On Graded 2-Absorbing and Graded Weakly 2-Absorbing Ideals of a Commutative Ring

Abstract

Let G be a group with identity e and R be a G-graded commutative ring with 1 6= 0. In this paper, we study the graded versions of 2-absorbing and weakly 2-absorbing ideals which are generalizations of the graded prime and graded weakly prime ideals, respectively. A graded proper ideal I of R is called a graded 2- absorbing (resp. graded weakly 2-absorbing) ideal if whenever abc ∈ I (resp. 0 != abc ∈ I) for homogeneous elements a,b, c ∈ R, then ab ∈ I or ac ∈ I or bc ∈ I. It is clear that a graded ideal which is a 2-absorbing ideal, is a graded 2-absorbing ideal, but we show that the converse is not true in general. It is proved that if I = ⊕g∈GIg is a graded weakly 2-absorbing ideal of R, then either I is a 2-absorbing ideal of R or I3g = (0) for all g ∈ G. It is also shown that if I = ⊕i∈GIg is a graded weakly 2-absorbing ideal of R, then for each g ∈ G, either Ig is a 2-absorbing Re-submodule of Rg or (Ig :Re Rg) 2 Ig = 0.