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 $Q$.

need your help please, thank you.

Solution 1:

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.