Subrings of fraction fields [closed]

Let $R$ be an integral domain and let $S$ be a ring with $R \le S \le \text{Frac}(R)$ (fraction field).

Question: Is there a multiplicatively closed subset $U \subseteq R\setminus \{0\}$ such that $S=R[U^{-1}]$ ?

There are lots of examples arising from integral closures. For example, $k[x^2,x^3] \subseteq k[x]$ with field of fractions $k(x)$ is no localization since $k[x^2,x^3]^* = k^* = k[x]^*$.

I've shown here that the ring $S=K[X]+YK(X)[Y]$ is not noetherian. Note that we have $K[X,Y]=R\subset S\subset Q(R)=K(X,Y)$. Since $R$ is noetherian, $S$ can't be a localization of $R$.