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.