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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 $n+1$.

The cases $n=0$ and $n=d$ can indeed be finite: a Noetherian ring has finitely many minimal primes, and a semilocal ring may have finitely many maximal ideals. We prove that every intermediate height occurs infinitely often.

Because $d$ is a finite integer equal to the supremum of prime-chain lengths, there is a chainEach ${\mathfrak{p}}_i$ has height exactly $i$: its displayed lower chain gives height at least $i$, while any longer lower chain could be extended by the remaining displayed primes and would contradict $\mathrm{ht}{\mathfrak{p}}_d=d$.

Suppose $0<n<d$. The three primesfall under prime ideals between a three-prime chain, so infinitely many primes $\mathfrak{q}$ satisfyEvery such $\mathfrak{q}$ has height exactly $n$: the lower inclusion gives height at least $n$, and height at least $n+1$ would, after appending ${\mathfrak{p}}_{n+1}$, contradict its height $n+1$. Therefore there are infinitely many height-$n$ primes. The assumed finiteness forcesThis is infinitude of intermediate-height prime ideals.