Cohen-Macaulay and regular rings

Question

I know this is a simple question but to make sure...:

$A$ is a commutative ring which is Cohen-Macaulay and for every maximal ideal $\mathfrak{m}$ in $A$ we have $\dim A_{\mathfrak{m}}=\dim A$. Is then $A$ regular?

This seems wrong... why?

Answer 1

Any reduced noetherian ring of dimension $1$ is Cohen-Macaulay, but they are not regular in general.

Example: $A=\mathbb C[x,y]/(y^2-x^3)$.

Answer 2

For instance, there are local CM rings that aren't regular.