On G-\((n,d)\)-rings
type: Item
instance of: scholarly article
title: On G-\((n,d)\)-rings
zbMATH Open document ID: 1253.13014
DOI: 10.1216/RMJ-2012-42-3-999
author: Khalid Ouarghi
author: Najib Mahdou
published in: Rocky Mountain Journal of Mathematics
publication date: 2012-08-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/0902.2237
full work available at URL: https://projecteuclid.org/euclid.rmjm/1344430869
review text: Let \(R\) denote a commutative ring. An \(R\)-module \(M\) is called \(n\)-presented whenever there is an exact sequence \(F_n \to \ldots \to F_0 \to M \to 0\) of finitely generated free \(R\)-modules \(F_i, i = 0,\ldots, n\). In his paper [Commun. Algebra 22, No. 10, 3997--4011 (1994; Zbl 0814.13010)] \textit{D. L. Costa} studied the following doubly filtered set of classes of rings: For non-negative integers \(n, d\) a commutative ring \(R\) is called \((n,d)\)-ring if the projective dimension satisfies \(\text{pd}_R M \leq d\) for each \(n\)-presented \(R\)-module \(M\). The authors extend Costa's idea in the following way: A commutative ring \(R\) is called G-\((n,d)\)-ring provided every \(n\)-presented \(R\)-module \(M\) has Gorenstein projective dimension \(\text{Gpd}_R M \leq d\), where Gorenstein projective dimension is introduced by \textit{E. E. Enochs} and \textit{O. M. G. Jenda} [Math. Z. 220, No. 4, 611--633 (1995; Zbl 0845.16005)]. The authors characterize \(n\)-coherent G-\((n,0)\)-rings and give various examples of G-\((n,d)\)-rings.
reviewed by: Peter Schenzel
Mathematics Subject Classification ID: 13D05
zbMATH DE Number: 6066671
zbMATH Keywords: Gorenstein projective dimension
zbMATH Keywords: Gorenstein flat dimension
zbMATH Keywords: coherent ring
zbMATH Keywords: \((n,d)\)-ring
MaRDI profile type: Publication
arXiv ID: 0902.2237
cites work: Stable module theory
cites work: Finitistic Dimension and a Homological Generalization of Semi-Primary Rings
cites work: Strongly Gorenstein projective, injective, and flat modules
cites work: Global Gorenstein dimensions
cites work: Global Gorenstein dimensions of polynomial rings and of direct products of rings
cites work: Rings over which all modules are strongly Gorenstein projective
cites work: Q4488055
cites work: Gorenstein dimensions
cites work: Parameterizing families of non-noetherian rings
cites work: Gorenstein injective and projective modules
cites work: On gorenstein injective modules
cites work: Relative homological algebra
cites work: Q4271869
cites work: Trivial extensions of Abelian categories. Homological algebra of trivial extensions of Abelian categories with applications to ring theory
cites work: Commutative coherent rings
cites work: Gorenstein homological dimensions.
cites work: Q4039904
cites work: Q4736162
cites work: Trivial Extensions Defined by Coherent-like Conditions
cites work: Q5603302
cites work: ON COSTA'S CONJECTURE
cites work: On 2-von Neumann Regular Rings#
cites work: Gorenstein dimensions in trivial ring extensions
cites work: On (strongly) Gorenstein (semi)hereditary rings
cites work: On (Strongly) Gorenstein Von Neumann Regular Rings
cites work: Q4431261
cites work: Q3882615
OpenAlex ID: W2963011998
description: scientific article; zbMATH DE number 6066671