On $ S $-principal right ideal rings

Let $ S $ be a multiplicative subset of a ring $ R $. A right ideal $ A $ of $ R $ is referred to as $ S $-principal if there exist an element $ s \in S $ and a principal right ideal $ aR $ of $ R $ such that $ As \subseteq aR \subseteq A $. A ring is referred to as an $ S $- principal right ideal ring ($ S $-PRIR) if every right ideal is $ S $-principal. This paper examines $ S $-PRIRs, which extend the notion of principal right ideal rings. Various examples, including several extensions of $ S $-PRIRs are investigated, and some practical results are proven. A noncommutative $ S $-PRIR that is not a principal right ideal ring is found, and the $ S $-variants of the Eakin-Nagata-Eisenbud theorem and Cohen's theorem for $ S $-PRIRs are proven.