The Study On Some Characters Of Gorenstein Homlogical Dimension

Homological Algebra is an important part of Algebra. The development of Homological Algebra brought out a great important in studying the group, Lie algebra and associative ring. One important research field of the recent ring theory is the homological dimension of the ring. From 1960s the study about the homological dimension of the ring over noncommutative notherian ring enriched the classical result about the homological dimension of the ring. Its theories and methods influenced the Algebra and other subjects.When R is a two-sided Noetherian ring, Auslander and Bridger[2]introduce in 1969 the G-dimension, G — dim_RM, for every finite, that is, finitely generated R-module M. They proved the inequality G — dim_RM ≤ pd_RM, with equality G — dim_R = pd_RM whenpd_RM < ∞. In 1995 Over a general ring R, Enochs and Jenda defined in [13] Gorenstein projective modules. In 200C L.W:Christensen proved in [6]that if R is two-sided Noetherian, and G is a finite Gorenstein projective module, then the new definition agree with that of Auslander and Bridger.And he also proved that a finite module over a notherian Ring is Gorenstein projective if and only if G — dim_R = 0. In 2004 in [17], Henrik.Holm studied the closely related about Gorenstein projective, Gorenstein injective and Gorenstein flat dimensions.This paper has two part. In the first part it devotes to research thedefinition and character of Gorenstein injective module, Goernstein flat module and Gorenstein projective module. And we also deduce the relation about Gorenstein injective module, Gorenstein flat module and Gorenstein projective module. In the second part through research the relation of left and right injective dimension we get the relation of left and right Gorenstein injective dimension.In section 1, preliminaries. It introduces the definition and the elementary properties of Gorenstein projective modules and Gorenstein flat modules. Then introduces the Gorenstein divided function. And the relation between Gorenstein EXT function and EXT function, the Gorenstein TOR function and TOR function, further more, it introduces the selforthogonal module of finite injective dimension.In section 2, it discusses the characters of Gorenstein injective module and it proves that for every module M with finite Gorenstein injective dimension admits a nice Gorenstein injective preenvelopes. Further more it discusses the relation between Gorenstein injective module and injective module.In section 3, it defines the definition of coflat, and similarly it also define the definition of Gorenstein coflat. It introduces the definition of Gorenstein coflat, Gorenstein coflat resolution, Gorenstein coflat dimension. Then it introduces the characters of Gorenstein coflat modules. At last it mainly discusses the relation of Gorenstein flat module Gorenstein injective module, Gorenstein projective module.In section 4, it discusses when A is a k-Gorenstein ring and a^a is a faithfully balanced selforthogonal bimodule and if the Gorenstein injective di-mension of u\ is finite then the Gorenstein injective dimension of a^ is also finite. Further more we get l.Gid\U) = r.