A First Course in Noncommutative Rings

by Lam, T. Y.; Axler, S.; Gehring, F. W.; Halmos, P. R.

• ISBN13: 9780387953250
• ISBN10: 0387953256
• Edition: 2nd
• Format: Paperback
• Copyright: 2001-06-01
• Publisher: Springer Verlag

Summary

This book, an outgrowth of the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-semester course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semisimple rings, Jacobson's theory of the radical, representation theory of groups and algebras, prime and semiprime rings, local and semilocal rings, perfect and semiperfect rings, etc. By aiming the level of writing at the novice rather than the connoisseur and by stressing the role of examples and motivation, the author has produced a text that is suitable not only for use in a graduate course, but also for self-study in the subject by interested graduate students. More than 400 exercises testing the understanding of the general theory in the text are included in this new edition.

Table of Contents

Preface to the Second Edition	vii
Preface to the First Edition	ix
Notes to the Reader	xvii
Wedderburn--Artin Theory	1	(47)
Basic Terminology and Examples	2	(23)
Exercises for §1	22	(3)
Semisimplicity	25	(5)
Exercises for §2	29	(1)
Structure of Semisimple Rings	30	(18)
Exercises for §3	45	(3)
Jacobson Radical Theory	48	(53)
The Jacobson Radical	50	(17)
Exercises for §4	63	(4)
Jacobson Radical Under Change of Rings	67	(11)
Exercises for §5	77	(1)
Group Rings and the J-Semisimplicity Problem	78	(23)
Exercises for §6	98	(3)
Introduction to Representation Theory	101	(52)
Modules over Finite-Dimensional Algebras	102	(15)
Exercises for §7	116	(1)
Representations of Groups	117	(24)
Exercises for §8	137	(4)
Linear Groups	141	(12)
Exercises for §9	152	(1)
Prime and Primitive Rings	153	(49)
The Prime Radical; Prime and Semiprime Rings	154	(17)
Exercises for §10	168	(3)
Structure of Primitive Rings; the Density Theorem	171	(20)
Exercises for §11	188	(3)
Subdirect Products and Commutativity Theorems	191	(11)
Exercises for §12	198	(4)
Introduction to Division Rings	202	(59)
Division Rings	203	(13)
Exercises for §13	214	(2)
Some Classical Constructions	216	(22)
Exercises for §14	235	(3)
Tensor Products and Maximal Subfields	238	(10)
Exercises for §15	247	(1)
Polynomials over Division Rings	248	(13)
Exercises for §16	258	(3)
Ordered Structures in Rings	261	(18)
Orderings and Preorderings in Rings	262	(8)
Exercises for §17	269	(1)
Ordered Division Rings	270	(9)
Exercises for §18	276	(3)
Local Rings, Semilocal Rings, and Idempotents	279	(56)
Local Rings	279	(17)
Exercises for §19	293	(3)
Semilocal Rings	296	(12)
Appendix: Endomorphism Rings of Uniserial Modules	302	(4)
Exercises for §20	306	(2)
The Theory of Idempotents	308	(18)
Exercises for §21	322	(4)
Central Idempotents and Block Decompositions	326	(9)
Exercises for §22	333	(2)
Perfect and Semiperfect Rings	335	(35)
Perfect and Semiperfect Rings	336	(11)
Exercises for §23	346	(1)
Homological Characterizations of Perfect and Semiperfect Rings	347	(12)
Exercises for §24	358	(1)
Principal Indecomposables and Basic Rings	359	(11)
Exercises for §25	368	(2)
References	370	(3)
Name Index	373	(4)
Subject Index	377