# A note on weak w-projective modules

## DOI:

https://doi.org/10.53733/336## Keywords:

projective module , weak $w$-projective module, $w$-flat, $GV$-torsion, finitely presented type, $DW$-ring, coherent ring, $w$-coherent ring## Abstract

Let $R$ be a ring. An $R$-module $M$ is a weak $w$-projective module if ${\rm Ext}_R^1(M,N)=0$ for all $N$ in the class of $GV$-torsion-free $R$-modules with the property that ${\rm Ext}^k_R(T,N)=0$ for all $w$-projective $R$-modules $T$ and all integers $k\geq1$. In this paper, we introduce and study some properties of weak $w$-projective modules. We use these modules to characterise some classical rings. For example, we will prove that a ring $R$ is a $DW$-ring if and only if every weak $w$-projective is projective; $R$ 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 $R$ 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.

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## How to Cite

*New Zealand Journal of Mathematics*,

*55*, 43–52. https://doi.org/10.53733/336