Topological Rings Satisfying Compactness ConditionsSpringer Science & Business Media, 6. pro 2012. - Broj stranica: 327 Introduction In the last few years a few monographs dedicated to the theory of topolog ical rings have appeared [Warn27], [Warn26], [Wies 19], [Wies 20], [ArnGM]. Ring theory can be viewed as a particular case of Z-algebras. Many general results true for rings can be extended to algebras over commutative rings. In topological algebra the structure theory for two classes of topological algebras is well developed: Banach algebras; and locally compact rings. The theory of Banach algebras uses results of Banach spaces, and the theory of locally compact rings uses the theory of LCA groups. As far as the author knows, the first papers on the theory of locally compact rings were [Pontr1]' [J1], [J2], [JT], [An], lOt], [K1]' [K2]' [K3], [K4], [K5], [K6]. Later two papers, [GS1,GS2]appeared, which contain many results concerning locally compact rings. This book can be used in two w.ays. It contains all necessary elementary results from the theory of topological groups and rings. In order to read these parts of the book the reader needs to know only elementary facts from the theories of groups, rings, modules, topology. The book consists of two parts. |
Sadržaj
| 1 | |
| 5 | |
| 8 | |
| 11 | |
The axioms of separation in topological groups | 15 |
Initial topologies Products of topological groups | 19 |
The coproduct topology on the algebraic direct sum | 28 |
Semidirect products of topological groups | 31 |
Trivial extensions | 149 |
Nil and nilpotence in the class of locally compact rings | 155 |
The WedderburnMalcev theorem for compact rings | 170 |
Topological products of primary compact rings | 172 |
Zero divisors in topological rings | 177 |
The group of units of a topological ring | 179 |
Boundedness in locally compact rings | 186 |
Simple topological rings | 200 |
The embedding of totally bounded groups in pseudocompact ones | 34 |
Metrization of topological groups | 35 |
The connected component of a topological group | 38 |
Quasicomponents of topological groups | 45 |
Complete topological groups | 59 |
Minimal topological groups | 71 |
Free topological groups | 75 |
The finest precompact topology on an Abelian group | 77 |
Ordered topological groups | 78 |
Topological groups of the second category | 80 |
Topological rings | 83 |
Neighborhoods of zero of a topological ring | 90 |
Subrings of topological rings | 93 |
Compact right topological rings | 94 |
The local structure of locally compact rings | 113 |
Structure of compact rings | 124 |
The separated completion of a topological ring | 143 |
Homological dimension of a compact ring | 203 |
Local direct sums of locally compact rings | 211 |
Ende RM | 230 |
Locally compact division rings | 243 |
Nonmetrizable compact domains | 253 |
Open subrings of topological division rings | 262 |
Tensor products of compact rings | 267 |
Pseudocompact topologies on the ring of polynomials | 272 |
The Lefschetz duality | 274 |
The uniqueness of compact ring topologies | 287 |
Totally bounded topological rings | 291 |
Representations of locally compact rings | 305 |
Open questions in topological groups and rings | 307 |
| 311 | |
| 325 | |
Ostala izdanja - Prikaži sve
Uobičajeni izrazi i fraze
A₁ affirm algebraic Assume the contrary base consisting canonical homomorphism Cauchy filter closed subgroup cofinite compact group compact right topological compact topology Consider contains continuous homomorphism contradiction COROLLARY Denote dense direct sum discrete division ring exists a neighborhood filter base finite subset follows fundamental system group G group topology hence idempotent Jacobson radical k₁ LCA group left ideal Lemma Let G linearly compact locally compact ring locally topologically nilpotent mapping module natural number neighborhood of zero nilpotent ideal nilring non-discrete non-zero obtain Obviously open ideal PROOF proved quotient ring R-module right ideal right topological ring ring topology ring with identity semisimple subgroup of G submodule subring subspace system of neighborhoods T₁ THEOREM topological Abelian group topological direct topological group topological product topological ring topological space topologically isomorphic topologically locally finite totally bounded totally disconnected two-sided ideal U₁ V₁ W₁