On weakly S-ρ-ideals in noncommutative rings

Let R be a noncommutative ring, and φ ≠ S ⊆ R be an m-system of R. In this paper, we extend the concept of weakly prime ideals in noncommutative rings by introducing weakly S-ρ-ideals. This generalization builds on the notions of right S-prime, weakly S-n-ideals and S-ρ-ideals. We present equivalent definitions and key properties of weakly S-ρ-ideals. We also explore with the method of idealization how the weakly S-ρ-ideal property transfers to related rings and construct such ideals. On the other hand, we give a generalization of S-ρ-ideals which were introduced in [12].