DC-projective dimensions, Foxby equivalence and SDC-projective modules

This is a study of Ding projective modules relative to a semidualizing module and related topics. Firstly, we study DC-projective dimensions and DC-projective modules under change of rings. Secondly, we establish a new version of the Foxby equivalence with respect to DC-projective modules and DC-injective modules. Thirdly, we characterize Ding projective modules in 𝒜C(R) and Ding injective modules in ℬC(R). At last, as applications, some new characterizations of perfect rings and quasi-Frobenius rings are given.