123 5. REPRESENTATIONS OF FINITE GROUPS 124 1. The elements of the theory 124 2. A theorem of Hurwitz 141 3. Applications to group theory 144 References 149 6. POLYNOMIAL IDENTITIES 150 1. A result on radicals 150 2. Standard identities 153 3. A theorem of Kaplansky 157 4. The Kurosh Problem for P.I. algebras 162 References 168 7. GOLDIE'S THEOREM 169 1. Ore's theorem 169 2. Goldie's theorems 171 3. Ultra-products and a theorem of Posner 179 References 186 8. THE GOLOD-SHAFAREVITCH THEOREM 187 References 194 SUBJECT INDEX 195 NAME INDEX 199 AFTERWORD by Lance Small 201

SUBJECT INDEX

Albert's Theorem, 103 Central simple algebra, 89 Albert-Brauer-Hasse-Ncether The- reverse of, 91 orem, 121 Centralizer, 94 Algebra, 14 double, 104 algebraic, 155 Centroid, 46 Artinian, 19 Character of representation, 125 central simple, 89 irreducible, 138 finite-dimensional semi-simple, orthogonality relation of, 134, 51 138 graded, 181 unit, 128 locally finite, 162 Class function, 126 radical of, 15 Class sum, 129 Algebraic algebra Cohomology group, 117 of bounded degree, 155 Commuting ring, 5 Algebraic extension, 76 Contragradient representation, Annihilator 134 ideal, 174 Crossed product, 108 left, 171 Cyclotomic polynomial, 66 right, 171 Artinian algebra, 19 Degree Artinian ring, 18 of division algebra, 119 semi-simple, 25 of homogeneous element, 181 simple, 48 of irreducible representation, torsion free, 30 127 Augmentation ideal, 58 Dense action, 40 Density theorem, 39,41 Brauer group, 93 Derivation, 100 Burnside Problem, 62, 162, 167, inner, 100 187, 193 Desargues' theorem, 69 Bounded, 62,168 Direct product, 52 for matrix groups, 66 Division algebra Burnside's theorem, 146 central, 91 degree of. 119

Cayley's theorem, 124 index of (er(D)), 119 Cayley-Hamilton theorem, 65 real, 102 Central division algebra, 91 splitting field of, 106 195 196 SUBJECT INDEX

Division ring Identity finite, 70, 102 polynomial, 153 Double Centralizer theorem, 104 standard, 154 Index of nilpotence, 32 Equivalent representations, 125 Inseparable extension, 76 Essential left ideal, 171 Irreducible constituent, 126 Extension multiplicity of, 126 algebraic, 76 Irreducible module, 5 inseparable, 76 Irreducible representation, 125 separable, 76 degree of, 127 character of, 135 Factor set, 108 condition, 108 Jacobson radical, 9 equivalence of, 110 Jacobson's theorem, 73 normalized, 110 Jacobson-Noether theorem, 78 product of, 11 Faithful module, 3 Kaplansky's theorem, 157 Filter, 179 Kothe conjecture, 21 ultra, 179 Kronecker product, 135 Frobenius' theorem, 146 Kurosh problem, 162, 167, 187, Frobenius' theorem on real 192 division algebras, 102 Large left ideal, 171 Goldie Ring, 171 Left Goldie's theorem, 176 Golod-Shafarevitch theorem, 188 annihilator, 171 Graded algebra, 181 essential ideal, 171 Group large ideal, 171 order, 170 locally finite, 62 quotient ring, 170 torsion, 62 Levitzki's theorem, 37 Group algebra, 26,28, 51 Linear representation, 131 of p-group, 58 Locally finite H»(G,K'), 111 algebra, 162 Homogeneous element, 181 group, 62 Hurwitz's theorem, 144 ideal, 163 radical, 163 Ideal locally finite, 163 Maschke's theorem, 26 nil, 13 Module, 1 nilpotent, 13,14 faithful, 3 quasi-regular, 12 irreducible, 5 regular right, 10 of semi-simple Artinian ring, Idempotent, 21 97,98 SUBJECT INDEX 197

representation, 124 Representation unitary, 2 character of, 125 Multiplication ring, 46 contragradient, 134 Multiplicity of representation, equivalence of, 125 126 irreducible, 125 linear, 131 Nil ideal, 13 module, 124 Nilpotent basis, 56 right regular, 26, 127 Nilpotent ideal, 13, 14 unit, 128 Nilpotence Right annihilator, 171 of element, 13 Right-quasi-regular, 12 of ideal, 13, 14 Right-quasi-inverse, 12 index of, 32 Ring Noetherian ring, 35, 171 Artinian, 18 Noether-Skolem theorem, 99 commuting, 5 of endomorphisms, 4 Order, 170 Goldie, 171 Ore condition, 170 multiplication, 46 Ore's theorem, 170 Noetherian, 35, 171 Orthogonality relations, 134, 138 prime, 44 primitive, 39 Pappus' theorem, 69 semi-prime, 172 Peirce decomposition, 32 semi-simple, 16 Polynomial identity, 153 semi-simple Artinian, 25 Posner's theorem, 181, 184 simple, 33 p-regular element, 131 subdirectly irreducible, 52 Prime ring, 44 torison free, 30 Primitive ring, 39

Schur's lemma, 5 Quasi-regular ideal, 12 Semi-prime ring, 172 Quaternions, 93, 96 Semi-simple Artinian ring, 25 Quotient ring, 170 finite dimensional algebra, 51 Radical, 9 Semi-simple ring, 16,54 algebra, 15 commutative, 54 Jacobson,9 Wedderburn theorem, 34 locally finite, 163 Separable extension, 76 Regular Simple algebra element, 170 central, 89 p-regular, 131 reverse of, 92 quasi, 12 Simple ring, 33 right ideal, 10 Wedderburn-Artin theorem, Regular representation, 26, 127 48 198 SUBJECT INDEX

