A First Course in Noncommutative Rings

Couverture
Springer Science & Business Media, 21 juin 2001 - 385 pages
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.
 

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Table des matières

WedderburnArtin Theory
1
1 Basic Terminology and Examples
2
Exercises for 1
22
2 Semisimplicity
25
Exercises for 2
29
3 Structure of Semisimple Rings
30
Exercises for 3
45
Jacobson Radical Theory
48
14 Some Classical Constructions
216
Exercises for 14
235
15 Tensor Products and Maximal Subfields
238
Exercises for 15
247
16 Polynomials over Division Rings
248
Exercises for 16
258
Ordered Structures in Rings
261
17 Orderings and Preorderings in Rings
262

4 The Jacobson Radical
50
Exercises for 4
63
M Jacobson Radical Under Change of Rings
67
Exercises for 5
77
6 Group Rings and the JSemisimplicity Problem
78
Exercises for 6
98
Introduction to Representation Theory
101
7 Modules Over FiniteDimensional Algebras
102
Exercises for 7
116
M Representations of Groups
117
Exercises for 8
137
9 Linear Groups
141
Exercises for 9
152
Prime and Primitive Rings
153
10 The Prime Radical Prime and Semiprime Rings
154
Exercises for 10
168
11 Structure of Primitive Rings the Density Theorem
171
Exercises for 11
188
12 Subdirect Products and Commutativity Theorems
191
Exercises for 12
198
Introduction to Division Rings
202
13 Division Rings
203
Exercises for 13
214
Exercises for 17
269
18 Ordered Division Rings
270
Exercises for 18
276
Local Rings Semilocal Rings and Idempotents
279
Exercises for 19
293
20 Semilocal Rings
296
Endomorphism Rings of Uniserial Modules
302
Exercises for 20
306
21 The Theory of Idempotents
308
Exercises for 21
322
22 Central Idempotents and Block Decompositions
326
Exercises for 22
333
Perfect and Semiperfect Rings
335
23 Perfect and Semiperfect Rings
336
Exercises for 23
346
24 Homological Characterizations of Perfect and Semiperfect Rings
347
Exercises for 24
358
25 Principal Indecomposables and Basic Rings
359
Exercises for 25
368
References
370
Name Index
373
Subject Index
377
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