Modules complemented with respect to a torsion theory

Abstract

This article introduces the concept of μ-complemented modules as follows: given a hereditary torsion theory in Mod-R with associated torsion functor μ we say that a module M is μ-complemented when every sub module is μ- dense in a summand of M. We present here some fundamental properties of this class of modules and study the relationship between them and extending modules. We apply these results, in particular, when μ is the torsion functor associated to the Goldie torsion theory and when μ is jansian. The question of which direct sums of μ-complemented R-modules are μ-complemented is treated here. We also obtain decomposition theorems for μ-complemented modules when either the ring or the module satisfies certain chain conditions.