Ding projective modules with respect to a semidualizing bimodule

Abstract

Let $R$ and $S$ be rings and ${}_SC_R$ a faithfully semidualizing bimodule. A left $S$-module $M$ is called \textit{Ding $C$-projective} if there exists an exact sequence of $C$-projective left $S$-modules \[ X=\cdots \rightarrow C\otimes_{R} P_{1}\rightarrow C\otimes_{R} P_{0}\rightarrow C\otimes_{R} P^{0}\rightarrow C\otimes_{R} P^{1}\rightarrow \cdots \] such that $M\cong\mbox{Coker\,}(C\otimes_{R} P_{1}\rightarrow C\otimes_{R} P_{0})$ and the complexes $\mbox{Hom}_S(C\otimes_{R}P, X)$ and $\mbox{Hom}_S(X,C\otimes_{R}F)$ are exact for any projective left $R$-module $P$ and any flat left $R$-module $F$. The properties of Ding $C$-projective modules and dimensions are given. Among others, the Foxby equivalences between some subclasses of the Auslander class and the Bass class are also investigated.