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.
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No, it is not true.
Consider a discrete valuation ring $R$ with uniformizing parameter $\pi$ and residue field $k$.
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 $Q$).