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