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).
