On a Generalization of Prime Submodules of a Module over a Commutative Ring

Let R he a commutative ring with identity, and n >= 1 an integer. A proper submodule N of an R-module M is called an n-prime submodule if whenever a(1)... a(n+1) m is an element of N for some non-units a(1), ..., a(n+1) is an element of R and m is an element of M, then m is an element of N or there are n of the a(i)'s whose product is in (N : M). In this paper, we study n-prime submodules as a generalization of prime submodules. Among other results, it is shown that if M is a finitely generated faithful multiplication module over a Dedekind domain R, then every n-prime submodule of M has the form m(1)... m(t) M for some maximal ideals m(1),..., m(t) of R with 1 <= t <= n.