Topological Rings (eBook)

S. Warner (Autor)

eBook Download: PDF

1993 | 1. Auflage
497 Seiten
Elsevier Science (Verlag)
9780080872896 (ISBN)

Infos

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.

Inhalt

Front Cover 1
Topological Rings 4
Copyright Page 5
Contents 10
Preface 8
Chapter I. Topological Rings and Modules 12
1. Examples of Topological Rings 12
2. Topological Modules, Vector Spaces, and Algebras 22
3. Neighborhoods of Zero 29
4. Subrings, Ideals, Submodules, and Subgroups 34
5. Quotients and Projective Limits of Rings and Modules 43
Chapter II. Metrizability and Completeness 55
6. Metrizable Groups 55
7. Completions of Commutative Hausdorff Groups 66
8. Completions of Topological Rings and Modules 75
9. Baire Spaces 79
10. Summability 82
11. Continuity of Inversion and Adversion 91
Chapter III. Local Boundedness 99
12. Locally Bounded Modules and Rings 99
13. Locally Retrobounded Division Rings 105
14. Norms and Absolute Values 113
15. Finite-dimensional Vector Spaces 123
16. Topological Division Rings 131
Chapter IV. Real Valuations 145
17. Real Valuations and Valuation Rings 145
18. Discrete Valuations 152
19. Extensions of Real Valuations 163
Chapter V. Complete Local Rings 177
20. Noetherian Modules and Rings 177
21. Cohen Subrings of Complete Local Rings 184
22. Complete Discretely Valued Fields 197
23. Complete Local Noetherian Rings 203
24. Complete Semilocal Noetherian Rings 208
Chapter VI. Primitive and Semisimple Rings 217
25. Primitive Rings 217
26. The Radical of a Ring 227
27. Artinian Modules and Rings 237
Chapter VII. Linear Compactness and Semisimplicity 243
28. Linearly Compact Rings and Modules 243
29. Linearly Compact Semisimple Rings 252
30. Strongly Linearly Compact Modules 262
31. Locdy Linearly Compact Semisimple Rings 267
32. Locally Compact Semisimple Rings 278
Chapter VIII. Linear Compactness in Rings with Radical 294
33. Linear Compactness in Rings with Radical 294
34. Lifting Idempotents 306
35. Locally Compact Rings 328
36. The Radical Topology 341
Chapter IX. Complete Local Noetherian Rings 370
37. The Principal Ideal Theorem 370
38. Krull Dimension and Regular Local Rings 379
39. Complete Regular Local Rings 388
40. The Japanese Property 402
Chapter X. Locally Centrally Linearly Compact Rings 409
41. Complete Discretely Valued Fields and Division Rings 409
42. Finite-dimensional Algebras 417
43 Locally Centrally Linearly Compact Rings 425
Chapter XI. Historical Notes 435
44. Topologies on Commutative Rings 435
45. Locally and Linearly Compact Rings 439
46. Category, Duality, and Existence Theorems 446
Bibliography 451
Errata 498
Index of Names 500
Index of Symbols and Definitions 503

Produkt-Details

• Erscheint lt. Verlag: 7.7.1993
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-13: 9780080872896 / 9780080872896