S-Prime Ideals, S-Noetherian Noncommutative Rings, and the S-Cohen’s Theorem

Let S be an m-system of a ring R. This paper presents the notion of a right S-prime ideal into noncommutative rings and provides some properties and equivalent definitions. We define a right S-idempotent ideal and an S-totally ordered set, and we show that every ideal of R is a right S-idempotent ideal, and the set of ideals in R is S-totally ordered if and only if every ideal in R is a right S-prime ideal. We also generalize the concept of an S-finite ideal and an S-Noetherian ring. Furthermore, we provide the S-versions of Cohen’s and Cohen–Kaplansky’s Theorems in a special case, and we demonstrate that the ring \(T_2(R)\) $T_2(R)$ of upper triangular matrices over an S-Noetherian ring R is right \(S_{T_2(R)}\) $S_{T_2(R)}$ -Noetherian.