On Weakly (2, ϱ)‐Ideals in Noncommutative Rings

In this paper, we investigate weakly (2, ϱ)‐ideals ( w ‐(2, ϱ)‐ideals) in noncommutative rings, where ϱ denotes a special radical. We demonstrate, supported by counterexamples, that these ideals serve as generalizations of weakly ϱ‐ideals and (2, ϱ)‐ideals. Furthermore, we examine the properties of these new ideals in ring extensions such as homomorphic images, idealizations, and product rings. Special attention is given to cases where the special radical coincides with the Nil or Jacobson radical, and a comparative analysis with the existing literature is provided. Our results extend the investigation of weakly (2, ϱ)‐ideals to noncommutative rings and highlight the significance of these ideals in the general context of ring theory.