Generalization of \(\rho \)-ideals associated with an m-system and a special radical class

Let \(\phi \ne S\subseteq R\) $\varphi \ne S\subseteq R$ be an m-system of a ring R, and let \(\rho \) $\rho$ be a special radical. This study introduces the concept of S- \(\rho \) $\rho$ -ideals in noncommutative rings. This notion extends the previously studied \(\rho \) $\rho$ -ideals and can also be seen as a generalization of the right S-prime ideals. We show how some properties associated with \(\rho \) $\rho$ -ideals have evolved into results within these generalizations. Relationships between S- \(\rho \) $\rho$ -ideals and other types of ideals like \(\rho \) $\rho$ -ideals, right S-prime ideals, and S-finite ideals are shown. We show the behaviour of this notion in related rings. The construction of \((S\boxplus M)\) $(S⊞M)$ - \(\rho \) $\rho$ -ideals in idealization rings is presented for an R-R-bimodule M. Additionally, we introduce S- \(\mathcal {P}\) $\mathcal{P}$ -ideals using Baer-McCoy radical \(\mathcal {P}\) $\mathcal{P}$ and examine their properties.