Y-gorenstein Injective Dimension And Relevant Problems

In this paper,we always assume that iR is a ring with unit 1,and M(R)is a module category consisting of all the left R-modules and we assume that y represents a class of modules that contain all injective modules and closes on both the direct product and the direct summand.We define a new relative homological module y-Gorenstein injective modules.The y-Gorenstein injective modules and their related problems are studied.In this paper,we define the y-Gorenstein injective module.We study the structural properties of the y-Gorenstein injective module.We mainly study the y-Gorenstein injective dimension of the modules and obtain some properties of the y-Gorenstein injective dimension.In the first chapter,we give the relevant background and necessary pre-liminary knowledge of the y-Gorenstein injective module.In Chapter 2,we define y-Gorenstein injective module.Then we dis-cuss y-Gorenstein injective modules and obtain some good properties and equivalent conditions of y-Gorenstein injective modules.We prove that the y-Gorenstein injective module class forms an injective resolving class.Further,we use y-Gorenstein injective resolution to define y-Gorenstein injective dimension.The functorial descriptions of y-Gorenstein injective modules are discussed in detail,and a series of equivalent conditions for a module with finite y-Gorenstein injective dimension are given.In chapter 3,we study the finitistic y-Gorenstein injective dimension of the ring,and prove that the finitistic y-Gorenstein injective dimension of the ring is equal to the finitistic injective dimension of the ring.In the homological algebra,the projective and injective modules have a very good duality relationship.On the basis of Gorenstein homological module studied by Enochs and Jenda,we discuss the relationship between the Gorenstein pro?jective modules and the Gorenstein injective modules over the n-Gorenstein ring,and we obtain an interesting result:there is an one to one correspon-dence between finite generated Gorenstein projective left-modules and fi-nite generated Gorenstein injective left R-modules over the communicative n-Gorenstein ring.Finally,we study the strongly y-Gorenstein injec-tive modules,and give some properties and equivalent characterizations of the strongly y-Gorenstein injective modules,and prove that:In a split short exact sequence,0→E→M→N→0,E∈y,N is a strongl y-Gorenstein injective module is equivalent to that M is a strongly y-Gorenstein injective module.