A Characterization of Prime Submodules

Let R be a commutative domain and let M be an R-module. It is proved that to every prime submodule of M there corresponds a prime ideal of R and a set of linear equations of a certain type, and conversely. In particular, in case M is a finitely generated R-module generated by n elements, for some positive integer n, then the prime submodules of M are given by prime ideals of R and certain finite systems of equations containing at most n equations.