Gordon Ring Theory

Proceedings of a Conference on Ring Theory Held in Park City, Utah, March 2-6, 1971
1. Auflage 2014
ISBN: 978-1-4832-7415-7
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 396 Seiten, Web PDF

Ring Theory provides information pertinent to the fundamental aspects of ring theory. This book covers a variety of topics related to ring theory, including restricted semi-primary rings, finite free resolutions, generalized rational identities, quotient rings, idealizer rings, identities of Azumaya algebras, endomorphism rings, and some remarks on rings with solvable units. Organized into 24 chapters, this book begins with an overview of the characterization of restricted semi-primary rings. This text then examines the case where K is a Hensel ring and A is a separable algebra. Other chapters consider establishing the basic properties of the four classes of projective modules, with emphasis on the finitely generated case. This book discusses as well the non-finitely generated cases and studies infinitely generated projective modules. The final chapter deals with abelian groups G that are injective when viewed as modules over their endomorphism rings E(G). This book is a valuable resource for mathematicians.

Autoren/Hrsg.

Gordon, Robert

Inhaltsverzeichnis

1;Front Cover;1
2;Ring Theory;4
3;Copyright Page;5
4;Table of Contents;6
5;CONTRIBUTORS;10
6;PREFACE;12
7;CHAPTER 1. RESTRICTED SEMIPRIMARY RINGS;16
7.1;1. Introduction;16
7.2;2. Main Results;16
7.3;3. Some Related Restricted Conditions;21
7.4;References;23
8;CHAPTER 2. ALGEBRAS WITH HOCHSCHILD DIMENSION = 1;24
8.1;Introduction;24
8.2;1. Relative Hochschild dimensions and relatively separable subalgebras;25
8.3;2. Algebras with dimension = 1 and singularly segregated algebras;29
8.4;3. Separable algebras over a Hensel ring;34
8.5;References;42
9;CHAPTER 3. HEREDITARILY AND COHEREDITARILY PROJECTIVE MODULES;44
9.1;1. Conventions and background; duality of modules;45
9.2;2. Lemmas and definitions;46
9.3;3. Applications and examples;53
9.4;4. Some problems and some results on infinitely generated modules;55
9.5;5. Strongly (a-) hereditary rings;60
9.6;6. Commutative rings;70
9.7;7. Further notes;76
9.8;References;76
10;CHAPTER 4. LIFTING MODULES AND A THEOREM ON FINITE FREE RESOLUTIONS;78
10.1;Summary;78
10.2;1. Discussion of the lifting problem, and statement of the main result;78
10.3;2. Some consequences of Theorem 1;82
10.4;3. A technique for lifting;83
10.5;4. Ideals with 3 generators and 3 relations;86
10.6;References;89
11;CHAPTER 5. ON THE AUTOMORPHISM SCHEMEOF A PURELY INSEPARABLE FIELD EXTENSION;90
11.1;1. The Horn scheme and representability;92
11.2;2. The equi-exponential case;98
11.3;3. An example;101
11.4;4. The truncated automorphism scheme;104
11.5;5. The Coset space by the truncated Aut scheme;113
11.6;References;120
12;CHAPTER 6. GENERALIZED RATIONAL IDENTITIES;122
12.1;1. Introduction;122
12.2;2. Universal skew fields of fractions;123
12.3;3. Inertia;126
12.4;4. Generalized rational identities;127
12.5;References;129
13;CHAPTER 7. K2 OF POLYNOMIAL RINGS AND OF FREE ALGEBRAS;132
13.1;References;138
14;CHAPTER 8. TRIVIAL EXTENSIONS OF ABELIAN CATEGORIES AND APPLICATIONS TO RINGS: AN EXPOSITORY ACCOUNT;140
14.1;Introduction;140
14.2;1. Foundations;142
14.3;2. Coherence;148
14.4;3. Homological dimension in Map(FG, B);150
