Torsion Theories Relative To A Module U

Hereditary torsion theory has been developed since the 1960 s and has been extensively studied by Golan ,Gabriel ,Diskon ,Stenstrom ,etc.In this thesis ,gc:n-eral torsion theory determined by a given module u is especially discussed .This kind of torsion theory is not necessarily hereditary ,but we can obtain some results corresponding to hereditary torsion theory .The article can be divided into four sections .In the first section ,some preliminaries are mentioned .In order to build a good foundation for later parts ,we introduce some basic concepts and some theorems which are mainly the results of hereditary torsion theory and can be found in [3],[7] .In the second section ,by constructing torsion theory (AU,BU] ,which is denoted as TU according to an arbitary given module u ,we discuss the structure of R ?Tors = (TU|U 6 R ?mod}and further characterize them .Therefore ,we generalize the results of hereditary torsion theory .In the third section ,as the generalization of projective modules and injcctive modules ,we introduce the concepts of ,4u-projective modules and ^4u-injective modules ,meanwhile ,we give several properties of them and the important relationship between them .In the fourth section ,we employ u-cocritial module and u-seniicritial module. A module M is called u-cocritial if M is u-torsionless and if every proper factor module of M is ru-torsion .A module M is called u-semicritical if there exists finite submodule variety {Mi, A/2, ??? Mn} of M ,such that n?=i Mj = 0 ,and Af/M, is u-cocritical .In this section ,we deal with the basic properties of u-cocritical module ,and compare the relationship between the u-cocritical module and u-semicritical module .Moreover ,under some given conditions ,we integrate them with hereditary torsion theory .