# Example of Gorenstein ring that has infinite Krull dimension?

Let $R$ be a Noetherian commutative ring. If $R$ is local, then $R$ is Gorenstein if $inj.dim(R)<\infty$. Otherwise $R$ is Gorenstein if $R_P$ Gorenstein for all prime ideals $P$.\

There is a simple result stating that $inj.dim(R)<\infty$ if and only if $R$ is Gorenstein and it has finite Krull dimension, and I proved it. My question is, is there any ring that is Gorenstein and has infinite Krull dimension?

• What about the Nagata example of Noetherian ring with infinite Krull dimension? – user26857 Jun 21 '18 at 11:15
• Found it. I'll check – chí trung châu Jun 21 '18 at 11:31
• Since regular rings are Gorenstein, your question was answered here. – Fred Rohrer Jun 30 '18 at 19:52
• Interesting... Some machine classified my answer as "trivial" and hence converted it to a comment. Whatever. (But the machine has not seen that I wrote complete nonsense at first. Ha!) – Fred Rohrer Jun 30 '18 at 19:53