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

TL;DR

This paper introduces and explores graded weakly 2-absorbing ideals in non-commutative graded rings, extending concepts previously studied mainly in commutative cases and establishing their properties in non-commutative settings.

Contribution

It generalizes the notion of graded weakly prime ideals to non-commutative graded rings and demonstrates that many properties from commutative cases also apply.

Findings

Many results from commutative graded rings hold in non-commutative cases.

Introduction of graded weakly 2-absorbing ideals in non-commutative graded rings.

Extension of existing theories to broader non-commutative contexts.

Abstract

For commutative graded rings, the concept of graded $2$-absorbing (graded weakly $2$-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 (graded weakly prime) ideals. Up to now, research on these topics mainly concentrated on commutative graded rings. On the other hand, graded prime ideals over non-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-commutative graded rings, the concept of graded $2$-absorbing ideals over non-commutative graded rings has been introduced and investigated by Abu-Dawwas, Shashan and Dagher (WSEAS Transactions on Mathematics, 19 (2020), 232-238). Recently, graded weakly prime ideals over non-commutative graded rings have been introduced and studied by Alshehry and Abu-Dawwas (Communications in Algebra, 49 (11) (2021), 4712-4723). In this article, we introduce and study the concept of graded weakly $2$-absorbing ideals as a generalization of graded weakly prime ideals in a non-commutative graded ring, and show that many of the results in commutative graded rings also hold in non-commutative graded rings.