P-Essential Maximal Submodules

Let be a commutative ring with identity and be a unitary left -module. The main goal of this work is to introduce and study the idea of a P-essential maximal submodule, which is a generalization for maximal submodules, as a proper submodule of an -Mod is called P-essential maximal (shortly Pe-max) if for any P-essential submodule of a module with implies that , where a sub mod of an -Mod is called P-essential in , if such that , implies . An ideal of a ring is called P-essential maximal ideal, if for any Pe-ideal of such that implies . And we introduce the concept P-essential local, an -Mod is called P-essential local module (for short Pe-local), if it has only one Pe-maximal submodule, which is contains each proper submodule of . And we research some relationships between P-essential maximal submodules and some other related ideas. Moreover, we give P-essential local module and P-essential radical module. Some properties and many results were proved as well as the behaviour of that concept with its localization was studied and shown