$w$-$\rm FP$-projective modules and dimension

TL;DR

This paper introduces the concept of $w$-FP-projective modules, generalizing $ m FP$-projective modules, and explores their properties and dimensions to characterize Noetherian $DW$ rings.

Contribution

It defines $w$-FP-projective modules, studies their properties, and relates their dimensions to classical homological dimensions, providing new tools for ring characterization.

Findings

Characterization of Noetherian $DW$ rings using $w$-$ m FP$-projective modules

Introduction of $w$-$ m FP$-projective dimensions for modules and rings

Relations established between $w$-$ m FP$-projective dimensions and classical homological dimensions

Abstract

Let $R$ be a ring. An $R$-module $M$ is said to be an absolutely $w$-pure module if and only if $\Ext^1_R(F,M)$ is a GV-torsion module for any finitely presented module $F$. In this paper, we introduce and study the concept of $w$-FP-projective module which is in some way a generalization of the notion of $\rm FP$-projective module. An $R$-module $M$ is said to be $w$-$\rm FP$-projective if $\Ext^1_R(M,N)=0$ for any absolutely $w$-pure module $N$. This new class of modules will be used to characterize (Noetherian) $DW$ rings. Hence, we introduce the $w$-$\rm FP$-projective dimensions of modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed. Illustrative examples are given.