A note on weak \(w\)-projective modules
type: Item
instance of: scholarly article
title: A note on weak \(w\)-projective modules
zbMATH Open document ID: 1547.13023
DOI: 10.53733/336
author: Refat Abdelmawla Khaled Assaad
published in: New Zealand Journal of Mathematics
publication date: 2024-06-27T00:00:00Z
review text: Let \(R\) a commutative ring and \(M\) an \(R\)-module. Then \(M\) is said to be a weak \(w\)-projective module if \(\mathrm{Ext}^{1}_{R}(M,N) = 0\) for all \(GV\)-torsion-free \(R\)-modules \(N\) with the property that \(\mathrm{Ext}^{k}_{R}(T,N) = 0\) for all \(w\)-projective \(R\)-modules \(T\) and all integers \(k\geq 1\). In the paper under review, the author studied some properties of certain classes of commutative rings using the notion of weak \(w\)-projective modules. Among others, he proved that a ring \(R\) is a \(DW\)-ring (i. e., every ideal is a \(w\)-ideal) if and only if every weak \(w\)-projective module is projective; a ring \(R\) is a von Neumann regular ring if and only if every \(FP\)-projective module is weak \(w\)-projective if and only if every finitely presented \(R\)-module is weak \(w\)-projective; and a ring \(R\) is \(w\)-semi-hereditary if and only if every finite type submodule of a free module is weak \(w\)-projective if and only if every finitely generated ideal of \(R\) is weak \(w\)-projective.
reviewed by: A. Mimouni
Mathematics Subject Classification ID: 13D05
Mathematics Subject Classification ID: 13D07
Mathematics Subject Classification ID: 13H05
zbMATH DE Number: 7872366
zbMATH Keywords: projective module
zbMATH Keywords: weak \(w\)-projective module
zbMATH Keywords: \(w\)-flat
zbMATH Keywords: \(GV\)-torsion
zbMATH Keywords: finitely presented type
zbMATH Keywords: \(DW\)-ring
zbMATH Keywords: coherent ring
zbMATH Keywords: \(w\)-coherent ring
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7872366