Definition:Krull Dimension of Ring

Definition
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 the subset relation:

where:
 * $\map {\mathrm {ht} } {\mathfrak p}$ is the height of $\mathfrak p$
 * $\Spec R$ is the prime spectrum of $R$

In particular, 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 $\dim$, if there is no confusion.

Also see

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