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 \neq M $ , and (ii) whenever $ rm \in P $ for some $ r\in R , m \in M \backslash P $ , then $ rM \subseteq P $ . An R-submodule N of M is called a semiprime submodule of M if (i) $ N \neq M $ , and (ii) whenever $ r^k m \in N $ for some $ r \in R , m \in M $ and natural number k, then $ rm \in 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.