Some Special Modules And Rings In Torsion Theory

In 1960s, S.E.Dickson gave a generalization of the properties of the classes of torsion and torsionfree Abelian groups to Abelian categories. Dickson's work was published in the Transactions of the Amerian Mathematical Society. In 2001, A.Beligiannis and I.Reiten generalized the torsion theory to triangulated categories. Now, torsion theory plays an important role in many areas of research, such as, ring and module theory, localization theory, algebra representation theory.This paper mainly introduced some special rings and modules relative to torsion theory.This paper is divided into three chapters.In the first chapter, we introduce the background of this paper and the definitions of torsion theory, preenvelope class, precover class.In the second chapter, we investigated the existence of Ï„-projective test modules and Ï„-flat test modules; the properties of left G-regular rings and the relations of T-pure injective modules, Ï„-cotorsion modules and Ï„-injective modules. The main results as following:Theorem2.1.8 r is a cohereditary torsion theory, G₀ = {I|I is a maximal left ideal of R and R/I ∈ F], if every left R-module has a r-projective cover , then N = ∏_I∈G0 R/I is a Ï„- projective test module.Theorem2.2.4 G_e={I|I ∈ G , I ≠ R and I is a essential left ideal of R}, then N = (?)_I∈Ge R/I is a Ï„-flat test module.Theorem2.3.2 A ring R is left G-regular if and only if every cyclic left ideal I = Rr in G, there exists a element r' ∈ R, such that rr'r = r.Theorem2.3.6 If every cyclic left ideal I = Ra in G, such th at R/aR is a Ï„-flat module, then R is a left G-regular ring.