–supplemented Modules Relative to a Torsion Theory

Let R be ring and M a right R-module. This article introduces the concept of τ −⊕-supplemented modules as follows: Given a hereditary torsion theory in Mod-R with associated torsion functor τ we say that a module M is τ −⊕-supplemented when for every submodule N of M there exists a direct summand K of M such that M = N +K and N ∩K is τ−torsion, and M is called completely τ −⊕-supplemented if every direct summand of M is τ −⊕supplemented. We present here some fundamental properties of these class of modules and study the decompositions of τ −⊕-supplemented modules under certain conditions on modules. The question of which direct sum of τ − ⊕supplemented modules are τ − ⊕-supplemented is treated here. The ring R is called right τ-perfect if every right R−module has a τ -projective cover, and the module M is strongly τ −⊕-supplemented when for every submodule N of M there exists a direct summand K of M such that M = N +K and N ∩K is small τ -torsion submodule of M . It is shown that R is right τ -perfect if and only if every projective right R-module is strongly τ − ⊕-supplemented, and R is right τ(R)-perfect [14] if and only if every projective right R-module is τ −⊕-supplemented.