\begin{definition}[Definition:Krull Dimension of Ring]
Let $\struct {R, +, \circ}$ be a commutative ring with unity.
The '''Krull dimension''' of $R$ is the supremum of lengths of chains of prime ideals, ordered by inclusion:
:$\map {\operatorname {dim_{Krull} } } R = \sup \set {n \in \N: \exists p_0, \ldots, p_n \in \Spec R: \mathfrak p_0 \subsetneqq \mathfrak p_1 \subsetneqq \cdots \subsetneqq \mathfrak p_n}$
where $\Spec R$ is the prime spectrum of $R$.
That is, the '''Krull dimension''' is $\infty$ if there exist arbitrarily long chains.
\end{definition}