Let $R \subseteq S$ be two Noetherian local rings (not necessarily regular) which are integral domains, with $m_RS=m_S$, namely, the ideal in $S$ generated by $m_R$ (= the maximal ideal of $R$) is $m_S$ (= the maximal ideal of $S$).
Further assume that $R$ and $S$ are $\mathbb{C}$-algebras, $R \subseteq S$ is flat and algebraic but not integral (algebraic non-integral means: Every element of $S$ satisfies a polynomial with coefficients in $R$, with non-invertible leading coefficient).
Could one find an example of such rings?
Unfortunately, the examples I find are integral, for example: $R=\mathbb{C}[x(x-1)]_{(x(x-1))}$, $S=\mathbb{C}[x]_{(x)}$.
Remarks:
(i) I am interested in both cases where $R$ and $S$ have the same fields of fractions or different fields of fractions.
(ii) Recall the following results, which are not applicable here, since I assume that $R \subseteq S$ is non-integral: If $A \subseteq B$ is integral and flat, then $A \subseteq B$ is faithfully flat, and if in addition, $Q(A)=Q(B)$ (same fields of fractions), then $A=B$.
Relevant questions: a, b and c. Also asked in MO, and quite immediately got an answer there; however, it seems that the answer there does not describe the special case $S=R[w]$, for some $w \in S$ (which I mentioned there in a comment after the answer).
Any hints and comments are welcome; thank you.
