N-almost prime submodules

Abstract

Let R be a commutative ring with identity. A proper submodule N of an R-module M will be called prime [resp. n-almost prime], if for r ∈ R and a ∈ M with ra ∈ N [resp. ra ∈ N \ (N: M) n−1 N], either a ∈ N or r ∈ (N: M). In this note we will study the relations between prime, primary and n-almost prime submodules. Among other results it is proved that: (1) If N is an n-almost prime submodule of an R-module M, then N is prime or N = (N: M)N, in case M is finitely generated semisimple, or M is torsion-free with dim R = 1. (2) Every n-almost prime submodule of a torsion-free Noetherian module is primary. (3) Every n-almost prime submodule of a finitely generated torsion-free module over a Dedekind domain is prime. (4) There exists a finitely generated faithful R-module M such that every proper submodule of M is n-almost prime, if and only if R is Von Neumann regular or R is a local ring with the maximal ideal m such that m 2 = 0. (5) If I is an n-almost prime ideal of R and F is a flat R-module with IF ≠ F, then IF is an n-almost prime submodule of F.