Topological Rings Book

	Preface
Ch. I	Topological Rings and Modules	1
1	Examples of Topological Rings	1
2	Topological Modules, Vector Spaces, and Algebras	11
3	Neighborhoods of Zero	18
4	Subrings, Ideals, Submodules, and Subgroups	23
5	Quotients and Projective Limits of Rings and Modules	32
Ch. II	Metrizability and Completeness	44
6	Metrizable Groups	44
7	Completions of Commutative Hausdorff Groups	55
8	Completions of Topological Rings and Modules	64
9	Baire Spaces	68
10	Summability	71
11	Continuity of Inversion and Adversion	80
Ch. III	Local Boundedness	88
12	Locally Bounded Modules and Rings	88
13	Locally Retrobounded Division Rings	94
14	Norms and Absolute Values	102
15	Finite-dimensional Vector Spaces	112
16	Topological Division Rings	120
Ch. IV	Real Valuations	134
17	Real Valuations and Valuation Rings	134
18	Discrete Valuations	141
19	Extensions of Real Valuations	152
Ch. V	Complete Local Rings	166
20	Noetherian Modules and Rings	166
21	Cohen Subrings of Complete Local Rings	173
22	Complete Discretely Valued Fields	186
23	Complete Local Noetherian Rings	192
24	Complete Semilocal Noetherian Rings	197
Ch. VI	Primitive and Semisimple Rings	206
25	Primitive Rings	206
26	The Radical of a Ring	216
27	Artinian Modules and Rings	226
Ch. VII	Linear Compactness and Semisimplicity	232
28	Linearly Compact Rings and Modules	232
29	Linearly Compact Semisimple Rings	241
30	Strongly Linearly Compact Modules	251
31	Locally Linearly Compact Semisimple Rings	256
32	Locally Compact Semisimple Rings	267
Ch. VIII	Linear Compactness in Rings with Radical	283
33	Linear Compactness in Rings with Radical	283
34	Lifting Idempotents	295
35	Locally Compact Rings	317
36	The Radical Topology	330
Ch. IX	Complete Local Noetherian Rings	359
37	The Principal Ideal Theorem	359
38	Krull Dimension and Regular Local Rings	368
39	Complete Regular Local Rings	377
40	The Japanese Property	391
Ch. X	Locally Centrally Linearly Compact Rings	398
41	Complete Discretely Valued Fields and Division Rings	398
42	Finite-dimensional Algebras	406
43	Locally Centrally Linearly Compact Rings	414
Ch. XI	Historical Notes	424
44	Topologies on Commutative Rings	424
45	Locally and Linearly Compact Rings	428
46	Category, Duality, and Existence Theorems	435
	Bibliography	440
	Errata	487
	Index of Names	489
	Index of Symbols and Definitions	492