Modules with bounded spectra

Let R be a commutative ring with identity and let M be an R-module. We examine the situation where for each prime ideal ρof R the set of all ρ-prime submodules of M is finite. In case R is Noetherian and M is finitely generated, we prove that this condition is equivalent to there being a positive integer n such that for every prime ideal ρ of R, the number of ρ-prime submodules of Mis less than or equal to n. We further show that in this case, there is at most one ρ-prime submodule for all but finitely many prime ideals ρ of R.