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Let  S  and  R  be rings,be a semidualizing bimodule andbe an integer orWe 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 thatis 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 pairsandare coproduct-closed and product-closed duality pairs and bothandare covering and preenveloping, whereanddenote 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.

communications in algebra

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