Topological Rings Satisfying Compactness Conditions [Hardcover]

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.1. Elements of topological groups.- 1. The definition of a topological group.- 2. Neighborhoods of elements of a topological group.- 3. Subgroups of a topological group.- 4. Morphisms and quotients of topological groups.- 5. The axioms of separation in topological groups.- 6. Initial topologies. Products of topological groups.- 7. The co-product topology on the algebraic direct sum.- 8. Semi-direct products of topological groups.- 9. The embedding of totally bounded groups in pseudo-compact ones.- 10. Metrization of topological groups.- 11. The connected component of a topological group.- 12. Quasi-components of topological groups.- 13. Complete topological groups.- 14. Minimal topological groups.- 15. Free topological groups.- 16. The finest precompact topology on an Abelian group.- 17. Ordered topological groups.- 18. Topologils&