# Flat and algebraic (non-integral) local rings extension $R \subseteq S$ with $m_RS=m_S$

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.

• Asking the same question on MSE and MO in such a short time period is generally frowned upon. You should probably also find some way to communicate that this question was answered at MO, and that you accepted the answer there. Commented Apr 14, 2021 at 20:15
• @KReiser, thank you for your comment. The link is relevant and informative. Ok, you are right, from now on if I will ask the same question both on MSE and MO and get a good answer in one of them, I will inform the other. (I now edited the above question: "... and quite immediately got an answer there"). Commented Apr 14, 2021 at 21:49