Let $R \subseteq S$ be two commutative $\mathbb{C}$-algebras such that:
(1) $R$ and $S$ are integral domains.
(2) $Q(R)=Q(S)$, namely, their fields of fractions are equal.
(3) $S=R[w]$, for some $w \in S$.
(4) $S$ is separable over $R$, namely, $S$ is a projective $S \otimes_R S$-module via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$.
Should such $S$ be flat over $R$? I guess no, so please it would be nice to see a counterexample.
The above is (almost) question 3 of this question.
Thank you very much!
Now also asked in MO.