Covers and hulls relative to torsion theories

The aim of this thesis is to examine fundamental properties of torsion theory and make a new generalization of CS modules by studying generalizations of some concepts in module theory relative to a torsion theory. This thesis consists of four chapter. In the first chapter fundamental definitions and theorems in module theory are given without proofs.In the first part of the second chapter definition and some properties of torsion theory are examined. In the next part, definitions and properties of τ-dense and τ-pure submodules, which will be basic notions of next chapters, are given. Moreover as a generalization of simple module which is an important module class in module theory τ-simple and τ-cocritical modules are introduced. After giving fundamental properties of these concepts, τ-maximal submodules which are generalizations of maximal submodules are introduced. Then the connection between τ-small submodules and τ-radical is established as in module theory. Furthermore a generalization of Nakayama Lemma which is an important theorem of module theory is given. In the third chapter injective and projective modules are defined relative to a torsion theory and generalizations of injective hulls and projective covers are examined in a torsion theory. Moreover in this chapter, the concept of τ-divisible module which coincides with injective module in (0,Mod-R) is given. In the last chapter τ-CS modules are defined by defining τ-closed and τ-complement submodules in order to obtain a new generalization of CS modules. Then properties of these new concepts are investigated.