A note on weak w-projective modules

TL;DR

This paper introduces weak w-projective modules over rings, explores their properties, and characterizes classical rings like DW-rings and Von Neumann regular rings using these modules.

Contribution

It defines weak w-projective modules and uses them to characterize various classical rings, providing new insights into ring theory.

Findings

A ring is a DW-ring iff every weak w-projective module is projective.

A ring is Von Neumann regular iff every FP-projective module is weak w-projective.

A ring is w-semi-hereditary iff every finitely generated ideal is weak w-projective.

Abstract

Let $R$ be a ring. An $R$-module $M$ is said to be a weak $w$-projective module if ${\rm Ext}_R^1(M,N)=0$ for all $N \in \mathcal{P}_{w}^{\dagger_\infty}$ (see, \cite{FLQ}). In this paper, we introduce and study some properties of weak $w$-projective modules. And we use these modules to characterize 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 is weak $w$-projective if and only if every finitely presented $R$-module is weak $w$-projective and $R$ is a $w$-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 a weak $w$-projective.