Some Dense Supplemented Modules Relative To A Torsion Theory

Hereditary torsion theories have been developed since the 1950's or 1960s and have been extensively studied by Golan, Gabriel, Disckson, etc. Using the classification of injective modules, Golan [9]formulated the theory. All kinds of concepts of the general modules were correspondingly generalized and many properties of hereditary torsion theories were obtained. In this thesis, we will inverstigate the relationship among some classes of modules with respect to a torsion theory, in particular, we study some dense supplemented modules relative to a torsion theory. This paper is divided into five sections:In section one, we mainly provide some basic concepts which can be found in [9].In section two, we inverstigate some dense supplemented modules with respect to a torsion theory , such as Ï„-dense supplemented module,Ï„-weakly dense supplemented module,r-confintely supplemented module, and Ï„-cofintely weakly supplemented module,etc.In section three, we indroduce the concept of Ï„-maximal submodule, and using it,Ï„-cofintely weakly supplemented module is characterized.In section four,we mainly discuss r-strongly local module which is a class of modules being directly connected with r-small submodules.In section five, we introduce the concept of generalization of Ï„-small submodule in which the generalization of some kinds of dense supplemented modules with respect to a torsion theory may be found too.