φ-2-Absorbing Submodule

Let r be a commutative ring with identity and m be a unitary r-module. a proper submodule n of m is 2− absorbing if r1, r2, r3 in r, m in m with r1r2r3m in m implies r1r2m in n or r1r3m in n or r2r3m in n. let φ : s(m) longrightarrow s(m) cup {ø} be a function where s(m) is the set of all submodules of m. we call a proper submodule n of m a φ-2-absorbing submodule if r1, r2, r3 2 r, m in m with r1r2r3m in n −φ(n) implies that r1r2m in n or r1r3m in n or r2r3m in n. we want to extend 2-absorbing ideals to φ-2-absorbing submodules and we show that φ-2-absorbing submodules enjoy analogs of many of the properties of 2-absorbing ideals