Injective cogenerators for a class of Gabriel topologies
type: Item
instance of: scholarly article
title: Injective cogenerators for a class of Gabriel topologies
zbMATH Open document ID: 0801.13007
author: Bangteng Xu
published in: Journal of Algebra
publication date: 1994-11-24T00:00:00Z
review text: Let \(R\) be a commutative ring. A linear ring topology \({\mathfrak T}\) on \(R\) is a Gabriel topology if, for every ideal \(J\) of \(R\), \(J' = \{r \in R : (J : r)\) is open\} open implies the openness of \(J\). It is known, that if \({\mathfrak T}\) is a Gabriel topology of \(R\), \(M\) is a maximal ideal, then the localization of \({\mathfrak T}\), \({\mathfrak T}_ M = \{I_ M : I \in {\mathfrak T}\}\) is a Gabriel topology of \(R_ M\). It is shown that if \({\mathfrak T}\) is a Gabriel topology of \(R\), and \(C[M]\) an injective cogenerator for \({\mathfrak T}_ M\), for every maximal ideal \(M\), then \(\prod C[M]\) \((M\) runs over maximal ideals of \(R)\) is an injective cogenerator for \({\mathfrak T}\).
Mathematics Subject Classification ID: 13J99
zbMATH DE Number: 571354
zbMATH Keywords: Gabriel topology
Wikidata QID: Q123346357
reviewed by: Witold Więsław
MaRDI profile type: Publication
full work available at URL: jabr.1993.1165
OpenAlex ID: W2081392335
DOI: 10.1006/JABR.1993.1165
description: scientific article; zbMATH DE number 571354