Gorenstein AC Homology With Respect To A Semidualizing Module

Let R be a commutative ring and C a semidualizing R-module.Firstly,C-absolutely clean modules and C-level modules are introduced and some properties of them are given,and some rings are characterized by these modules.Secondly,C-Gorenstein AC-injective modules are introduced and some properties of them are investigated.We show that the class of all C-Gorenstein AC-injective modules is injectively resolving.We give some equivalent characterizations for C-Gorenstein AC-injective dimension of R-modules.We construct a complete hereditary cotorsion theory by the class of all C-Gorenstein AC-injective modules.Thirdly,We consider the stability of the category of C-Gorenstein AC-injective and show that the class of all C-Gorenstein AC-injective modules and the class of all two-degree C-Gorenstein AC-injective modules happen to coinside.At last,we study the Foxby equivalence between the class of all C-Gorenstein AC-injective mod-ules in AC(R)and the class of all Gorenstein AC-injective modules in BC(R).