= Krull dimension of a principal ideal domain
{c}
{title2=$\dim R=1\quad(R\text{ not a field})$}
In a <principal ideal domain>, every nonzero <prime ideal> is maximal. Indeed, write it as $(p)$ with $p\ne0$. 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 $(0)\subsetneq\mathfrak m$ is a <prime chain> of length one, so the <Krull dimension> is one. A <field> instead has <Krull dimension> zero.
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