$w$-$\mathrm{FP}$-projective modules and dimension

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 $\mathrm{FP}$-projective module. An $R$-module $M$ is said to be $w$-$\mathrm{FP}$-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 $w$-$\mathrm{FP}$-projective dimensions of modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed. Illustrative examples are given.