This paper presents a comprehensive study of right \({\textbf {S}}\) -maximal ideals in noncommutative rings, providing a natural extension of classical ideal theory through the framework of \({\textbf {m}}\) -systems. We establish deep connections among right \({\textbf {S}}\) -maximal, S-comaximal, and \({\textbf {S}}\) -prime ideals, and introduce new concepts such as the \({\textbf {S}}\) -Jacobson radical and \({\textbf {S}}\) -invertible elements to further develop the structural theory of rings. A key contribution is the introduction of \({\textbf {S}}\) -local rings and the formulation of an \({\textbf {S}}\) -version of Nakayama’s Lemma. In addition, we propose two complementary definitions of left \({\textbf {S}}\) -primitive ideals–one ideal-theoretic and the other annihilator-based–and demonstrate their intrinsic relationship to \({\textbf {S}}\) -prime ideals. Our results not only unify and generalize existing notions but also open promising avenues for further research, particularly in exploring the interplay between different forms of \({\textbf {S}}\) -primitivity.
