Some Generalizations Of Strongly Universal Gorenstein Homological Modules

Gorenstein algebras play an important role in many research fields, such as representation theory of algebras and algebraic geometry, and has been wildly investigated by many mathematician. In this thesis, we mainly study a new kind of Gorenstein homological modules, say strongly universal Gorenstein homological module, and consider the generalization of these modules in the category of complexes. This thesis consists of the following four parts.In the first chapter, we recall the background and main works.In Chapter 2, we introduce the notions and some well-known results. Be-ing similar to strongly universal Gorenstein projective modules, we investigate basic properties of strongly universal injective and flat modules, and give a characterization of strongly universal injective and flat modules.Chapter 3, deals with the notions of n-strongly universal Gorenstein pro-jective, injective and flat modules, and study their properties. Moreover, we discuss the relations among the projective (resp. injective or flat) modules, strongly universal Gorenstein projective (resp. injective or flat) modules, u-niversal Gorenstein projective (resp. injective or flat) modules and n-strongly universal Gorenstein projective (resp. injective or flat) modules.In the final chapter, considering the generalization of the above Gorenstein homological modules in the category of complexes, we introduce the definition of strong Gorenstein projective and injective complexes, universal Gorenstein projective and injective complexes and strongly universal Gorenstein projective and injective complexes, and study their basic properties.