Topological Rings Satisfying Compactness Conditions

The main aim of this text is to introduce the beginner to the theory of topological rings. Whilst covering all the essential theory of topological groups, the text focuses on locally compact, compact, linearly compact, hereditarily linear compact and bounded topological rings. The text also contains new, unpublished results on topological rings, for example the nilideals of topological rings, trivial extensions of special type, rings with a unique compact topology, compact right topological rings and the results from groups of units of topological rings.

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."

Introduction. Notation. 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. 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. Topological groups of the second category. 19. Inverse limits of topological groups. 2. Topological rings. 1. The notion of a topological ring. 2. Neighborhoods of zero of a topological ring. 3. Subrings of topological rings. 4. Compact right topological rings. 5. The local structure of locally compact rings. 6. Structure of compact rings. 7. The separated completion of a topological ring. 8. Trivial extensions. 9. Ni1 and nilpotence in the class of locally compact rings. 10. The Wedderburn-Mal'cev theorem for compact rings. 11. Topological products of primary compact rings. 12. Zero divisors in topological rings. 13. The group of units of a topological ring. 14. Boundedness in locally compact rings. 15. Simple topological rings. 16. Homological dimension of a compact ring. 17. Local direct sums of locally compact rings. 18. Radicals in the class of locally compact rings. 19. Endc(RM). 20. Locally compact division rings. 21. Non-metrizable compact domains. 22. Open subrings of topological division. 23. Tensor products of compact rings. 24. Pseudo-compact topolo on the ring of polynomials. 25. The Lefschetz duality. 26. The uniqueness of compact ring topologies. 27. Totally bounded topological rings. 28. Representations of locally compact rings. 29. Open questions in topological groups and rings. Bibliography. Index.