Gorenstein Projective Modules And Generalized Gorenstein Dimension

Coherent rings, Noether rings and Gorenstein rings are three kinds of important rings in ring theory. There is a close link among these three types of ring:Noether ring is a special coherent ring while Gorenstein ring is a special Noether ring. The development of modern ring theory is always associated with the modules and category of modules. In recent years, the module and the module category are always important objects in some field of algebra. Meanwhile, homological algebra has played an important role in the development of ring theory, which provided a powerful tool for the study of the ring theory. In this article, we used homological method to define and calculate the various homological dimensions, and discussed their properties. Gorenstein projective module and generalized Gorenstein dimension are mainly investigated in the thesis, the discussion about varies dimensions is the focus of analysis. The elementary tolls are homological formulas and push-out diagrams and long exact sequences in the thesis.This paper is divided into three parts:The first part is introduction, which not only surveied the background and development of objects but also descripted the main contents of this paper.In the second part, we investigated Gorenstein projective module on the Gorenstein ring. Using Gorenstein projective module, we characterized Gorenstein ring in this paper. By the push-out diagrams, we got a theorem that when R was n-Gorenstein ring, for any moudule RM, there was an exact sequence: 0→K→E→M→0, where pd(K)≤n-1 and E was a Gorenstein projective module, which showed a clear relationship between n-Gorenstein ring and Gorenstein projective module. In addition, we obtained another theorem:there was an exact sequence:0→M→K'→E'→0, where pd(K')≤n and E'was a Gorenstein projective. Then we proved the equivalence between these two conclusions by the push-out diagrams. In a certain sense, we generalized the corresponding conclusion of Gorenstein projective module. In the third part, we considered the self-orthogonal module on the coherent ring, and mainly discussed FP-injective dimension as well as generalized Gorenstein dimension. We obtained a sufficient condition to show that l.FP-idR(ω)<∞implied G-dimv(M)<∞, where M∈mod R, which generalized the result of Huang and Tang about the relationship between FP-injective dimension and generalized Gorenstein dimension. In addition, we got that the left-orthogonal dimension was equal to the generalized dimension when G-dimω(M) was finite.