Pseudo-prime radical submodule

Abstract

Let R be acommutative ring with identity ,and M be a unitary R-module .Aproper submodule N of M is called pseudo-prime submodule if whenever abm∈N for a,b∈R and m∈M ,then either Introduce pseudo-prime radical of asubmodule N as ageneralization of aprime –radical of asubmoudule N where a pseudo-prime radical of asubmodule N denoted byPP-radM (N) is define to the as the intersection of all pseudo prime submodule of M that contain N. also ,weIntroduce the concept pseudo –prime radical submodule where aproper submodule N of M which satisfies PP-radM (N) = N is called pseudo –prime radical submodule ofM.We give some basic properties of these concepts.