Infinitude of intermediate-height prime ideals
= Infinitude of intermediate-height prime ideals
If a Noetherian ring has finite <Krull dimension> $d$, then it has infinitely many prime ideals of each height $n$ with $0<n<d$. In a prime chain of maximal length, apply <prime ideals between a three-prime chain> to the terms of heights $n-1$, $n$, and $n+1$.