On 2-absorbing Primary Submodules of Modules over Commutative Rings

All rings are commutative with 1 6= 0, and all modules are unital. The purpose of this paper is to investigate the concept of 2-absorbing primary submodules generalizing 2-absorbing primary ideals of rings. Let M be an R-module. A proper submodule N of an R-module M is called a 2-absorbing primary submodule of M if whenever a, b ∈ R and m ∈M and abm ∈ N , then am ∈M -rad(N) or bm ∈M -rad(N) or ab ∈ (N :R M). It is shown that a proper submodule N of M is a 2-absorbing primary submodule if and only if whenever I1I2K ⊆ N for some ideals I1, I2 of R and some submodule K of M , then I1I2 ⊆ (N :R M) or I1K ⊆M -rad(N) or I2K ⊆M -rad(N). We prove that for a submodule N of an R-module M if M -rad(N) is a prime submodule of M , then N is a 2-absorbing primary submodule of M . If N is a 2-absorbing primary submodule of a finitely generated multiplication R-module M , then (N :R M) is a 2-absorbing primary ideal of R and M -rad(N) is a 2-absorbing submodule of M .