Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 101 / 4 / d

Let $A$ be a principal ideal domain that is not a field. A PID is Noetherian and is a unique factorization domain. Every nonzero prime ideal is generated by a prime element. Ifthen $a\mid p$, 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.

Conversely, let $A$ be a Noetherian UFD of Krull dimension one. Every nonzero prime is minimal among nonzero primes, and the forward argument in part (c) makes it principal. The zero ideal is principal as well, so part (b) shows that $A$ is a PID. A field has Krull dimension zero, hence $A$ is not a field.