Do isomorphic quotient fields imply isomorphic rings?

Question

Let $Q(R)$ denote the quotient field (or field of fractions) of an integral domain, $R$.

If $R$ and $S$ are integral domains such that $Q(R)\cong Q(S)$, does this imply that $R\cong S$?

I am not sure about this. I was thinking the answer is yes, but I couldn't prove it. Perhaps there exist a counterexample?

Answer 1

No.

Hint: Think about $\Bbb Z$ and $\Bbb Z[\frac12]$, for example.

Answer 2

No: $\mathbb Z$ and $\mathbb Q$ have the same quotient field but are not isomorphic.

A less trivial example is $\mathbb Z[X]$ and $\mathbb Q[X]$, whose quotient field is $\mathbb Q(X)$.

Answer 3

Any integral domain $R$ and any localization of $R$ say at a prime ideal $\mathcal{p}$ have always the same quotient field.