In this paper, we study some properties of $G_C$-flat $R$-modules, where $C$ is a semidualizing module over a commutative ring $R$ and we investigate the relation between the $G_C$-yoke with the $C$-yoke of a module as well as the relation between the $G_C$-flat resolution and the flat resolution of a module over $GF$-closed rings. We also obtain a criterion for computing the $G_C$-flat dimension of modules.

Let $𝒲$ be a self-orthogonal class of left $R$-modules. We introduce a class of modules, which is called strongly $𝒲$-Gorenstein modules, and give some equivalent characterizations of them. Many important classes of modules are included in these modules. It is proved that the class of strongly $𝒲$-Gorenstein modules is closed under finite direct sums. We also give some sufficient conditions under which the property of strongly $𝒲$-Gorenstein module can be inherited by its submodules and quotient modules....