Summary: Let \(R\) be a commutative ring with identity and \(M\) be a unital \(R\)-module. A proper submodule \(N\) of \(M\) with \(N:_RM=\mathfrak p\) is said to be prime or \(\mathfrak p\)-prime \((\mathfrak p\) a prime ideal of \(R)\) if \(rx\in N\) for \(r\in R\) and \(x\in M\) implies that either \(x\in N\) or \(r\in\mathfrak p\). 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 rank of a submodule of a Noetherian module and we give an upper bound for it.