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

Abstract This paper presents a comprehensive study of right $${\textbf {S}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> -maximal ideals in noncommutative rings, providing a natural extension of classical ideal theory through the framework of $${\textbf {m}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>m</mml:mi> </mml:math> -systems. We establish deep connections among right $${\textbf {S}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> -maximal, S -comaximal, and $${\textbf {S}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> -prime ideals, and introduce new concepts such as the $${\textbf {S}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> -Jacobson radical and $${\textbf {S}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> -invertible elements to further develop the structural theory of rings. A key contribution is the introduction of $${\textbf {S}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> -local rings and the formulation of an $${\textbf {S}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> -version of Nakayama’s Lemma. In addition, we propose two complementary definitions of left $${\textbf {S}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> -primitive ideals–one ideal-theoretic and the other annihilator-based–and demonstrate their intrinsic relationship to $${\textbf {S}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> -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 $${\textbf {S}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> -primitivity.