On w-copure projective modules

Let $R$ be a commutative ring. An $R$-module $M$ is said to be $w$-split if Ext$_{R}^1(M,N)$ is a GV-torsion $R$-module for all $R$-modules $N$. It is known that every projective module is $w$-split, but the converse is not true in general. In this paper, we study the w-split dimension of a flat module. To do so, we introduce and study the so-called $w$-copure (resp., strongly $w$-copure) projective modules which is in some way a generalization of the notion of copure (resp., strongly copure) projective modules. An $R$-module $M$ is said to be $w$-copure projective (resp., strongly $w$-copure projective) if Ext$_{R}^1(M,N)$ (resp., Ext$_{R}^n(M,N)$) is a GV-torsion $R$-module for all flat $R$-modules $N$ and any $n\geq1$.