On 2-$$\varrho $$ ideals in noncommutative rings

Abstract In this study, we define a broader class of ideals in non-commutative rings by considering the special radical of the 2-nil ideals introduced in commutative rings Yetkin Celikel E, Bulletin of the Belgian Mathematical Society Simon Stevin, 28(3) (2021). In the mentioned work, it was shown that in Noetherian integral domain rings, every 2-nil ideal is prime. Here, we demonstrate that in non-commutative rings, every 2- $$\mathcal {P}$$ P ideal of an Artinian and prime ring is prime. We also investigated the properties of 2- $$\varrho $$ ϱ -ideals in ring constructions such as direct product rings, homomorphic images, and idealization extensions. Moreover, in contrast to the work on commutative rings, some results have also been obtained concerning idealization and we explored the connections of this new class of ideals with various ideal-theoretic structures, including prime ideals, primary-like ideals, and 2-absorbing ideals.