On ϕ-2-Absorbing Primary Submodules

Abstract

Let R be a commutative ring with identity and M be a unitary R-module. Let \(\phi :S(M)\rightarrow S(M)\cup \{\emptyset \}\) be a function, where S(M) is the set of submodules of M. We say that a proper submodule N of M is a ϕ-2-absorbing primary submodule if rsx∈N∖ϕ(N) implies rx∈N, or sx∈N, or \(rs\in \sqrt {(N:M)}\), where r,s∈R and x∈M. In this paper, we study ϕ-2-absorbing primary submodules and we prove some basic properties of these submodules. Also, we give a characterization of ϕ-2-absorbing primary submodules and we investigate ϕ-2-absorbing primary submodules of some well-known modules.