Goldie extending property on the class of exact submodules

We define a module M to be (Formula presented.) -extending if for each exact submodule X of M there exists a direct summand D of M such that (Formula presented.) is essential in both X and D. We investigate (Formula presented.) -extending modules and locate the implications between the other extending properties. We study decomposition theory and extensions for (Formula presented.) -extending concept. We show that if a ring is right (Formula presented.) -extending, then so its essential overring. It is shown that (Formula presented.) -extending property is inherited by its rational hull. We provide examples by making special choices of left exact preradicals.