Rowen Ring Theory, 83

Student Edition
1. Auflage 2012
ISBN: 978-0-08-092548-6
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 688 Seiten

This is an abridged edition of the author's previous two-volume work, Ring Theory, which concentrates on essential material for a general ring theory course while ommitting much of the material intended for ring theory specialists. It has been praised by reviewers:**'As a textbook for graduate students, Ring Theory joins the best....The experts will find several attractive and pleasant features in Ring Theory. The most noteworthy is the inclusion, usually in supplements and appendices, of many useful constructions which are hard to locate outside of the original sources....The audience of nonexperts, mathematicians whose speciality is not ring theory, will find Ring Theory ideally suited to their needs....They, as well as students, will be well served by the many examples of rings and the glossary of major results.'**--NOTICES OF THE AMS

Department of Mathematics and Computer Science

Autoren/Hrsg.

Rowen, Louis H.

Inhaltsverzeichnis

1;Front Cover ;1
2;Ring Theory;4
3;Copyright Page ;5
4;Table of Contents;6
5;Foreword to the Student Edition;16
6;Errata from the Original Two-Volume Edition;18
7;Introduction: An Overview of Ring Theory;20
8;Table of Principal Notation;26
9;Chapter 0. General Fundamentals;30
9.1;0.0 Preliminary Foundations;30
9.2;0.1 Categories of Rings and Modules;40
9.3;0.2 Finitely Generated Modules, Simple Modules, and Noetherian and Artinian Modules;43
9.4;Exercises;48
10;Chapter 1. Construction of Rings;50
10.1;1.1 Matrix Rings and Idempotents;50
10.2;1.2 Polynomial Rings;63
10.3;1.3 Free Modules and Rings;71
10.4;1.4 Products and Sums;82
10.5;1.5 Endomorphism Rings and the Regular Representation;91
10.6;1.6 Automorphisms, Derivations, and Skew Polynomial Rings;99
10.7;1.7 Tensor Products;111
10.8;1.8 Direct Limits and Inverse Limits;120
10.9;1.9 Graded Rings and Modules;126
10.10;1.10 Central Localization (also, cf., §2.12.9ff.);130
10.11;Exercises;139
11;Chapter 2. Basic Structure Theory;148
11.1;2.1 Primitive Rings;149
11.2;2.2 The Chinese Remainder Theorem and Subdirect Products;161
11.3;2.3 Modules with Composition Series and Artinian Rings;164
11.4;2.4 Completely Reducible Modules and the Socle;175
11.5;2.5 The Jacobson Radical;178
11.6;2.6 Nilradicals;198
11.7;2.7 Semiprimary Rings and Their Generalizations;209
11.8;2.8 Projective Modules (An Introduction);222
11.9;2.9 Indecomposable Modules and LE-Modules;233
11.10;2.10 Injective Modules;241
11.11;2.11 Exact Functors;250
11.12;2.12 The Prime Spectrum;258
11.13;Exercises;274
12;Chapter 3. Rings of Fractions and Embedding Theorems;300
12.1;3.1 Classical Rings of Fractions;301
12.2;3.2 Goldie's Theorems and Orders in Artinian Quotient Rings;309
12.3;3.3 Localization of Nonsingular Rings and Their Modules;327
12.4;3.4 Noncommutative Localization;334
12.5;3.5 Left Noetherian Rings;344
12.6;Exercises;369
13;Chapter 4. Categorical Aspects of Module Theory;386
13.1;4.1 The Morita Theorems;386
13.2;Exercises;394
14;Chapter 5. Homology and Cohomology;398
14.1;5.0 Preliminaries about Diagrams;398
14.2;5.1 Resolutions and Projective and Injective Dimension;404
14.3;5.2 Homology, Cohomology, and Derived Functors;423
14.4;5.3 Separable Algebras and Azumaya Algebras;444
14.5;Exercises;457
15;Chapter 6 Rings with Polynomial Identities and Affine Algebras;464
15.1;6.1 Rings with Polynomial Identities;465
15.2;6.2 Affine Algebras;486
15.3;6.3 Affine PI-Algebras;500
15.4;Exercises;519
16;Chapter 7. Central Simple Algebras;526
16.1;7.1 Structure of Central Simple Algebras;526
16.2;7.2 The Brauer Group;540
16.3;Exercises;544
17;Chapter 8. Rings from Representation Theory;548
17.1;8.1 General Structure Theory of Group Algebras;548
17.2;8.2 Noetherian Group Rings;562
17.3;8.3 Enveloping Algebras;572
17.4;8.4 General Ring Theoretic Methods;584
17.5;Exercises;605
18;Dimensions for Modules and Rings;612
19;Major Ring- and Module-Theoretic Results Proved in Volume I (Theorems and Counterexamples; also cf. "Characterizations");614
20;Major Theorems and Counterexamples for Volume II;622
21;The Basic Ring-Theoretic Notions and Their Characterizations;632
22;References;636
23;Index;644

Leseproben

Table of Principal Notation
Note: ! after page reference means the symbol is used differently in another part of the text. , fA 1 {Ai:i?I} 1! Sym(n) 2 HomR(M,N) 3 N = M, M < N 3 A R 3 M/N 4 R/I 4 Z(R) 5 CR(A) 5 (M) 6 (S, =) 6 ?, ? 6 12 R- 12 R--R' 13 Hom(A, —) 13 Hom(—, A) 14 Rop 14 Asi 15 RS 15 Ann S 16 Ann x 16 ACC, DCC 18 dij 21 Mn(R) 21 eij 21 24 R[S] 35 R[?] 35 C{X} 36 R[?1, …, ?t] 37 R(S) 39 R[[?]] 42 C{{X}} 42 ? Mi 44 M(I) 44 (F; X) 46 < S> 48 Ri 53 Mi 53 M' ? M ? M? 54 Ai/F 56 Ai 60! 60 EndR M, End MT 64 [r1, r2] 71 R[?; s, d] 74 R[?; s] 74 (K, s, d) 78 (1, 2) 80 R[[?; s]] 81 R * G 82 M ? N 83 T ?R — 85 (Ai; fij) 92 ?Ai 92 ?Ai 94 ?,M? 96 Ms 97 M? n 100 TR(M) 100 ?(M) 101 RP 102! S- 1R = RS 102! R[s- 1] 107 M = M0 > … > Mt = 0 135 l(M) 137 V(S) 144 Na(R) 173 N(R) 175 {(xi, fi): i ? I} 198 M' M 201 E(M) 218 M#, f# 224 Spec(R) 229 (A) 229 LO(P) 241 GU(—, P) 241 S- 1R 276 S- 1M 279 Ann'S 280 ? 299 300! 309! QF(M) 314 p(M) 317 G(M), G(R) 328 328 ˜ 330 K-dim 332 G-dim 339 (R, R', M, M', t, t') 361 coker 369 pd(M) 377 gl.dim 381 M[?; s] 382 sM 382 FFR 385 K0(R) 386 (; (dn)) 394 , 395 Z(A), B(A), H(A) 396 Ln(T) 400 Rn(T) 403 Torn 404 Extn 404 w.dim 410 Re 416 Hn, Hn 416 J 416 MR 417 Der(M) 417 Ann' J 426 Cn, Sn 438 gn 444 Specn(R), Specn(Z) 452 Cn{Y} 453 (R) 454 C{r1, …, rt} 457 GS(n) 463 G(R) 464 GK...