# Definition:Krull Dimension of Ring

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## 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 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.

## 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

## Source of Name

This entry was named for Wolfgang Krull.