Finiteness conditions for graded modules (Gr-\(\Sigma\) (\(\Delta\) )- injective modules)
type: Item
instance of: scholarly article
title: Finiteness conditions for graded modules (Gr-\(\Sigma\) (\(\Delta\) )- injective modules)
zbMATH Open document ID: 0587.16001
DOI: 10.1016/0021-8693(86)90072-4
author: Constantin Năstăsescu
author: Serban Raianu
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: This paper is mainly concerned with providing ungraded finiteness properties from graded finiteness properties for \({\mathbb{Z}}\)-graded rings. It is shown in Section 2 of the paper that a graded R-module M which is \(\kappa^ g\)-Noetherian with respect to a graded localization \(\kappa^ g\) is in fact \(\kappa\)-Noetherian with respect to the ungraded localization \(\kappa\) generated by \(\kappa^ g\). The section also contains a strengthening of Hopkins' theorem for Grothendieck categories and this allows to prove a graded version of the relative Hopkins- Levitzki theorem. In Section 3 the authors study gr-\(\Sigma\) (\(\Delta)\)- injective modules and establish similar results i.e. one can forget about the grading and obtain ungraded \(\Sigma\) (\(\Delta)\)-injectivity properties. As an application it is mentioned that \(E_{R[X]}(Q[X])\) is a \(\Sigma\) (\(\Delta)\)-injective R[X]-module whenever Q is a \(\Sigma\) (\(\Delta)\)-injective R-module. In a Section 4 the authors study some dual results concerning \(\Sigma^*(\Delta^*)\)-projective modules.
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 18E15
zbMATH DE Number: 3941723
zbMATH Keywords: graded finiteness properties
zbMATH Keywords: \({\mathbb{Z}}\)-graded rings
zbMATH Keywords: graded localization
zbMATH Keywords: Grothendieck categories
zbMATH Keywords: Hopkins-Levitzki theorem
zbMATH Keywords: injective modules
Wikidata QID: Q123361013
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90072-4
OpenAlex ID: W2084240055
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description: scientific article; zbMATH DE number 3941723