Phi-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 r 1, r2, r3 2 R, m 2 M with r1r2r3m 2 M implies r1r2m 2 N or r1r3m 2 N or r2r3m 2 N. Let ' : S(M) −! S(M) [ {ø} be a function where S (M) is the set of all submodules of M. We call a proper submodule N of M a '-2-absorbing submodule if r1, r2, r3 2 R, m 2 M with r 1r2r3m 2 N −'(N) implies that r1r2m 2 N or r1r3m 2 N or r2r3m 2N. We want to extend 2-absorbing ideals to '-2-absorbing submodules and we show that-2-absorbing submodules enjoy analogs of many ofthe properties of 2-absorbing ideals.