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](https://stacks.math.columbia.edu/tag/0AG0) 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](https://en.wikipedia.org/wiki/Separable_algebra) 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](https://math.stackexchange.com/questions/2280481/example-of-a-local-cohen-macaulay-domain-which-is-not-regular), [b](https://math.stackexchange.com/questions/895673/cohen-macaulay-and-regular-rings), [c](https://mathoverflow.net/questions/390132/flat-and-algebraic-non-integral-local-rings-extension-r-subseteq-s-with-m), [d](https://math.stackexchange.com/questions/1320729/when-does-the-regularity-of-a-implies-the-regularity-of-aw?rq=1).


Thank you very much! I have asked the above question [here](https://math.stackexchange.com/questions/4109035/local-rings-r-subsetneq-s-with-r-regular-and-s-cohen-macaulay-non-regula), with no comments (yet).



Local rings $R \subsetneq S$ with $R$ regular and $S$ Cohen-Macaulay, non-regular
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 $S=R[w]/(w^2 - xy)$.


