The Localization Technique Of Torsion Theory

Abstract:Localization at multiplicative closed sets is a well-known clas-sical process for commutative rings. Since1940s, noncommucative localization has become a vigorously active area of research. In this dissertation, we study noncommucative localization by the (hereditary) torsion theory. The dissertation is composed of three chapters. In Chapter1, we introduce preliminaries, includ-ing some basic concepts, lemmas and notations. In Chapter2, we introduce the notion of T-morphisms and present some basic properties of the hom-sets of T-morphisms and of the functor hom. We give the construction of the quotient category of torsion (the quotient category determined by the hereditary torsion theory T). Then the module of quotient of a given module is determined. We also construct the kernel, image, cokernel for each T-morphism. Furthermore, exact sequences in a full subcategory of the torsion quotient category are also discussed. Chapter3devotes to the application. We characterize the T-monic, T-epic, T-finitely generated, etc. We also give the detailed form of hom-sets of T-morphisms over a special hereditary torsion theory.