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 \(T_2(R)\) of upper triangular matrices over an S-Noetherian ring R is right \(S_{T_2(R)}\)-Noetherian.

令S为环R的m系统。本文提出了非交换环中右S素数理想的概念，并提供了一些属性和等价定义。我们定义一个右S-幂等理想和一个S-全序集合，并且证明R的每个理想都是右S-幂等理想，并且R中的理想集合是S-全序当且仅当每个理想R是右S素数理想。我们还推广了S有限理想和S诺特环的概念。此外，我们在特殊情况下提供了科恩定理和科恩-卡普兰斯基定理的S版本，并且证明了S -诺特环R上的上三角矩阵的环\(T_2(R)\)是正确的\(S_ {T_2(R)}\) -Noetherian。