On 2-Absorbing Submodules

Abstract

In this paper, we introduce the concept of 2-absorbing submodules as a generalization of 2-absorbing ideals. Let R be a commutative ring and M an R-module. A proper submodule N of M is called 2-absorbing if whenever a, b ∈ R, m ∈ M and abm ∈ N, then am ∈ N or bm ∈ N or ab ∈ N:RM. Let N be a 2-absorbing submodule of M. It is shown that N:RM is a 2-absorbing ideal of R and either Ass R(M/N) is a totally ordered set or Ass R(M/N) is the union of two totally ordered sets. Furthermore, it is shown that if M is a finitely generated multiplication module over a Noetherian ring R, and Ass R(M/N) a totally ordered set, then N is 2-absorbing whenever N:RM is a 2-absorbing ideal of R. Also, the 2-absorbing avoidance theorem is proved.