14.5;4. Pseudo-duality;156
14.6;5. Applications to dominant dimension and Gorenstein rings;160
14.7;6. Representation dimension of finite dimensional algebras;162
14.8;References;165
15;CHAPTER 9. HIGHER K-FUNCTORS;168
15.1;References;173
16;CHAPTER 10. PROPERTIES OF THE IDEALISER;176
16.1;References;183
17;CHAPTER 11. STRUCTURE AND CLASSIFICATION OF HEREDITARY NOETHERIAN PRIME RINGS;186
17.1;Introduction;186
17.2;1. Preliminaries;189
17.3;2. A Model for Almost Dedekind Rings;193
17.4;3. Omission of Primes;197
17.5;4. The Canonical Form;201
17.6;5. The Canonical Form (continued);204
17.7;6. Structure of Pseudo-Dedekind Rings;207
17.8;7. Invariants for Similarity;211
17.9;8. Invariants for Formal Conjugacy;221
17.10;9. Invariants and Conjugacy;227
17.11;References;242
18;CHAPTER 12. ON THE REPRESENTATION OF MODULESBY SHEAVES OF MODULES OF QUOTIENTS;246
18.1;References;249
19;CHAPTER 13. SOME REMARKS ON RINGS WITH SOLVABLE UNITS;250
19.1;References;255
20;CHAPTER 14. QUASI-SIMPLE MODULES AND WEAK TRANSITIVITY;256
20.1;1. Introduction;256
20.2;2. Quasi-simple modules;257
20.3;3. The weak radical of a ring;260
20.4;4. Prime rings with maximal annihilators;261
20.5;References;263
21;CHAPTER 15. PRIME RIGHT IDEALS AND RIGHT NOETHERIAN RINGS;266
21.1;1. Introduction;266
21.2;2. Proof of the Theorem;266
21.3;References;270
22;CHAPTER 16. QUOTIENT RINGS;272
22.1;1. Localizations in categories of modules;272
22.2;2. A characterization of the subcategory D(V) with V injective;277
22.3;3. Quotient rings;280
22.4;4. Epimorphisms in the category of rings;282
22.5;5. A condition for exactness of the subcategory D(V) and a weak version of Nakayama's conjecture;285
22.6;6. Injective modules;287
22.7;7. Flat modules;290
22.8;8. Flat epimorphisms in the category of rings;292
22.9;9. The double centralizer property of modules;296
22.10;References;299
23;CHAPTER 17. ON THE IDENTITIES OF AZUMAYA ALGEBRAS;302
23.1;1. Generalized polynomial identities;302
23.2;2. Generalized versus ordinary polynomial identities;307
23.3;References;310
24;CHAPTER 18. BETTI NUMBERS AND REFLEXIVE MODULES;312
24.1;Introduction;312
24.2;I. Preliminaries;312
24.3;2. BNSI Rings;313
24.4;3. Constructing BNSI Rings;318
24.5;4. QF Rings and Betti Numbers;320
24.6;Addendum;322
24.7;References;322
25;CHAPTER 19. IDEALIZER RINGS;324
25.1;1. Preparatory results;324
25.2;2. Main results;326
25.3;3. Applications;330
25.4;References;332
26;CHAPTER 20. PERFECT PROJECTORS AND PERFECT INJECTORS;334
26.1;Introduction;334
26.2;1. Preliminaries;335
26.3;2. Some properties of (Gp, Fp, Hp);337
26.4;3. Perfect projectors;339
26.5;4. Perfect injectors;342
26.6;5. Semi-perfect and perfect modules;346
26.7;References;347
27;CHAPTER 21. LINEARLY COMPACT MODULES AND LOCAL MORITA DUALITY;348
27.1;1. Introduction;348
27.2;2. Linearly compact modules and rings;348
27.3;3. Local Morita duality;353
27.4;References;361
28;CHAPTER 22. IDEALS IN FINITELY-GENERATED PI-ALGEBRAS;362
28.1;References;366
29;CHAPTER 23. INTRODUCTION TO GROUPS OF SIMPLE ALGEBRAS;368
29.1;Introduction;368
29.2;References;376
30;CHAPTER 24. MODULES OVER PIDs THAT ARE INJECTIVEOVER THEIR ENDOMORPHISM RINGS;378
30.1;1. Introduction;378
30.2;2. Duality;379
30.3;3. Pure-injectives;382
30.4;4. Injectives;384
30.5;5. Questions and examples;386
30.6;References;387
31;PROBLEMS;388