Topologically semisimple and topologically perfect topological rings

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 discre...

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Autores: Positselski, Leonid|||0000-0001-8836-3911, Šťovíček, Jan|||0000-0002-0834-9376
Formato: artículo
Fecha de publicación:2022
País:España
Recursos:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:264544
Acesso em linha:https://ddd.uab.cat/record/264544
https://dx.doi.org/urn:doi:10.5565/PUBLMAT6622202
Access Level:acceso abierto
Palavra-chave:Topological rings
Discrete modules
Contramodules
Semisimple abelian categories
Perfect decompositions
Projective covers
Topological perfectness
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spelling Topologically semisimple and topologically perfect topological ringsPositselski, Leonid|||0000-0001-8836-3911Šťovíček, Jan|||0000-0002-0834-9376Topological ringsDiscrete modulesContramodulesSemisimple abelian categoriesPerfect decompositionsProjective coversTopological perfectnessExtending 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. 22022-01-0120222022-01-01Articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/264544https://dx.doi.org/urn:doi:10.5565/PUBLMAT6622202reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengopen accesshttp://purl.org/coar/access_right/c_abf2Aquest material està protegit per drets d'autor i/o drets afins. Podeu utilitzar aquest material en funció del que permet la legislació de drets d'autor i drets afins d'aplicació al vostre cas. Per a d'altres usos heu d'obtenir permís del(s) titular(s) de drets.https://rightsstatements.org/vocab/InC/1.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:2645442026-06-06T12:50:31Z
dc.title.none.fl_str_mv Topologically semisimple and topologically perfect topological rings
title Topologically semisimple and topologically perfect topological rings
spellingShingle Topologically semisimple and topologically perfect topological rings
Positselski, Leonid|||0000-0001-8836-3911
Topological rings
Discrete modules
Contramodules
Semisimple abelian categories
Perfect decompositions
Projective covers
Topological perfectness
title_short Topologically semisimple and topologically perfect topological rings
title_full Topologically semisimple and topologically perfect topological rings
title_fullStr Topologically semisimple and topologically perfect topological rings
title_full_unstemmed Topologically semisimple and topologically perfect topological rings
title_sort Topologically semisimple and topologically perfect topological rings
dc.creator.none.fl_str_mv Positselski, Leonid|||0000-0001-8836-3911
Šťovíček, Jan|||0000-0002-0834-9376
author Positselski, Leonid|||0000-0001-8836-3911
author_facet Positselski, Leonid|||0000-0001-8836-3911
Šťovíček, Jan|||0000-0002-0834-9376
author_role author
author2 Šťovíček, Jan|||0000-0002-0834-9376
author2_role author
dc.subject.none.fl_str_mv Topological rings
Discrete modules
Contramodules
Semisimple abelian categories
Perfect decompositions
Projective covers
Topological perfectness
topic Topological rings
Discrete modules
Contramodules
Semisimple abelian categories
Perfect decompositions
Projective covers
Topological perfectness
description 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.
publishDate 2022
dc.date.none.fl_str_mv 2
2022-01-01
2022
2022-01-01
dc.type.none.fl_str_mv Article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/264544
https://dx.doi.org/urn:doi:10.5565/PUBLMAT6622202
url https://ddd.uab.cat/record/264544
https://dx.doi.org/urn:doi:10.5565/PUBLMAT6622202
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
eu_rights_str_mv openAccess
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instname:Universitat Autònoma de Barcelona
instname_str Universitat Autònoma de Barcelona
reponame_str Dipòsit Digital de Documents de la UAB
collection Dipòsit Digital de Documents de la UAB
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