# Local rings $R \subsetneq S$ with $R$ regular and $S$ Cohen-Macaulay, non-regular

Let $$R \subseteq S$$ be local rings with maximal ideals $$m_R$$ and $$m_S$$. Assume that:

(1) $$R$$ and $$S$$ are (Noetherian) integral domains.

(2) $$\dim(R)=\dim(S) < \infty$$, where $$\dim$$ is the Krull dimension.

(3) $$R$$ is regular (hence a UFD).

(4) $$S$$ is Cohen-Macaulay.

(5) $$R \subseteq S$$ is simple, namely, $$S=R[w]$$ for some $$w \in S$$.

(6) $$R \subseteq S$$ is free.

(7) $$R \subseteq S$$ is integral, namely, every $$s \in S$$ satisfies a monic polynomial over $$R$$.

(8) $$m_RS=m_S$$, namely, the extension of $$m_R$$ to $$S$$ is $$m_S$$.

(9) It is not known whether the fields of fractions of $$R$$ and $$S$$, $$Q(R)$$ and $$Q(S)$$, are equal or not.

(10) It is not known if $$R \subseteq S$$ is separable or not.

Remark: It is known that if a (commutative) integral domains ring extension $$A \subseteq B$$ is integral+flat, then it is faithfully flat, and if also $$Q(A)=Q(B)$$, then $$A=B$$. This is why I did not want to assume that $$Q(R)=Q(S)$$, since in this case $$R=S$$ immediately.

Question: Is it true that, assuming (1)-(10) imply that $$S$$ is regular or $$R=S$$?

Example: $$R=\mathbb{C}[x(x-1)]_{x(x-1)}$$ and $$S=\mathbb{C}[x]_{(x)}$$, with $$R \neq S$$ and $$S$$ is regular.

Non-example: $$R=\mathbb{C}[x^2]_{(x^2)}$$ and $$S=\mathbb{C}[x^2,x^3]_{(x^2,x^3)}$$, but condition (8) is not satisfied.

Relevant questions, for example: a, b, c, d.

Thank you very much! I have asked the above question here, with no comments (yet).

Condition (8) implies that the map is unramified, and since it is also flat by (6) (which by the way implies $$S$$ is CM by Miracle Flatness), it is etale, and so $$S$$ is regular. Note you don't need (5) here.
Without condition (8) you can take $$R=\mathbf{C}[[x,y]]$$ and $$S=R[w]/(w^2 - xy)$$.