= Prime ideals between a three-prime chain
If $R$ is Noetherian and $\mathfrak p_0\subsetneq\mathfrak p_1\subsetneq\mathfrak p_2$ are prime ideals, there are infinitely many primes strictly between $\mathfrak p_0$ and $\mathfrak p_2$. After quotienting by $\mathfrak p_0$ and localizing at $\mathfrak p_2$, this reduces to a local Noetherian domain of dimension at least two. If it had only finitely many height-one primes, <prime avoidance> would give a nonzero element of the maximal ideal outside all of them, contradicting the <Krull principal ideal theorem>.
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