Popesco-Gabriel theorem for \({\mathcal L}(V)\)
type: Item
instance of: scholarly article
title: Popesco-Gabriel theorem for \({\mathcal L}(V)\)
zbMATH Open document ID: 0592.16016
DOI: 10.1016/0021-8693(85)90094-8
author: Toyonori Kato
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let R be a ring with identity, \({\mathcal M}_ R\) the category of unital right R-modules, V an injective module in \({\mathcal M}_ R\), and let \({\mathcal L}(V)\) be the full subcategory of \({\mathcal M}_ R\) consisting of all modules of V-dominant dimension \(\geq 2\). The author proves the following ring theoretical version of the Popesco-Gabriel Theorem: Let V be an injective right R-module, \(U\in {\mathcal L}(V)\), \(S=End(U_ R)\), and Q the V-localization of R. Then the following conditions are equivalent: (1) U is a generator for \({\mathcal L}(V)\). (2) V is U-distinguished in \({\mathcal M}_ Q\). (3) The Hom-functors \[ Hom(U_ R,-)_ S: {\mathcal L}(V)\rightleftarrows {\mathcal L}(W): Hom(U^*_ S,-)_ R \] are equivalences, where \(W_ S=Hom(U_ R,V)_ S\) and \({}_ RU^*_ S=_ RHom(U_ R,Q)_ S\). (4) \(Hom(Q/I,V_ R)=0\) with I the trace ideal of U in Q.
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16B50
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 18E15
zbMATH DE Number: 3950756
zbMATH Keywords: category of unital right R-modules
zbMATH Keywords: injective module
zbMATH Keywords: V-dominant dimension
zbMATH Keywords: Popesco-Gabriel Theorem
zbMATH Keywords: V-localization
zbMATH Keywords: generator
zbMATH Keywords: equivalences
reviewed by: T. W. Hungerford
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90094-8
OpenAlex ID: W2067748591
cites work: Rings of U-dominant dimension \(\geq 1\)
cites work: Dominant modules
cites work: U-distinguished modules
cites work: Localizations in categories of modules. I
cites work: Q5331540
cites work: On splitting of module categories
cites work: Q5599869
description: scientific article; zbMATH DE number 3950756