Gabriel dimension for graded rings.
type: Item
instance of: scholarly article
title: Gabriel dimension for graded rings.
zbMATH Open document ID: 1114.16037
author: M. Asensio
author: Constantin Năstăsescu
author: Blas Torrecillas
published in: Applied Categorical Structures
publication date: 2007-01-25T00:00:00Z
review text: Let \(G\) be a group with identity element \(e\), and let \(R=\bigoplus_{\sigma\in G}R_\sigma\) be a ring graded by \(G\), such that the grading has finite support. Using colocalization for Grothendieck categories with a family of projective generators, it is proved that if the category \(R_e\)-mod has Gabriel dimension, then so does the category \(R\)-gr of \(G\)-graded \(R\)-modules. Then using graded Clifford theory, it is shown that if a graded module \(M\) has Gabriel dimension \(\alpha\) in \(R\)-gr, then \(M\) has Gabriel dimension \(\leq\alpha\) in \(R\)-mod. Finally, modular lattices are used to show that if \(R\)-mod has Gabriel dimension, then so does \(R\)-gr.
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 18E15
Mathematics Subject Classification ID: 16P60
zbMATH DE Number: 5118666
zbMATH Keywords: graded rings
zbMATH Keywords: Grothendieck categories
zbMATH Keywords: projective generators
zbMATH Keywords: Gabriel dimension
zbMATH Keywords: modular lattices
reviewed by: Sorin Dascalescu
MaRDI profile type: Publication
full work available at URL: s10485-006-9042-7
OpenAlex ID: W2127113693
cites work: Q3220700
cites work: Krull dimension
cites work: Graded Semiartinian Rings: Graded Perfect Rings
cites work: Localization on graded modules, relative Maschke's theorem and applications
cites work: Relative graded Clifford theory
cites work: Q3029134
DOI: 10.1007/S10485-006-9042-7
description: scientific article; zbMATH DE number 5118666