$w$-$\mathrm{F}\mathrm{P}$-projective modules and dimension

Let $R$ be a ring. An $R$-module $M$ is said to be an absolutely $w$-pure module if and only if ${\text{Ext}}_R^1(F,M)$ 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 $\mathrm{F}\mathrm{P}$-projective module. An $R$-module $M$ is said to be $w$-$\rm FP$-projectiveif$\Ext^1_R(M,N)=0$foranyabsolutely$w$-puremodule$N$. This new class of modules will be used to characterize (Noetherian) $DW$ rings. Hence, we introduce the $w$-$\mathrm{F}\mathrm{P}$-projective dimensions of modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed. Illustrative examples are given.