Gorenstein Hereditary Rings

Abstract

In this paper, we introduce the Gorenstein hereditary and Gorenstein semi-hereditary modules and study the properties of the radical of Gorenstein projective modules over left artin rings. As an application, we characterize when Gorenstein projective modules are projective and give some characterizations of the Gorenstein hereditary rings by radical and Gorenstein projective modules over the endomorphism algebra of a module, respectively. Meanwhile we study the projective complexes over Gorenstein hereditary rings. In the last part of the paper, we study the heredity, Gorenstein heredity and quasi-heredity of the Morita context ring $\Lambda_{(0,0)}$ which is a left artin algebra.