A SHORT NOTE ON PRIME SUBMODULES

Let R be a commutative ring with identity and M be a unital R-module. A proper submodule N of M with N :R M = p is said to be prime or p-prime (p a prime ideal of R) if rx 2 N for r 2 R and x 2 M implies that either x 2 N or r 2 p. In this paper we study a new equivalent conditions for a minimal prime submodules of an R-module to be a nite set, whenever R is a Noetherian ring. Also we introduce the concept of arithmetic rank of a submodule of a Noetherian module and we give an upper bound for it.