About Gorenstein ring

Question

Is it true that in a (non-local) Gorenstein ring, every maximal ideal has the same height? It seems a little strange, but I don't see any reason why it shoudn't.

Answer 1

No, it is not true.

Consider a discrete valuation ring $R$ with uniformizing parameter $\pi$ and residue field $k$.
Take for your Gorenstein ring the polynomial ring $A=R[X] $: since $R$ is regular, so is $A$ and thus $A$ is a fortiori Gorenstein.
Now consider the ideals $\mathfrak p=(\pi, X)$ and $\mathfrak q=(\pi X-1)$.
We have $A/\mathfrak p=k$, a field, and $A/\mathfrak q=Frac(R)$, a field too. So both $\mathfrak p$ and $\mathfrak q$ are maximal ideals. However their heights are different: $ht(\mathfrak p)=2$ (use $dim A=2$) and $ht(\mathfrak q)=1$ (use Krull's principal ideal theorem)

Answer 2

I think the answer is no. Take the product of two local Gorenstein rings of different dimensions; or take any Gorenstein ring $A$ and two prime ideals $P, Q$ of different heights, and no inclusion relation between them and localize at $A\setminus (P\cap Q)$ (you get a semi-local ring whose maximal ideals are given by $P$ and $Q$).