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For commutative graded rings, the concept of graded $2$-absorbing (gradedweakly $2$-absorbing) ideals was introduced and examined by Al-Zoubi,Abu-Dawwas and \c{C}eken (Hacettepe Journal of Mathematics and Statistics, 48(3) (2019), 724-731) as a generalization of the concept of graded prime (gradedweakly prime) ideals. Up to now, research on these topics mainly concentratedon commutative graded rings. On the other hand, graded prime ideals overnon-commutative graded rings have been introduced and examined by Abu-Dawwas,Bataineh and Al-Muanger (Vietnam Journal of Mathematics, 46 (3) (2018),681-692). As a generalization of graded prime ideals over non-commutativegraded rings, the concept of graded $2$-absorbing ideals over non-commutativegraded rings has been introduced and investigated by Abu-Dawwas, Shashan andDagher (WSEAS Transactions on Mathematics, 19 (2020), 232-238). Recently,graded weakly prime ideals over non-commutative graded rings have beenintroduced and studied by Alshehry and Abu-Dawwas (Communications in Algebra,49 (11) (2021), 4712-4723). In this article, we introduce and study the conceptof graded weakly $2$-absorbing ideals as a generalization of graded weaklyprime ideals in a non-commutative graded ring, and show that many of theresults in commutative graded rings also hold in non-commutative graded rings.

Graded Weakly $2$-Absorbing Ideals over Non-Commutative Rings