Topologically semisimple and topologically perfect topological rings

0558266 - MÚ 2023 RIV ES eng J - Journal Article
Positselski, Leonid - Šťovíček, J.
Topologically semisimple and topologically perfect topological rings.
Publicacions Matematiques. Roč. 66, č. 2 (2022), s. 457-540. ISSN 0214-1493. E-ISSN 0214-1493
Institutional support: RVO:67985840
Keywords : discrete modules and contramodules * projective covers * modules with perfect decomposition
OECD category: Pure mathematics
Impact factor: 1.1, year: 2022
Method of publishing: Open access
https://dx.doi.org/10.5565/PUBLMAT6622202

Extending the Wedderburn-Artin theory of semisimple associative rings to the realm of topological rings with right linear topology, we show that the abelian category of left contramodules over such a ring is split (equivalently, semisimple) if and only if the abelian category of discrete right modules over the same ring is split (equivalently, semisimple). An extension of the Bass theory of left perfect rings to the topological realm is formulated as a list of conjecturally equivalent conditions, many equivalences and implications between which we prove. In particular, all the conditions are equivalent for topological rings with a countable base of neighborhoods of zero. We establish a connection between the concept of a topologically perfect topological ring and the theory of modules with perfect decomposition and show that a countably generated module Sigma-coperfect over its endomorphism ring has a perfect decomposition, partially answering a question of Angeleri Hugel and Saorin.
Permanent Link: http://hdl.handle.net/11104/0331998