Subdirectly irreduciblering, 52 Subdirect sum ofrings, 52 Ultra products, 179 Wedderburn theorems, 48 NAME INDEX

Albert, Α. Α., 89, 103, 121, 123 Jacobson, Ν., 1,9, 12, 38,41,68, Amitsur, S. Α., 15,28,37,38, 89, 70, 73, 78, 79, 81, 82, 88, 105, 123, 150, 155, 168, 179 106,123,168 Artin, E., 38, 48, 87, 88, 89, 123 Jain, S. K.,87 Auslander, M., 28, 38, 89, 123 Kaplansky, I., 58, 63, 68, 77, 79, Belluce, R., 87 88,150,157,166,168,179,185 Bergman, G., 40,67 Kothe,G.,21 Brauer, R., 89, 93, 94, 107, 121, Kurosh, Α., 15,38,150,162,165, 123,130,149 167,187,192 Burnside, W., 56, 62, 63, 64, 65, Levitzki, J., 35, 36, 38, 150, 155, 66,145,146,162,168,187,193 168 Chase, S., 89,123 Martindale, W., 88 Chevalley, C, 41 Maschke, H., 25, 26, 51, 124 Curtis, C, 149 McLaughlin, J., 28,38 Deuring, M., 123 Nagata, M., 88 Dickson, L. E., 89,103 Nakayama, T., 88 Eckmann.B., 141,149 Nesbitt, C, 38,123 Faith, C. C, 87,169 Noether, E., 78, 79, 89, 96, 99, Goldie, A. W., 44,169, 171, 172, 103, 104, 105, 106, 107, 121 Novikov, 63 173, 175, 176, 177, 178, 183, Ore, O., 169,170 186 Passmann, P. S., 28,38 Goldman, O., 123 Posner, E., 179,181,184,186 Golod, E. S., 63,68,187,188,194 Procesi, C, 58, 67, 68, 169, 171, Hall, M., 149 186 Harris, B., 58,67 Reiner, I., 149 Harrison, D., 89, 123 Rickart, G. E., 28 Hasse, H., 89,121 Rosenberg, Α., 89,123 Herstein, I. N., 28,35,36,38,58, Sasiada, E., 33,38 67,68, 79, 87, 88,149,169,186 Schur,I.,5,6, 7,40,43,47 Hopkins, C, 20 Serre,J.P.,89 Hurwitz, Α., 141,143 Shafarevitch, I. R., 63, 68, 187, Ikeda.M.,88 188,194 199 200 NAME INDEX

Skolem, T., 96, 99, 103, 104, 107 Villamayor, O., 28,38 Small, Lance, 35, 36, 38, 58, 68, Wedderburn, J. Η. M., 34, 39, 169,171,186 48,49,56,58,59,68,69, 70, 72, Thompson, John, 148,149 73, 88, 89, 91, 93, 97, 102, 106, Thrall, R., 38,123 107, 114, 122, 161, 169, 184 Tuzuku,T.,88 Wielandt, H., 146,149 Utumi, Y., 36, 38,169 Zelinsky, D., 89,123 Afterword

Noncommutative Rings is a classic. It is fair to say that almost every practicing ring theorist has, at some time, studied portions of this book. Herstein's style and grace make ring theory especially attractive. In this reprinting we have not changed text—only corrected typos. There are no additional chapters or remarks in the body of the text. However, we do want to add a few remarks as a guide to the current literature and an indication of some of the enor- mous development in ring theory since Noncommutative Rings appeared. The theory of rings satisfying a polynomial identity had a huge "pay-off" when Amitsur, in 1971, produced a finite- dimensional algebra that is not a crossed product. Polynomial identity theory itself was dramatically changed by Formanek's and Razmyslov's discovery of central polynomials around the same time. For an account of all of this, see Rowen's book [4]. Noetherian ring theory experienced a dramatic development with substantial applications to the study of enveloping algebras of finite-dimensional Lie algebras. The vade mecum of Noetherian rings is the volume of McConnell and Robson [2]. For a crisp exposition, see also the monograph of Goodearl and Warfield [1]. Ring theory and its methods have had an impact in other areas as well. "Actions" on rings and algebras, modem invari- ant theory, are active areas of interest; see the recent monograph of Montgomery [3].

201 202 AFTERWORD

Finally, I wish to thank George Bergman and his students, Susan Montgomery and Adrian Wadsworth, for compiling lists of typos, etc. that were used in correcting the text of Noncom- mutative Rings.

Lance W. Small

References

1. K. R. Goodearl and R. B. Warfield, Jr., An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts 16, Cambridge, New York, 1989. 2. J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, John Wiley & Sons, Chichester, 1987. 3. Susan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics 82, American Mathematical Society, Providence, RI, 1993. 4. Louis Rowen, Polynomial identities in ring theory, Academic Press, New York, 1980. AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 15 15

Noncommutative Rings

Noncommutative Rings provides a cross-section of ideas, techniques, Noncommutative Rings I. N. Herstein and results that give the reader an idea of that part of algebra which concerns itself with noncommutative rings. In the space of 200 pages, Herstein covers the Jacobson radical, semisimple rings, commuta- NONCOMMUTATIVE tivity theorems, simple algebras, representations of nite groups, polynomial identities, Goldie's theorem, and the Golod–Shafarevitch theorem. Almost every practicing ring theorist has studied portions of RINGS this classic monograph.

—Fabio Mainardi, MAA Reviews M A PRESS / MAA AMS

4-color Process 215 pages spine: 3/8" finish size: 5.5" X 8.5" 50 lb stock