A note on weak w-projective modules

Let $R$ be a ring. An $R$-module $M$ is said to be a weak $w$-projective module if ${\mathrm{E}\mathrm{x}\mathrm{t}}_R^1(M,N)=0$ for all $N\in {\mathcal{P}}_w^{{†}_{\mathrm{\infty }}}$ (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.