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.