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 ∈ n for r ∈ r and x ∈ m implies that either x ∈ n or r ∈ p. in this paper we study a new equivalent conditions for a minimal prime submodules of an r-module to be a finite set, whenever r is a noetherian ring. also we introduce the concept of arithmetic rankof a submodule of a noetherian module and we give an upper bound for it.