Modules With Finite Gorenstein-Projective Dimension

Gorenstein homological algebras have been investigated widely by many authors and play important roles in the representation theory and the rela-tive homology theory. Being similar to the projective modules (resp. injective modules, flat modules), Gorenstein projective modules (resp. Gorenstein in-jective modules, Gorenstein flat modules) are the most fundamental classes of modules and hence the Gorenstein projective dimension (resp. Gorenstein in-jective dimension, Gorenstein flat dimension) are the most important relative homological dimensions. Moreover, one notice that the module classes with fi-nite projective dimension (resp. injective dimension, flat dimension) and their Ext-orthogonal classes form complete cotorsion pairs, respectively, and hence they were investigated widely in the cotorsion theory and theory of envelopes and covers. In 2008, Jia-Qun Wei introduced the definition of Ext- orthogonal and Tor-orthogonal classes of modules by the classes of modules with finite Gorenstein-projective dimensions and Ext and Tor functors. In this thesis, our main work is on their properties. We characterize some relative homological dimensions of modules and rings, and investigate envelopes and covers of some special modules by using cotorsion theory. This paper consists of the following three parts.In the first part, we introduce some related background, development, and our main work.In the second part, we introduce the notions of Gn-injective modules and Gn-flat modules, and give some basic properties and exchange relations of these modules on some different rings. On the other hand, we introduce the defi-nition of GI-injective dimensions of module and rings by using the classes of Gorenstein-projective modules and Ext functor. Meanwhile, we obtain some basic properties of GI-injective dimension and some equivalent char-acterizations on GI-injective dimension and GI-injective dimension at most n on noetherian rings.In the first part of the third chapter, we give the definition of the v-dimension of rings, and characterize the ring of limited l.v-dimensions. Sec-ondly, we investigate some properties of envelopes and covers of Gn-injective modules and Gn-flat modules by the classes of modules with finite Gorenstein-projective dimensions. Finally, the FGPn-precover and Fn-precover are in-vestigated, and some rings on which each Gn-injective module is injective are characterized.