Modules with chain conditions for finite matrix subgroups
type: Item
instance of: scholarly article
title: Modules with chain conditions for finite matrix subgroups
zbMATH Open document ID: 0876.16018
author: Wolfgang Zimmermann
published in: Journal of Algebra
publication date: 1997-11-09T00:00:00Z
review text: The author presents new ways of describing modules with maximum resp. minimum condition for finite matrix subgroups and shows that these chain conditions can be characterized by two well-known natural transformations. The first one is \(\mu\colon\text{Hom}_S(P,M)\otimes_RA\to\text{Hom}_S(P,M\otimes_R A)\) given by \(\mu(\varphi\otimes a)(x)=\varphi(x)\otimes a\) for \(\varphi\in\text{Hom}_S(P,M)\), \(a\in A\) and \(x\in P\). Here \(_SM_R\) is a bimodule and \(_SP\) and \(_RA\) are modules. Dual results are proven by studying the homomorphism \(\nu\colon c\otimes_R\text{Hom}_S(M,U)\to\text{Hom}_S(\text{Hom}_R(C,M),U)\) with \(\nu(c\otimes\varphi)(g)=\varphi(g(c))\) for \(c\in C\), \(\varphi\in\text{Hom}_S(M,U)\), and \(g\in\text{Hom}_R(C,M)\), where \(_SM_R\) is a bimodule and \(C_R\) and \(_SU\) are modules.
Mathematics Subject Classification ID: 16P70
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 1013965
zbMATH Keywords: pure-injective modules
zbMATH Keywords: finite matrix subgroups
zbMATH Keywords: chain conditions
zbMATH Keywords: natural transformations
reviewed by: Xue Weimin
MaRDI profile type: Publication
full work available at URL: jabr.1996.6882
OpenAlex ID: W2052091725
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DOI: 10.1006/JABR.1996.6882
description: scientific article; zbMATH DE number 1013965