Relative Weak Injective and Weak Flat Modules with Respect to a Semidualizing Module

Let R be a commutative ring, and let C be a semidualizing R-module. We introduce the notion of finitely presented C-injective modules, finitely presented C-flat modules, weak C-injective modules and weak C-flat modules. Some properties of these modules are investigated. It is proved that over weak C-injective ring R, if for a super finitely presented R-module M, $${\text {Hom}}_{R}(M,R)$$ is super finitely presented and $${\text {Hom}}_{R}(M,R) \in \mathcal {A}_C(R)$$ , then the following statements hold: