Purely Goldie Extending Modules

An R-module M is extending if every submodule of M is essential in a direct summand of M. Following Clark, an R-module M is purely extending if every submodule of M is essential in a pure submodule of M. It is clear purely extending is generalization of extending modules. Following Birkenmeier and Tercan, an R-module M is Goldie extending if, for each submodule X of M, there is a direct summand D of Msuch that XβD. In this paper, we introduce and study class of modules which are proper generalization of both the purely extending modules and G-extending modules. We call an R-module M is purely Goldie extending if, for each X≤M, there is a pure submodule P of Msuch that XβP. Many characterizations and properties of purely Goldie extending modules are given. Also, we discuss when a direct sum of purely Goldie extending modules is purely Goldie extending and moreover we give a sufficient condition to make this property of purely Goldie extending modules is valid.