# Noetherian rings and prime ideals

Let $R$ be a noetherian ring and $P\subset Q$ be prime ideals. I'm trying to prove that if there exists another prime ideal $P_1$ such that $P\subset P_1\subset Q$ and $P\ne P_1\ne Q$, then there are infinitely many prime ideals betwen $P$ and $Q$.

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@Lepidopterist Actually I know the proof of Krull's but I don't see a connection here. –  user60184 Mar 19 '13 at 1:57
Mod out by P and localize at Q... now suppose there are only finitely many prime ideals. Then $R$ is Artinian, but it's also an integral domain... so it's a field. –  Dylan Wilson Mar 19 '13 at 2:44
Why does it follow that $R$ is Artinian? –  mbrown Mar 19 '13 at 2:51
Take the quotient of $R$ by $P$, then localized at the image of $Q$. You then have a noetherian local ring of dimension $\ge 2$. Any prime ideal of height $1$ of this local ring will induce a prime ideal of $R$ strictly included between $P$ and $Q$, and two such prime ideals will induce two distinct prime ideals.
Now in a noetherian ring of dimension $\ge 2$, there are always infinitely many prime ideals of height $1$, see my answer to this question.