On commutative Noetherian rings which have the s.p.a.r. property

Abstract

Let R be a commutative ring with 1 and M be an unitary R-module. Prime and semiprime submodules of M are defined as follows. An R-submodule P of M is called a prime submodule of M if (i) P ≠ M, and (ii) whenever rm ∈ P for some r ∈ R, m ∈ M\P, then rM ⊆ P. An R-submodule N of M is called a semiprime submodule of M if (i) N ≠ M, and (ii) whenever rk m ∈ N for some r ∈ R, m ∈ M and natural number k, then rm ∈ N. It is clear that an intersection of prime submodules of M is a semiprime submodule of M. In this paper, we give a characterization of a commutative Noetherian ring R with property that, every semiprime submodule of an R-module is an intersection of prime submodules.