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

Abstract

Let R be 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 a1 · · · an+1m 2 N for some non-units a1, . . . , an+1 2 R and m 2 M, then m 2 N or there are n of the ai’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 m1 · · ·mtM for some maximal ideals m1, . . . ,mt of R with 1 t n.