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