CLASSES OF EXTENDING MODULES ASSOCIATED WITH A TORSION THEORY

Abstract

A module M over an associative ring R is said to be extending if ev- ery submodule of M is essential in some direct summand of M. In this paper we consider τ-extending modules, an analogue of extending that corresponds to a torsion theory τ on the modules over R. We present some fundamental properties of τ-extending modules, relate them to ex- tending modules, and give some results regarding when direct sums of τ -extending modules are τ -extending. Examples are provided to illus- trate our results. For τG the Goldie torsion theory and τ a larger torsion theory, the τG-extending modules coincide with the τ-extending modules.