FP n -injective and FP n -flat modules with respect to a semidualizing bimodule

Abstract

Let S and R be rings, be a semidualizing bimodule and be an integer or We introduce and study C-FP n -injective and C-FP n -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 is a faithfully semidualizing bimodule. We give some equivalent characterizations of left n-coherent rings in terms of C-FP n -flat left S-modules and C-FP n -injective left R-modules. Then we show that the pairs and are coproduct-closed and product-closed duality pairs and both and are covering and preenveloping, where and denote the classes of C-FP n -flat left S-modules and C-FP n -injective left R-modules respectively. Finally, we investigate Foxby equivalence relative to C-FP n -injective and C-FP n -flat modules.