Topological Rings Satisfying Compactness Conditions

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.