Generalization of ρ-ideals associated with an m-system and a special radical class

Let phi not equal S subset of R be an m-system of a ring R, and let. be a special radical. This study introduces the concept of S-rho-ideals in noncommutative rings. This notion extends the previously studied rho-ideals and can also be seen as a generalization of the right S-prime ideals. We show how some properties associated with rho-ideals have evolved into results within these generalizations. Relationships between S-rho-ideals and other types of ideals like rho-ideals, right S-prime ideals, and S-finite ideals are shown. We show the behaviour of this notion in related rings. The construction of (S boxed plus M)-rho-ideals in idealization rings is presented for an R- R-bimodule M. Additionally, we introduce S-P-ideals using Baer-McCoy radical P and examine their properties.