关键词: Gorenstein 投射性, Gorenstein 内射性, QF-环, Artin 环

Abstract: The Gorenstein projective and Gorenstein injective properties of modules over rings were studied. Firstly，under the framework of an Artin ring，the sufficient and necessary conditions for any left R-module to be a Gorenstein projective module and a Gorenstein injective module were discussed by the characterization condition of a special ring. Then，the Gorenstein projective and Gorenstein injective properties of modules over a general ring were described in terms of the embeddability of the right R-module to free Rmodules and projective modules，as well as the zero-dependent lifting chain. The results of the existing literature were partially extended.

Key words: Gorenstein projectivity, lGorenstein injectivity, lQF-ring, Artin ring