Colocalization And Relative Cover Envelope

Torsion theories have been developed since the 1950's and have been extensively studied by Golan, Gabriel, Dickson, Stenstrom, et al. In this thesis, the cohereditary torsion theory is studied, and the colocalizations of modules over it are constructed, the properties of injectivity and projectivity over certain torsion theory are discussed.In the first section, some preliminaries are mentioned. In order to build a good foundation for later parts, some basic concepts and some theorems which are mainly the results of torsion theory can be found in [3].In the second section, the cohereditary torsion theory is studied. R-Ctors, which is constructed by all the cohereditary torsion pairs, is a lattice over Jansian R-modules category. Some lattice's properties of R-Ctors are given.In the third section, by using F-projective module, the concrete form of colocalization of module M relative cohereditary torsion pair (T, F) is constructed. Some categorical properties of P/c(K), which is denoted as pσ(M),are given.In the fourth section, the corresponding concepts of cover and hull through Q gets the properties of τ-injective, τ-projective, τ-divisible and τ-codivisible are introduced. The properties of τ-torsion injective hull, τ-torsion projective cover, τ-torsionfree divisible hull and τ-torsionfree codivisible cover are discussed, τ-injective hull E_τ(M) is characterized by using them.