Goldie extending property on projection invariant submodules

Abstract

A module M is called extending (or CS) if every submodule of M is essential in a direct summand of M. A module M is called Goldie extending if and only if for each submodule N of M there exists a direct summand D of M such that N ?D is essential in both N and D. In this work, we define a module M to be PG-extending if and only if for each projection invariant submodule X of M there exists a direct summand D of M such that X ? D is essential in both X and D. We investigate the relation between this new class of module and several known generalizations of the extending modules, such as C11, P I-extending and Goldie extending. Our focus is the behaviour of the PG-extending modules with respect to direct sums and direct summands. Moreover, we show that if S is a right essential overring of R and RR is PG-extending, then SR and SS are PG-extending. © 2018, Academic Publishing House. All rights reserved.