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 (in $R$) leading coefficient.



> 
> **Question:** 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](https://math.stackexchange.com/questions/4093536/a-sandwich-theorem-for-local-rings/4093553?noredirect=1#comment8461064_4093553), [b](https://math.stackexchange.com/questions/4092004/ideals-in-the-localization-r-p?noredirect=1#comment8457172_4092004) and [c](https://math.stackexchange.com/questions/4092716/definition-of-homomorphism-of-local-rings/4093922#4093922).
Also asked in [MSE](https://math.stackexchange.com/questions/4101305/local-rings-r-subseteq-s-with-m-rs-m-s).


Any hints and comments are welcome; thank you.



Flat and algebraic (non-integral) local rings extension $R \subseteq S$ with $m_RS=m_S$
In general if $R$ is a local ring, then its henselization $R^h$ is flat and "algebraic" over $R$, but rarely integral. The intuition is that the henselization is built out of localizations of etale extensions of $R$. Both localizations and etale extensions are flat and "algebraic" in your sense, but localizations are rarely integral.


To make a precise statement, first note that it suffices to find etale ring extensions $R\rightarrow R'\rightarrow S$ with $R$ normal, $S/R$ an etale local homomorphism of local rings, and $S/R'$ not integral.


Indeed, if we have found $R,R',S$ as above, then [$S/R'$ non-integral implies $S/R$ non-integral](https://stacks.math.columbia.edu/tag/035D). Moreover, [etale morphisms preserve normality](https://stacks.math.columbia.edu/tag/025P), so $S$ must be normal, hence a domain since $S$ is local so $\text{Spec }S$ is connected. Finally, [etale ring maps are locally standard etale](https://stacks.math.columbia.edu/tag/00ue), so every element of $S$ is "algebraic" over $R$.


Here's a sketch of how to produce an example: Start with a connected normal scheme $Y$ over $\mathbb{C}$ and a finite flat map $f : X\rightarrow Y$ with $X$ irreducible. In characteristic 0, $f$ is generically etale, so let $y\in Y$ be a point above which $f$ is etale. Let $R := \mathcal{O}\_{Y,y}$, let $X\_R := X\times\_Y\text{Spec }\mathcal{O}\_{Y,y}$, and write $X\_R = \text{Spec }R'$. Then $R'/R$ is finite etale. On the other hand, $R'$ is connected, since the generic point of $X$ maps to the generic point of $Y$ which lies in $\text{Spec }R$, so the generic point of $X$ lies in $X\_R$ and specializes to every closed point of $X\_R$, so $R'$ is a normal domain. However, if $X$ has multiple closed points lying over $y$, then $R'$ is not local, in which case let $\mathfrak{m}$ be a maximal ideal of $R'$, and let $S := R'\_{\mathfrak{m}}$. Then $S/R$ becomes an etale local homomorphism of local rings with $S/R'$ non-integral, as desired.


For an explicit example, you can take $Y = \text{Spec }\mathbb{C}[y]$, and
$$X := \text{Spec }\mathbb{C}[x,y]/(y^2-(x-a)(x-b)(x-c))$$
where $a,b,c\in\mathbb{C}$ are distinct. Then take $y\in Y$ to be the point $(y = 0)$. In particular, this shows that the henselization of $\mathbb{C}[y]\_{(y)}$ also satisfies your conditions.


