Gorenstein flat and Gorenstein injective dimensions of simple modules

Abstract

Let R be a right GF-closed ring with finite left and right <br />Gorenstein global dimension. We prove that if I is an ideal of <br />R such that R/I is a semi-simple ring, then the Gorensntein flat <br />dimensnion of R/I as a right R-module and the Gorensntein <br />injective dimensnnion of R/I as a left R-module are identical. <br />In particular, we show that for a simple module S over a <br />commutative Gorensntein ring R, the Gorenstein flat dimension of <br />S equals to the Gorenstein injective dimension of S.