Topological Rings

Topological Rings

by S. Warner
     
 

ISBN-10: 0444894462

ISBN-13: 9780444894465

Pub. Date: 07/21/1993

Publisher: Elsevier Science

This text brings the reader to the frontiers of current research in topological rings. The exercises illustrate many results and theorems while a comprehensive bibliography is also included.

The book is aimed at those readers acquainted with some very basic point-set topology and algebra, as normally presented in semester courses at the beginning graduate level

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Overview

This text brings the reader to the frontiers of current research in topological rings. The exercises illustrate many results and theorems while a comprehensive bibliography is also included.

The book is aimed at those readers acquainted with some very basic point-set topology and algebra, as normally presented in semester courses at the beginning graduate level or even at the advanced undergraduate level. Familiarity with Hausdorff, metric, compact and locally compact spaces and basic properties of continuous functions, also with groups, rings, fields, vector spaces and modules, and with Zorn's Lemma, is also expected.

Product Details

ISBN-13:
9780444894465
Publisher:
Elsevier Science
Publication date:
07/21/1993
Series:
North-Holland Mathematics Studies Series, #178
Pages:
512
Product dimensions:
1.19(w) x 6.14(h) x 9.21(d)

Table of Contents

Preface
Ch. ITopological Rings and Modules1
1Examples of Topological Rings1
2Topological Modules, Vector Spaces, and Algebras11
3Neighborhoods of Zero18
4Subrings, Ideals, Submodules, and Subgroups23
5Quotients and Projective Limits of Rings and Modules32
Ch. IIMetrizability and Completeness44
6Metrizable Groups44
7Completions of Commutative Hausdorff Groups55
8Completions of Topological Rings and Modules64
9Baire Spaces68
10Summability71
11Continuity of Inversion and Adversion80
Ch. IIILocal Boundedness88
12Locally Bounded Modules and Rings88
13Locally Retrobounded Division Rings94
14Norms and Absolute Values102
15Finite-dimensional Vector Spaces112
16Topological Division Rings120
Ch. IVReal Valuations134
17Real Valuations and Valuation Rings134
18Discrete Valuations141
19Extensions of Real Valuations152
Ch. VComplete Local Rings166
20Noetherian Modules and Rings166
21Cohen Subrings of Complete Local Rings173
22Complete Discretely Valued Fields186
23Complete Local Noetherian Rings192
24Complete Semilocal Noetherian Rings197
Ch. VIPrimitive and Semisimple Rings206
25Primitive Rings206
26The Radical of a Ring216
27Artinian Modules and Rings226
Ch. VIILinear Compactness and Semisimplicity232
28Linearly Compact Rings and Modules232
29Linearly Compact Semisimple Rings241
30Strongly Linearly Compact Modules251
31Locally Linearly Compact Semisimple Rings256
32Locally Compact Semisimple Rings267
Ch. VIIILinear Compactness in Rings with Radical283
33Linear Compactness in Rings with Radical283
34Lifting Idempotents295
35Locally Compact Rings317
36The Radical Topology330
Ch. IXComplete Local Noetherian Rings359
37The Principal Ideal Theorem359
38Krull Dimension and Regular Local Rings368
39Complete Regular Local Rings377
40The Japanese Property391
Ch. XLocally Centrally Linearly Compact Rings398
41Complete Discretely Valued Fields and Division Rings398
42Finite-dimensional Algebras406
43Locally Centrally Linearly Compact Rings414
Ch. XIHistorical Notes424
44Topologies on Commutative Rings424
45Locally and Linearly Compact Rings428
46Category, Duality, and Existence Theorems435
Bibliography440
Errata487
Index of Names489
Index of Symbols and Definitions492

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