Gorenstein flatness and injectivity over Gorenstein rings

Let R be a Gorenstein ring. We prove that if I is an ideal of R such that R/I is a semi-simple ring, then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical. In addition, we prove that if R → S is a homomorphism of rings and _S E is an injective cogenerator for the category of left S-modules, then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical. We also give some applications of these results. This work was partially supported by the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20060284002), the National Natural Science Foundation of China (Grant No. 10771095) and the Natural Science Foundation of Jiangsu Province of China (Grant No. BK2007517)