$R$ an integral domain and $S\subset R$, prove that $F(R)=F(S) \implies R=S$?

Question

Let $R$ be an integral domain, and $S$ be a unital subring of $R$.

If the two field of fractions $F(R)=F(S)$, does it imply that $R=S$.

Im pretty sure it does, but can't think of a proof.

Answer 1

The statement you wrote down is false. Here is a counterexample:

$\mathbb{Z}$ and $\mathbb{Q}$ have the same fraction field, i.e.

$$F(\mathbb{Z})= F(\mathbb{Q})= \mathbb{Q}$$

but $\mathbb{Z} \neq \mathbb{Q}$.

Answer 2

Counterexample:

Consider any element $f\in\mathbf Z$, $f\ne 0,1,-1$. Then $\;\mathbf Z\varsubsetneq \mathbf Z_f$, yet $\mathbf Q$ is their common field of fractions.