Strongly Gorenstein Homological Properties With Respect To A Semidualizing Module

With respect to a semidualizing module C,firstly,we introduce the strongly Gorenstein C-projective(C-injective,C-flat)modules and study some properties class of modules.Their equivalent characterizations are given and proved that:(1)Let M+ be a strongly Gorenstein C-flat module.Then M+ is a strongly Gorenstein C-injective module.(2)Let R be a Noether ring.If M+ is a strongly Gorenstein C-injective module,then M is a strongly Gorenstein C-flat module.Secondly,some properties between M and Ext(F1,M),Ext(M,F2)are studied,where F1,F2 is a C-flat module.Finally,we study that:(1)Give the connections between strongly Gorenstein C-projective(C-injective,C-flat)modules properties for R-module M and strongly Gorenstein C-projective(C-injective,C-flat)modules properties for R-modules M.(2)Give the connections between strongly Gorenstein C-projective(C-injective,C-flat)modules properties for R-modules M and strongly Gorenstein S-1C-projective(S-1C-injective,S-1C-flat)modules properties for S-1 R-modules S-1M.(3)Give the connections between strongly Gorenstein C-projective(C-injective,C-flat)modules properties for R-modules M and strongly Gorenstein C[x]-projective(C[x]-injective,C[x]-flat)modules properties for R[x]-modules M[x].