Generalised reduced modules
type: Item
instance of: scholarly article
title: Generalised reduced modules
author: Annet Kyomuhangi
author: David Ssevviiri
publication date: 2023-03-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/2205.13252
review text: An \(R\)-module \(M\) is reduced if for all \(a\in R\) and \(m\in M,a^{2}m=0\) implies that \(am=0\). An equivalent formulation of a reduced module is in terms of the \(a\)-torsion functor \(\Gamma _{a}\) which assigns to an \(R\)-module \(M\) a submodule \(\Gamma _{a}(M)=\{m\in M\mid a^{k}m=0\) for some \(k\in\mathbb{Z}^{+}\}\). Then \(M\) is reduced iff \(a\Gamma _{a}(M)=0\) for all \(a\in R.\) Here the authors present a generalisation of this notion. For a positive integer \(t\) and \(a\in R\), an \(R\)-module \(M\) is \(a^{t}\)\textit{-reduced} if for every positive integer \(k\geq t\) and \(m\in M\), \(a^{k}m=0\) implies that \(a^{t}m=0\) (or equivalently, \(a^{t}\Gamma _{a}(M)=0)\) and \(M\) is \textit{universally} \(a^{t}\)\textit{-reduced} if it is \(a^{t}\)-reduced for all \(a\in R\). A number of characterizations and properties of these notions are given and the relationship between such modules and the corresponding ring is clarified.
reviewed by: Stefan Veldsman
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16S90
Mathematics Subject Classification ID: 13D07
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16D80
zbMATH DE Number: 7658704
zbMATH Keywords: reduced modules
zbMATH Keywords: reduced rings
zbMATH Keywords: regular rings
zbMATH Keywords: locally nilradical
MaRDI profile type: Publication
OpenAlex ID: W3210171026
cites work: ON REDUCED MODULES
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cites work: The locally nilradical for modules over commutative rings
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cites work: Torsion functors, small or large
cites work: Completion, Čech and Local Homology and Cohomology
DOI: 10.1007/S12215-021-00686-8
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 7658704