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