Gorenstein Injective Dimension

In this thesis, we investigate some problems about Gorenstein injective modules and Gorenstcin injective dimensions of modules. This dissertation consists of six sections, the first and second sections are introduction and preliminaries respectively.In the third section, we introduce the concept of GI-closed Ring, then give a characterization of Gorcnstein injective modules, by investigate the extension problem of the class of Gorcnstcin injective modulcs,wc give a characterization of GI-closed Ring, and give such a result: On GI-closed ring, GI(R) is closed under direct summands.In section four, we first give such a result: If R is n-Gorenstcin, so is any localization S^-1R. Then by investigate the localization of Gorcnstein injective modules, we give some results which arc paraller to injective modules, meanwhile, we give such a result: If R is a commutative n-Gorcnstein ring, then GD(R)≥GD(S^-1 R).In section five. motivated by the notation of relative injective dimensions of [10], we introduce the concept of relative Gorcnstcin injective dimensions, by investigation, we acquire a lot of results which arc parallcr to the classical injective dimensions.In section six, we investigate the Gorcnstcin injective dimensions in direct products of rings, the main result is the following:Let R =ΠR_i be a direct product of rings and let M = M₁ (?)M₂(?)……(?)M_n be adecomposition of an R-module into R_i modules. Then we have an equality...