Gorenstein χ-injective Modules And Gorenstein χ-flat Modules

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