Modules Supplemented Relative to A Torsion Theory

This article introduces the concept of a-supplemented module as follows: Given a hereditary torsion theory in Mod R with associated torsion functor ,w e say that a module M is -supplemented when for every submodule N of M there exists a direct summand K of M such that K N and N=K is -torsion module. We present here some fundamental properties of this class of modules and study the decompositions of -supplemented modules under certain conditions on modules. The question of which direct sum of -supplemented R-modules are -supplemented is treated here.