Structural properties of S-Maximal ideals and their applications in noncommutative rings

Abstract This paper presents a comprehensive study of right $${\textbf {S}}$$ S -maximal ideals in noncommutative rings, providing a natural extension of classical ideal theory through the framework of $${\textbf {m}}$$ m -systems. We establish deep connections among right $${\textbf {S}}$$ S -maximal, S -comaximal, and $${\textbf {S}}$$ S -prime ideals, and introduce new concepts such as the $${\textbf {S}}$$ S -Jacobson radical and $${\textbf {S}}$$ S -invertible elements to further develop the structural theory of rings. A key contribution is the introduction of $${\textbf {S}}$$ S -local rings and the formulation of an $${\textbf {S}}$$ S -version of Nakayama’s Lemma. In addition, we propose two complementary definitions of left $${\textbf {S}}$$ S -primitive ideals–one ideal-theoretic and the other annihilator-based–and demonstrate their intrinsic relationship to $${\textbf {S}}$$ 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 $${\textbf {S}}$$ S -primitivity.