Gorenstein ?-injective Modules And Gorenstein ?-flat Modules

Let R be a ring and ? a class of left R-modules.In this thesis,we first introduce the notion of Gorenstein ?-injective modules,some equivalent characterizations and properties of such modules are investigated under the condition that ? is syzygy closed,and some equivalent characterizations of R is a left self ?-injective ring are given via Gorenstein ?-injective modules.Secondly,the notion of Gorenstein ?-flat modules is introduced,some equivalent characterizations and properties of such modules are investigated under the condition that ? is syzygy closed,and the relations of Gorenstein ?-injective modules and Gorenstein ?-flat modules are discussed.Finally,we introduce the concept of strongly Gorenstein ?-injective modules and strongly Gorenstein ?-flat modules.We show that each Gorenstein ?-flat(?-injective)module is a direct summand of a strongly Gorenstein ?-flat(?-injective)module if ? is syzygy closed(and ? contained in the class of finitely presented left R-modules).