Universal Gorenstein Homological Methods And Its Applications

Since 1960s, relative homological algebra, especially, Gorensteinhomological algebra has been studied extensively and has become a vigorouslyactive area of research. In this dissertation, we study relative homologicalalgebra by using of universal Gorenstein homological methods. The dissertationis composed of five chapters. In Chapter 1, we introduce and study universalGorenstein projective, injective and ?at modules and show some of theirgeneral properties. The relations between the universal Gorenstein projective(resp., injective, ?at) modules and other modules are discussed. Then weinvestigate the rings over which every module is universal Gorenstein projective(resp., injective, ?at). We also discussed the connection of universal Gorensteininjective and ?at modules. In Chapter 2, we introduce the notion of GorensteinFP-injective modules and present some basic properties of the class of modules.The relations between the Gorenstein FP-injective modules and other modulesare discussed. Then we characterize the (coherent) rings over which all modulesare Gorenstein FP-injective modules. In the last section, strongly GorensteinFP-injective modules are also investigated. In Chapter 3, we introduce theconcepts of GI-injective and GI-?at modules and give some properties of thesemodules. It is discussed that GI-injective dimensions of modules and ringsand some applications of GI-injective modules in commutative rings. Thenwe investigate the ?at dimensions of Gorenstein injective modules in terms ofGI-?at modules. In particular, we give some characterizations of GI-injectiveand GI-?at dimensions of simple modules and show some connections ofthem. Chapter 4 establishes the fundamental results of Gorenstein cotorsiondimension of modules and rings. It is shown that some properties of Gorensteincotorsion modules. Then we study the global Gorenstein cotorsion dimensionsover coherent rings. We also present some description of Gorenstein cotorsion(pre)envelopes over a right coherent ring. Chapter 5 devotes to the study ofGorenstein yoke modules. For a right coherent ring R, we prove that a leftR-module is a Gorenstein n-yoke module if and only if it is an n-yoke module.Many results are developed, some generalizing known results.