Gorenstein Projective Modules And Relative Homological Dimensions

In relative homological algebra, Gorenstein modules and relative homological dimen-sions are important objects, and they have been studied by many authors [8,14,16,22,29, 42].Gorenstein projective (injective, flat) modules were introduced in[20,21,24] by Enochs, Jenda and Torrecillas, these concepts generalize the classical projective (injec-tive, flat) modules and these notions can also be used to define the Gorenstein projective (injective, flat) dimension for arbitrary modules. The Gorenstein projective dimension coincides with the classical G-dimension over commutative Noetherian rings introduced by Auslander and Bridger in [3] for finitely generated modules. In, Gorenstein global dimension was introduced, and it was used to give characterizations of some rings. The investigation of the relative homological dimensions will give us more information about some classical rings and homological dimensions.In this paper, we mainly discuss the characterizations and properties of Gorenstein projective modules and the relative homological dimensions.This paper is divided into four chapters.In Chapter 1, some background and main results are given.In Chapter 2, we study the characterization and the property of Gorenstein projective modules over arbitrary rings. Firstly, using strongly Gorenstein projective modules, we give a characterization of Gorenstein projective modules, then we prove that the class of Gorenstein projective modules is closed under transfinite extensions. Secondly, we prove that strongly Gorenstein projective modules of countable type are Gorenstein flat. Finally, we supply a sufficient condition such that the class of Gorenstein projective modules is precovering.In Chapter 3, using relative homological dimensions, we will give some characteriza- tions of IF rings. Firstly, we study relations between the Gorenstein projective dimension of finitely presented modules and classical homological dimensions. Then we will charac-terize the IF rings.In Chapter 4, we study the characterization of finitely generated modules of finite Gorenstein projective dimension over a Noetherian ring, and using tilting modules, we provide a characterization of these modules in this chapter.