7c-Gorenstein Projective, ;Cc-Gorenstein Injectiveand Wcr-Gorenstei丨丨 Flat Modules

The authors introduce and investigate the Tc-Gorenstein projective, Lc- Gorenstein injective and Hc-Gorenstein flat modules with respect to a semidualizing module C which shares the common properties with the Gorenstein projective, injective and flat modules, respectively. The authors prove that the classes of all the Tc-Gorenstein projective or the Hc-Gorenstein flat modules are exactly those Gorenstein projective or flat modules which are in the Auslander class with respect to C, respectively, and the classes of all the Lc-Gorenstein ＇injective modules are exactly those Gorenstein injective modules which are in the Bass class, so the authors get the relations between the Gorenstein projective, injective or flat modules and the C-Gorenstein projective, injective or flat modules. Moreover, the authors consider the Tc（R）-projective and Lc（R）-injective dimensions and Tc（R）-precovers and Lc（R）-preenvelopes. Fiually, the authors study the Hc-Gorenstein flat modules and extend the Foxby equivalences.

T_C-Gorenstein Projective,L_C-Gorenstein Injective and H_C-Gorenstein Flat Modules

The authors introduce and investigate the $mathcal{T}_C $ -Gorenstein projective, $mathcal{L}_C $ -Gorenstein injective and $mathcal{H}_C $ -Gorenstein flat modules with respect to a semidualizing module C which shares the common properties with the Gorenstein projective, injective and flat modules, respectively. The authors prove that the classes of all the $mathcal{T}_C $ -Gorenstein projective or the $mathcal{H}_C $ -Gorenstein flat modules are exactly those Gorenstein projective or flat modules which are in the Auslander class with respect to C, respectively, and the classes of all the $mathcal{L}_C $ -Gorenstein injective modules are exactly those Gorenstein injective modules which are in the Bass class, so the authors get the relations between the Gorenstein projective, injective or flat modules and the C-Gorenstein projective, injective or flat modules. Moreover, the authors consider the $mathcal{T}_C $ (R)-projective and $mathcal{L}_C $ (R)-injective dimensions and $mathcal{T}_C $ (R)-precovers and $mathcal{L}_C $ (R)-preenvelopes. Finally, the authors study the $mathcal{H}_C $ -Gorenstein flat modules and extend the Foxby equivalences.