Modules graded by \(G\)-sets: Duality and finiteness conditions
type: Item
instance of: scholarly article
title: Modules graded by \(G\)-sets: Duality and finiteness conditions
zbMATH Open document ID: 0887.16028
author: L. Grunenfelder
author: Daniel Bulacu
author: Sorin Dascalescu
published in: Journal of Algebra
publication date: 1998-01-07T00:00:00Z
review text: Let \(G\) be a group, \(A\) a finite \(G\)-set, \(R=\bigoplus_{g\in G}R_g\) a \(G\)-graded ring, and denote by \((G,A,R)\text{-gr}\) the category of \(A\)-graded left \(R\)-modules. If \(M\) is an object of this category and \(N\) is an \(R\)-submodule of \(M\), then let \((N)^g\) denote the smallest \(A\)-graded submodule of \(M\) containing \(N\). The paper deals with the following question: which finiteness conditions on \(N\) (that is, ascending and descending chain conditions, existence of Krull dimension) are inherited by \((N)^g\)? In case \(A=G\), such an investigation uses a well-known duality theorem of M. Cohen and S. Montgomery. The authors show that in general such a technique does not provide much, so they prove, by using different methods, the above properties on \(N\) are inherited by \((N)^g\).
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16P60
zbMATH DE Number: 1088619
zbMATH Keywords: \(G\)-sets
zbMATH Keywords: graded rings
zbMATH Keywords: finiteness conditions
zbMATH Keywords: dualities
zbMATH Keywords: categories of graded modules
reviewed by: Andrei Marcus
MaRDI profile type: Publication
full work available at URL: jabr.1997.7081
OpenAlex ID: W1980467315
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DOI: 10.1006/JABR.1997.7081
description: scientific article; zbMATH DE number 1088619