$S$-Noetherian Rings and Their Extensions

Let $R$ be an associative ring with identity, $S$ a multiplicative subset of $R$, and $M$ a right $R$-module. Then $M$ is called an $S$-Noetherian module if for each submodule $N$ of $M$, there exist an element $s \in S$ and a finitely generated submodule $F$ of $M$ such that $Ns \subseteq F \subseteq N$, and $R$ is called a right $S$-Noetherian ring if $R_R$ is an $S$-Noetherian module. In this paper, we study some properties of right $S$-Noetherian rings and $S$-Noetherian modules. Among other things, we study Ore extensions, skew-Laurent polynomial ring extensions, and power series ring extensions of $S$-Noetherian rings.