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,
where 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][1], [b][2] and [c][3].
Also asked in [MSE][4].

Any hints and comments are welcome; thank you.


  [1]: https://math.stackexchange.com/questions/4093536/a-sandwich-theorem-for-local-rings/4093553?noredirect=1#comment8461064_4093553
  [2]: https://math.stackexchange.com/questions/4092004/ideals-in-the-localization-r-p?noredirect=1#comment8457172_4092004
  [3]: https://math.stackexchange.com/questions/4092716/definition-of-homomorphism-of-local-rings/4093922#4093922
  [4]: https://math.stackexchange.com/questions/4101305/local-rings-r-subseteq-s-with-m-rs-m-s