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).