note on weak w-projective modules

Let R be a ring. An R-module M is a weak w-projective module if ExtR¹(M,N)=0 for all N in the class of $GV$-torsion-free R-modules with the property that ExtᵏR(T,N)=0 for all w-projective R-modules T and all integers k≥1. 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.