Let $P, Q$ be prime ideals of the commutative Noetherian ring $R$(with unity) such that $ P \subset Q$. Show that, if there exists one prime ideal of $R$ strictly between $P$ and $Q$ then there are infinitely many.
I just study Krull's Principal ideal Theorem and immediately after I got this exercise. I think I should use Krull's Principal ideal theorem.
Can I do something with integral domain $R/P$?
My Attempt : Consider R/P which is Noetherian and integral domain. Hence we can assume R is Noetherian domain and P is zero prime ideals. If possible let there exist finitely many prime ideals between 0 and Q. Now what to do? Can I use prime avoidance theorem now? But how?