On φ-β-Absorbing Submodules

In this paper, we extend the concept of β-absorbing submodules to φ-β-absorbingsubmodules over a commutative ring with nonzero identity which is a generalization of 2-absorbing submodules. Let S(M) be the set of all submodules of M and φ : S(M) → S(M)∪{∅} be a function. A proper submodule P of M is called a φ-β-absorbing submodule, if for each r, s ∈ R and m ∈ M with rsm ∈ P\φ(P), then rs + rs ∈ (P : M) or r(m + m) ∈ P or s(m + m) ∈ P. Some of the properties and characterizations of φ-β-absorbing submodules are investigated.