# Definition:Krull Dimension

## Contents

## Definition

### Krull Dimension of a Ring

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.

### Krull Dimension of a Topological Space

Let $T$ be a topological space.

Its **Krull dimension** $\operatorname {dim_{Krull}} \left({T}\right)$ is the supremum of lengths of chains of closed irreducible sets of $T$, ordered by inclusion.

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

## Source of Name

This entry was named for Wolfgang Krull.