Z-quasi prime submodules and the Z-radicals of submodules

Abstract

Let R be a commutative ring with identity and N be a proper submodule N of R-moduLe M is called prime if whenever rx ∈ N; r ∈ R , x ∈ M , implies either x ∈ N or r ∈ [N : M], where [N : M] = {r ∈ R : rM ⊆ N}.  In this paper we say that N is Z-quasi prime submodule of R-moduLe M, if [N : (x)] is a Z-prime ideal of R for each x ∈ M.  We prove some result of this type of submodules.