$R$ is a noetherian domain and $K=\operatorname{Frac}(R)$. If $S$ subring s.t. $R\subset S\subset K$, is $S$ noetherian?

Question

$\textbf{Q:}$ Suppose $R$ is noetherian domain and $K=\operatorname{Frac}(R)$. If $S$ is a subring of $K$ s.t. $R\subset S\subset K$, is $S$ noetherian? I suspect this will be the case in general. One can consider $(R:_R I)$ with $S$-ideal $I\subset S$. There is no good reason that $(R:_R I)\neq 0$. Furthermore, there is no good reason that $S$ being localization of $R$ in general unless $R$ is Bézout.