The Extending Condition Relative to Sets of Submodules

Abstract

A module M is called an extending (or CS) module provided that every submodule of M is essential in a direct summand of M. We call a module C{script}-extending if every member of the set C{script} is essential in a direct summand where C{script} is a subset of the set of all submodules of M. Our focus is the behavior of the C{script}-extending modules with respect to direct sums and direct summands. By obtaining various well-known results on extending modules and generalizations as corollaries of our results, we show that the C{script}-extending concept provides a unifying framework for many generalizations of the extending notion. Moreover, by applying our results to various sets C{script}, including the projection invariant submodules, the projective submodules, and torsion or torsion-free submodules of a module, we obtain new results including a characterization of the projection invariant extending Abelian groups. © 2014 Copyright Taylor and Francis Group, LLC.