2022 Topologically semisimple and topologically perfect topological rings
Leonid Positselski, Jan Šťovíček
Author Affiliations +
Publ. Mat. 66(2): 457-540 (2022). DOI: 10.5565/PUBLMAT6622202

Abstract

Extending the Wedderburn–Artin theory of (classically) 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). Our results in this direction complement those of Iovanov–Mesyan–Reyes. An extension of 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 conditions are equivalent for topological rings with a countable base of neighborhoods of zero and for topologically right coherent topological rings. Considering the rings of endomorphisms of modules as topological rings with the finite topology, we establish a close connection between the concept of a topologically perfect topological ring and the theory of modules with perfect decomposition. Our results also apply to endomorphism rings and direct sum decompositions of objects in certain additive categories more general than the categories of modules; we call them topologically agreeable categories. We show that any topologically agreeable split abelian category is Grothendieck and semisimple. We also prove that a module Σ-coperfect over its endomorphism ring has a perfect decomposition provided that either the endomorphism ring is commutative or the module is countably generated, partially answering a question of Angeleri Hügel and Saorín.

Citation

Download Citation

Leonid Positselski. Jan Šťovíček. "Topologically semisimple and topologically perfect topological rings." Publ. Mat. 66 (2) 457 - 540, 2022. https://doi.org/10.5565/PUBLMAT6622202

Information

Received: 7 January 2022; Accepted: 25 May 2020; Published: 2022
First available in Project Euclid: 22 June 2022

MathSciNet: MR4443747
zbMATH: 1504.16077
Digital Object Identifier: 10.5565/PUBLMAT6622202

Subjects:
Primary: 16W80
Secondary: 16K40 , 16L30 , ‎16N40 , 18E10

Keywords: contramodules , discrete modules , perfect decompositions , projective covers , semisimple abelian categories , topological perfectness , topological rings

Rights: Copyright © 2022 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.66 • No. 2 • 2022
Back to Top