Generalized Morita equivalence for linearly topologized rings
type: Item
instance of: scholarly article
title: Generalized Morita equivalence for linearly topologized rings
zbMATH Open document ID: 0661.16036
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1988-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108098
review text: Two topological rings (A,\(\sigma)\) and (R,\(\tau)\) having bases at zero consisting of right ideals \((=\) linearly topologized) are called similar if the categories \(Mod\)-A\({}_{\sigma}\) and \(Mod\)-R\({}_{\tau}\) of discrete (A,\(\sigma)\)- and (R,\(\tau)\)-modules are equivalent. The author observes that every linearly topologized ring is similar to its Hausdorff completion. For some notions which are well known in the case of discrete rings their topological modifications are given [for example, for quasi- progenerator (cf. \textit{K. R. Fuller}, J. Algebra 29, 528-550 (1974; Zbl 0306.16020)), functors Hom and tensor product]. Using these modifications a characterization of similarities is given for complete linearly topologized rings, which is analogous to the description obtained for discrete rings. Moreover, it is proved that if two commutative complete linearly topologized rings are similar then there exists an (algebraic) isomorphism \(\alpha\) : \(A\to B\) such that an ideal \(I\subset A\) is closed if and only if \(\alpha\) (I) is closed in B.
Mathematics Subject Classification ID: 16W80
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 4079683
zbMATH Keywords: Morita equivalence
zbMATH Keywords: topological rings
zbMATH Keywords: Hausdorff completion
zbMATH Keywords: complete linearly topologized rings
author: Enrico Gregorio
reviewed by: Mihail I. Ursul
MaRDI profile type: Publication
cites work: Q4056148
cites work: Q5609457
cites work: Density and equivalence
cites work: Linear kompakte Moduln und Ringe
cites work: Representable Dualities between Finitely Closed Subcategories of Modules
cites work: Q3248050
description: scientific article; zbMATH DE number 4079683