A First Course in Noncommutative RingsSpringer Science & Business Media, 21 thg 6, 2001 - 385 trang A First Course in Noncommutative Rings, 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 th 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. |
Nội dung
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 |
| 377 | |
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Thuật ngữ và cụm từ thông dụng
a₁ abelian artinian ring assume automorphism B₁ central idempotents commutative ring conjugacy class conjugate Corollary cyclic decomposition defined denotes direct sum division algebra division ring domain element endomorphism equivalent example Exercise exists fact finite group finite-dimensional group G group ring Hint homomorphism idempotents implies indecomposable infinite integer irreducible isomorphism J-semisimple Jacobson radical k-algebra kG-module left artinian left ideal left primitive ring left resp Lemma Let G local ring M₁ matrix minimal left ideal module multiplication Neumann regular nil ideal nilpotent noncommutative nontrivial nonzero polynomial prime ideal primitive idempotents primitive ring Proof Proposition prove quaternions quotient R-module R/rad representation result right artinian right ideal right R-module ring theory semilocal ring semiperfect ring semiprime semiprime ring semisimple ring simple left R-module simple R-module simple ring splitting field subgroup submodule subring surjective Theorem unique Wedderburn zero