Is the ring $A[x_1, \ldots, x_n]$ Cohen-Macaulay? Does the dimension formula hold?

Question

Let $A[x_1, \ldots, x_n]$ be a polynomial ring over a Noetherian, commutative ring, $A$. Is the polynomial ring Cohen-Macaulay?

If not, does it follow the dimension formula, $ \mathrm{dim} (A[x_1, \ldots, x_n]) = \mathrm{dim} (A[x_1, \ldots, x_n]/\mathfrak{p}) + \mathrm{ht} (\mathfrak{p} ), $ for a prime ideal, $\mathfrak{p}$ in $A[x_1, \ldots, x_n]$.

Answer 1

1. It's well known that $A[x_1, \ldots, x_n]$ is CM iff $A$ is CM.

2. Let $A=\mathbb Z_{(2)}$. Then $\dim\mathbb Z_{(2)}[X]=2$, $\dim\mathbb Z_{(2)}[X]/(2X-1)=0$ and $\operatorname{ht}(2X-1)=1$.