n-weak injective and n-weak flat modules under m-semidualizing bimodules

Let [Formula: see text] and [Formula: see text] be rings, the [Formula: see text] two non-negative integers and [Formula: see text] an (faithfully) [Formula: see text]-semidualizing bimodule. In this paper, we introduce, via special super finitely presented module [Formula: see text] with respect [Formula: see text], the concepts of [Formula: see text]-[Formula: see text]-weak injective and [Formula: see text]-[Formula: see text]-weak flat modules, and then we obtain some characterizations of two classes of these modules, namely [Formula: see text] and [Formula: see text] which are larger than that of [Formula: see text]-weak injective and [Formula: see text]-weak flat modules, respectively. Also, we investigate Foxby equivalence relative to these modules. Then over any arbitrary ring, we show some nice properties from the classes [Formula: see text] and [Formula: see text].