Krull dimension of a principal ideal domain ($\dim R=1(R\text{ not a field})$)

In a principal ideal domain, every nonzero prime ideal is maximal. Indeed, write it as $(p)$ with $p\ne 0$. If $(p)\subseteq (a)$, then $p=ab$. Primality gives $a\in (p)$, hence $(a)=(p)$, or $b\in (p)$, say $b=pc$, in which case cancellation gives $1=ac$ and $(a)=R$.

Thus every prime chain has length at most one. If the principal ideal domain is not a field, choose a nonzero nonunit $a$ and a maximal ideal $\mathfrak{m}\supseteq (a)$. Then $(0)⊊\mathfrak{m}$ is a prime chain of length one, so the Krull dimension is one. A field instead has Krull dimension zero.