Generalization of the $S$-Noetherian concept

summary:Let $A$ be a commutative ring and ${\mathcal{S}}$ a multiplicative system of ideals. We say that $A$ is ${\mathcal{S}}$-Noetherian, if for each ideal $Q$ of $A$, there exist $I\in {\mathcal{S}}$ and a finitely generated ideal $F\subseteq Q$ such that $IQ\subseteq F$. In this paper, we study the transfer of this property to the polynomial ring and Nagata’s idealization