Localizations and adjointness
type: Item
instance of: scholarly article
title: Localizations and adjointness
zbMATH Open document ID: 0536.16006
DOI: 10.1007/BF01157402
author: A. I. Kashu
published in: Mathematical Notes
publication date: 1983-01-01T00:00:00Z
review text: The results are the logical continuation of the author's paper [Mat. Issled. 65, 71-87 (1982; Zbl 0492.16006)]. The homomorphisms \(\Phi_ M:M\to Hom_ S(U,U\otimes_ RM)\) and \(\Psi_ N:U\otimes_ RHom_ S(U,N)\to N\) for \({}_ SU_ R\) some S-R-module, \(M\in_ R{\mathcal M}\) the category of left R-modules and \(N\in_ S{\mathcal M}\) the category of left S-modules are under consideration. The author shows, that the homomorphism \(\Phi_ M\) is an \(r_ 0\)- localization of M \((r_ 0\) is the radical defined by \(r_ 0(M)=\ker \Phi_ M)\) if and only if \(\Psi_{U\otimes_ RM}\) is an isomorphism. The homomorphism \(\Psi_ N\) is an \(s_ 0\)-colocalization of a module N \((s_ 0\) is an idempotent preradical, for which \(s_ 0(N)=im \Psi_ N)\) if and only if \(\Phi_{Hom_ S(U,N)}\) is an isomorphism.
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16B50
Mathematics Subject Classification ID: 16Nxx
Mathematics Subject Classification ID: 16S90
zbMATH DE Number: 3851301
zbMATH Keywords: localization
zbMATH Keywords: radical
zbMATH Keywords: colocalization
zbMATH Keywords: idempotent preradical
reviewed by: L.Sh.Ioffe
MaRDI profile type: Publication
cites work: Q3953925
cites work: Q3859960
cites work: Flat modules, injective modules and quotient rings
cites work: Colocalization and localization
cites work: Torsion theories, additive semantics, and rings of quotients. With an appendix by H. H. Storrer on torsion theories and dominant dimensions
description: scientific article; zbMATH DE number 3851301