Commutative rings in which every finitely generated ideal is quasi-projective
type: Item
instance of: scholarly article
title: Commutative rings in which every finitely generated ideal is quasi-projective
zbMATH Open document ID: 1226.13014
published in: Journal of Pure and Applied Algebra
publication date: 2011-08-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/0810.0359
review text: The main purposes of the paper are to define the notion of an fqp-ring and to investigate the correlation between the fqp-property and Prüfer conditions. Let \(R\) be a commutative ring. Let \(M\) be an \(R\)-module. An \(R\)-module \(V\) is said to be \(M\)-projective if the map \(\text{Hom}_R(V,M) \rightarrow \text{Hom}_R(V,M/N)\) is surjective for any submodule \(N\) of \(M\). It is said that \(V\) is quasi-projective if it is \(V\)-projective. An fqp-ring is defined as a ring in which every finitely generated ideal is quasi-projective. In section\,2, it is shown that a finitely generated \(R\)-module is quasi-projective if and only if it is a direct summand of \((R/I)^n\) for some ideal \(I\) of \(R\) and integer \(n\geq0\). In section\,3, the following theorems are given. (Theorem\,3.2) We have the following implications: \(R\) is an arithmetical ring \(\Rightarrow\) \(R\) is an fqp-ring \(\Rightarrow\) \(R\) is a Gaussian ring. And the implications are irreversible in general. (Theorem\,3.11) The weak global dimension of an fqp-ring is equal to \(0,1\) or \(\infty\). (Theorem\,3.14) Let \(R\) be a local ring. Then \(R\) is an fqp-ring if and only if \(R\) is a Prüfer ring and the total quotient ring of \(R\) is an fqp-ring. In section\,4, the fqp-property in trivial ring extensions is studied.
reviewed by: Yoichi Aoyama
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13C13
zbMATH DE Number: 5941008
zbMATH Keywords: quasi-projective
zbMATH Keywords: fqp-ring
zbMATH Keywords: Prüfer condition
author: Jawad Abuhlail
author: Mohammad Jarrar
author: S. Kabbaj
MaRDI profile type: Publication
OpenAlex ID: W2096219124
arXiv ID: 0810.0359
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DOI: 10.1016/J.JPAA.2011.02.008
description: scientific article