Topologically semisimple and topologically perfect topological rings. (English) Zbl 1504.16077
In this paper, the authors study complete, Hausdorff, associative topological rings with a right linear topology, i.e., topological rings having a base of neighborhoods of zero consisting of right ideals.
The aim of the paper is to extend some important results of rings without topology to topological rings with a right linear topology. The classical Wedderburn-Artin theory about associative semisimple rings is extended to topological rings. The Bass’ theory of left perfect rings is extended partially to right linear topological rings. Moreover, the authors provide sufficient conditions under which a topological ring is realized as a ring of endomorphisms of some module. In this article, notions and results from category theory are used.
The aim of the paper is to extend some important results of rings without topology to topological rings with a right linear topology. The classical Wedderburn-Artin theory about associative semisimple rings is extended to topological rings. The Bass’ theory of left perfect rings is extended partially to right linear topological rings. Moreover, the authors provide sufficient conditions under which a topological ring is realized as a ring of endomorphisms of some module. In this article, notions and results from category theory are used.
Reviewer: Mihail I. Ursul (Oradea)
MSC:
| 16W80 | Topological and ordered rings and modules |
| 16K40 | Infinite-dimensional and general division rings |
| 16L30 | Noncommutative local and semilocal rings, perfect rings |
| 16N40 | Nil and nilpotent radicals, sets, ideals, associative rings |
| 18E10 | Abelian categories, Grothendieck categories |
Keywords:
contramodules; discrete modules; perfect decompositions; projective covers; semisimple abelian categories; topological perfectness; topological ringsReferences:
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