FPn-injective and FPn-flat modules with respect to a semidualizing bimodule

Let S and R be rings, SCR be a semidualizing bimodule and n≥0 be an integer or n=∞. We introduce and study C-FPn-injective and C-FPn-flat modules as a common generalization of some known modules such as C-injective (resp. C-FP-injective, C-weak injective) and C-flat (resp. C-projective, C-weak flat) modules. Suppose that SCR is a faithfully semidualizing bimodule. We give some equivalent characterizations of left n-coherent rings in terms of C-FPn-flat left S-modules and C-FPn-injective left R-modules. Then we show that the pairs (FFCn(S),FICn(Sop)) and (FICn(R),FFCn(Rop)) are coproduct-closed and product-closed duality pairs and both FFCn(S) and FICn(R) are covering and preenveloping, where FFCn(S) and FICn(R) denote the classes of C-FPn-flat left S-modules and C-FPn-injective left R-modules respectively. Finally, we investigate Foxby equivalence relative to C-FPn-injective and C-FPn-flat modules.