Communications of the
Korean Mathematical Society CKMS

Let $G$ be a group and $R$ a $G$-graded ring$.$ In 2023, Al-Shorman et al. introduced the concept of graded $J$-ideals of a $G$ -graded commutative ring and several characterizations of this concept were investigated. The notion of gr-$n$-ideals in a $G$-graded commutative ring was introduced by Al-Zoubi et al. and studied. In this note, we introduce the concept of a graded $\rho$-ideal (gr-$\rho$-ideal) for a graded special radical $\rho_{G}$ and show that the results of graded $J$-ideals and graded $n$-ideals are special cases of a more general situation. We prove that most of the results in graded commutative rings of the above-mentioned papers are satisfied for non-commutative $G$-graded rings as a special case.

Keywords: Graded $n$-ideal, graded J-ideal, graded special radical, graded radical ideal

MSC numbers: 16N20, 16N40, 16N80, 16L30, 16W50, 16D80