Z- PRIMESUBMODULES.

LetR be a commutative ring with identity and Mbe a unitary R-module, a submodulN of an R-module M is called prime if for each x?M,r?R such that rx?N implies thateither x?Nor r?[N:M]. In this paper we say that Nis a Z-prime submodule of an R-module Mif for eachx?M,f?M^*=Hom(M,R), such that f(x).x?Nimplies that either x?Norf(x)?[N:M], where [N:M]={r: r?RandrM?N}. We study also aZ-prime module in which (0) is a Z-prime submoduleinM. We give many properties of Z-prime submodule and Z-prime module.