= Solution
Let $A$ be a <principal ideal domain> that is not a field. A PID is <Noetherian ring>[Noetherian] and is a <unique factorization domain>. Every nonzero prime ideal is generated by a prime element. If
$$
(p)\subseteq(a)\subsetneq A,
$$
then $a\mid p$, so primality of $p$ makes $a$ a unit or an associate of $p$. The proper alternative is $(a)=(p)$, and every nonzero prime is therefore maximal. Since $A$ has a nonzero prime ideal, its <Krull dimension> is exactly one.
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