Introductory Lectures on Rings and ModulesCambridge University Press, 1999 M04 22 - 238 páginas The focus of this book is the study of the noncommutative aspects of rings and modules, and the style will make it accessible to anyone with a background in basic abstract algebra. Features of interest include an early introduction of projective and injective modules; a module theoretic approach to the Jacobson radical and the Artin-Wedderburn theorem; the use of Baer's criterion for injectivity to prove the structure theorem for finitely generated modules over a principal ideal domain; and applications of the general theory to the representation theory of finite groups. Optional material includes a section on modules over the Weyl algebras and a section on Goldie's theorem. When compared to other more encyclopedic texts, the sharp focus of this book accommodates students meeting this material for the first time. It can be used as a first-year graduate text or as a reference for advanced undergraduates. |
Contenido
RINGS | 1 |
11 Basic definitions and examples | 2 |
12 Ring homomorphisms | 20 |
13 Localization of integral domains | 34 |
14 Unique factorization | 43 |
15 Additional noncommutative examples | 50 |
MODULES | 63 |
21 Basic definitions and examples | 64 |
32 The Jacobson radical | 147 |
33 Semisimple Artinian rings | 155 |
34 Orders in simple Artinian rings | 161 |
REPRESENTATIONS OF FINITE GROUPS | 171 |
41 Introduction to group representations | 172 |
42 Introduction to group characters | 187 |
43 Character tables and orthogonality relations | 196 |
APPENDIX | 207 |
22 Direct sums and products | 78 |
23 Semisimple modules | 88 |
24 Chain conditions | 97 |
25 Modules with finite length | 102 |
26 Tensor products | 109 |
27 Modules over principal ideal domains | 121 |
28 Modules over the Weyl algebras | 127 |
STRUCTURE OF NONCOMMUTATIVE RINGS | 137 |
31 Prime and primitive ideals | 138 |
A2 Zorns lemma | 211 |
A3 Matrices over commutative rings | 213 |
A4 Eigenvalues and characteristic polynomials | 216 |
A5 Noncommutative quotient rings | 221 |
A6 The ring of algebraic integers | 226 |
| 229 | |
LIST OF SYMBOLS | 231 |
| 233 | |
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a₁ abelian group Ann(M Artinian ring ascending chain assume automorphism basis c₁ called chain condition characters of G coefficients commutative ring completely reducible completes the proof complex numbers composition series conjugacy classes contains corresponding cyclic defined definition denoted direct sum division ring divisor EndR(M Endz Example exists FG-module field F finite dimensional finite group function group homomorphism idempotent identity element implies integral domain invertible irreducible characters isomorphic Jacobson radical ker(ƒ left ideal left order left R-module Lemma Let F Let G linear transformation M₁ mapping matrix ring maximal ideal minimal submodules Mn(R module RM n x n noncommutative nonzero element one-to-one polynomial ring positive integer prime ideal primitive principal ideal domain Proposition Prove quotient field R-homomorphism regular element representation ring homomorphism scalar simple left simple modules space over F subgroup subring subset subspace tensor product unique factorization vector space zero