GOLDIE EXTENDING PROPERTY ON THE CLASS OF z-CLOSED SUBMODULES

Abstract

In this article, we define a module M to be G(z)-extending if and only if for each z-closed submodule X of M there exists a direct summand D of M such that X boolean AND D is essential in both X and D. We investigate structural properties of G(z)-extending modules and locate the implications between the other extending properties. We deal with decomposition theory as well as ring and module extensions for G(z-)extending modules. We obtain that if a ring is right G(z)-extending, then so is its essential overring. Also it is shown that the G(z)-extending property is inherited by its rational hull. Furthermore it is provided some applications including matrix rings over a right G(z)-extending ring.