On 2-Absorbing Primary Submodules of Modules over Commutative Rings

All rings are commutative with 1 ≠ 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.