$(n,d)$-injective and $(n,d)$-flat modules under a special semidualizing bimodule

Let $S$ and $R$ be rings, $n, d\geq 0$ be two integers or $n=\infty$. In this paper, first we introduce special (faithfully) semidualizing bimodule $_S(K_{d-1})_R$, and then introduce and study the concepts of $K_{d-1}$-$(n,d)$-injective (resp. $K_{d-1}$-$(n,d)$-flat) modules as a common generalization of some known modules such as $C$-injective, $C$-weak injective and $C$-$FP_n$-injective (resp. $C$-flat, $C$-weak flat and $C$-$FP_n$-flat) modules. Then we obtain some characterizations of two classes of these modules, namely $\mathcal{I}_{K_{d-1}}^{(n,d)}(R)$ and $\mathcal{F}_{K_{d-1}}^{(n,d)}(S)$. We show that the cleasses $\mathcal{I}_{K_{d-1}}^{(n,d)}(R)$ and $\mathcal{F}_{K_{d-1}}^{(n,d)}(S)$ are covering and preenveloping. Also, we investigate Foxby equivalence relative to the classes of this modules. Finally over $n$-coherent rings, we prove that the classes $\mathcal{I}_{ K_{d-1}}^{(n,d)}(R)_{<\infty}$ and $\mathcal{F}_{ K_{d-1}}^{(n,d)}(S)_{<\infty}$ are closed under extentions, kernels of epimorphisms and cokernels of monomorphisms. Keywords: $K_{d-1}$-$(n,d)$-injective module; $K_{d-1}$-$(n,d)$-flat module; Foxby equivalence; special semidualizing bimodule.