Definition:Krull Dimension of Ring

Definition
Let $\left({R, +, \circ}\right)$ be a commutative ring with unity.

The Krull dimension of $R$ is the supremum of lengths of chains of prime ideals, ordered by inclusion:
 * $\operatorname{dim_{Krull}} \left({R}\right) = \sup\{n\in\N : \exists p_0, \ldots, p_n \in \operatorname{Spec}(R) : \mathfrak p_0 \subsetneqq \mathfrak p_1 \subsetneqq \cdots \subsetneqq \mathfrak p_n\}$

where $\operatorname{Spec}(R)$ is the prime spectrum of $R$.

That is, the Krull dimension is $\infty$ if there exist arbitrarily long chains.

Also denoted as
The Krull dimension can also be denoted $\operatorname{K-dim}$ or simply $\dim$, of there is no confusion.

Also see

 * Krull's Theorem which proves the existence of a prime ideal