Table Of Content

Graduate A Igeb ta: Noncornmutatiye view Louis Halle Rowefl Graduate Studies in Mathematics Volume 91 Graduate Algebra: Noncom mutative View Graduate Algebra: Noncom mutative View Louis Halle Rowen Graduate Studies in Mathematics Volume 91 American Mathematical Society Providence, Rhode Island Editorial Board Walter Craig Nikolai Ivanov Steven G. Krantz David Saltman (Chair) 2000 Mathematics Subject Classification. Primary 16-01, 17-01; Secondary 17Bxx, 20Cxx, 20Fxx. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-91 Library of Congress Cataloging-in-Publication Data Rowen, Louis Halle. Graduate algebra : commutative view / Louis Halle Rowen. p. cm. - (Graduate studies in mathematics, ISSN 1065-7339 : v. 73) Includes bibliographical references and index. ISBN 978-0-8218-0570-1 (alk. paper) 1. Commutative algebra. 2. Geometry, Algebraic. 3. Geometry, Affine. 4. Commutative rings. 5. Modules (Algebra). I. Title. II. Series. QA251.3.R677 2006 512'.44-dc22 2006040790 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint -permi ss [email protected]. © 2008 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http : //www. ams. org/ 10987654321 131211 100908 To the memory of my beloved mother Ruth Halle Rowen, April 5, 1918 - January 5, 2007 Contents Introduction xiii List of symbols xvii Prerequisites xxiii Part IV. The Structure of Rings 1 Introduction 3 Chapter 13. Fundamental Concepts in Ring Theory 5 Matrix rings 7 Basic notions for noncommutative rings 14 Direct products of rings 16 The structure of Hom(M, N) 19 Representations of rings and algebras 21 The regular representation of an algebra 25 Supplement: Polynomial rings 26 Appendix 13A. Ring constructions using the regular representation 28 Chapter 14. Semisimple Modules and Rings and the Wedderburn-Artin Theorem 33 Semisimple modules 33 Semisimple rings 37 The Wedderburn-Artin Theorem 40 vii viii Contents Supplement: Rings with involution 43 Chapter 15. The Jacobson Program Applied to Left Artinian Rings 45 Primitive rings and ideals 46 The Jacobson radical 50 The structure of left Artinian rings 50 Supplement: The classical theory of finite-dimensional algebras 54 Appendix 15A: Structure theorems for rings and algebras 55 Appendix 15B. Kolchin's Theorem and the Kolchin Problem 60 Chapter 16. Noetherian Rings and the Role of Prime Rings 63 Prime rings 64 Rings of fractions and Goldie's Theorems 67 Applications to left Noetherian rings 77 The representation theory of rings and algebras: An introduction 78 Supplement: Graded and filtered algebras 82 Appendix 16A: Deformations and quantum algebras 83 Chapter 17. Algebras in Terms of Generators and Relations 87 Free algebraic structures 88 The free group 93 Resolutions of modules 99 Graphs 100 Growth of algebraic structures 104 Gel'fand-Kirillov dimension 109 Growth of groups 114 Appendix 17A. Presentations of groups 121 Groups as fundamental groups 122 Appendix 17B. Decision problems and reduction procedures 124 Appendix 17C: An introduction to the Burnside Problem 134 Chapter 18. Tensor Products 137 The basic construction 138 Tensor products of algebras 147 Contents ix Applications of tensor products 150 Exercises - Part IV 161 Chapter 13 161 Appendix 13A 164 Chapter 14 165 Chapter 15 167 Appendix 15A 170 Appendix 15B 171 Chapter 16 173 Appendix 16A 179 Chapter 17 180 Appendix 17A 184 Appendix 17B 187 Appendix 17C 187 Chapter 18 189 Part V. Representations of Groups and Lie Algebras 193 Introduction 195 Chapter 19. Group Representations and Group Algebras 197 Group representations 197 Modules and vector spaces over groups 202 Group algebras 204 Group algebras over splitting fields 211 The case when F is not a splitting field 216 Supplement: Group algebras of symmetric groups 218 Appendix 19A. Representations of infinite groups 228 Linear groups 230 Appendix 19B: Algebraic groups 238 The Tits alternative 244 Chapter 20. Characters of Finite Groups 249 Schur's orthogonality relations 250 The character table 254 Arithmetic properties of characters 257