Topologies and rings which arise from Artinian modules over a commutative ring
type: Item
instance of: scholarly article
title: Topologies and rings which arise from Artinian modules over a commutative ring
zbMATH Open document ID: 0836.13018
author: Ikuo Nishitani
published in: Journal of Algebra
publication date: 1996-04-29T00:00:00Z
review text: Let \(R\) be a commutative ring with identity, \(M\) be a certain unital \(R\)- module and \(I\) an ideal in \(R\). -- A linear topology \({\mathcal D}\) on \(R\) is called: (1) Artinian, if \(R/J\) is Artinian for every open ideal \(J\) in \(R\); (2) \(M\)-topology, if the set \(\{(0 :_R N) |N\) is a finitely generated submodule of \(M\}\) forms a base of neighbourhoods of zero in \({\mathcal D}\); (3) weakly adic, if there exists an ideal \(J\) in \(R\) such as the set of all subsets, every of which is a closure of some set \(J^n\), forms a base of neighbourhoods of zero in \({\mathcal D}\). It is proved that there exists an \(R\)-module \(X\) such that: -- a topology \({\mathcal D}\) is an Artinian topology iff it is an \(M\)- topology for some submodule \(M\) of \(X\); -- an Artinian topology \({\mathcal D}\) is weakly adic iff the corresponding module \(M\) is Artinian. Some properties of a completion of a topological ring equipped with weakly adic or \(M\)-topology are investigated.
Mathematics Subject Classification ID: 13J20
Mathematics Subject Classification ID: 13J10
Mathematics Subject Classification ID: 13E10
zbMATH DE Number: 823520
zbMATH Keywords: Artinian topology
zbMATH Keywords: completion of a topological ring
reviewed by: Vladimir Arnautov
MaRDI profile type: Publication
full work available at URL: jabr.1995.1314
OpenAlex ID: W2006281533
DOI: 10.1006/JABR.1995.1314
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description: scientific article; zbMATH DE number 823520