Definition:Krull Dimension

Definition

Krull Dimension of a 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 the subset relation:

\(\ds \map {\operatorname {dim_{Krull} } } R\)

\(=\)

\(\ds \sup \set {\map {\mathrm {ht} } {\mathfrak p} : \mathfrak p \in \Spec R}\)

\(\ds \)

\(=\)

\(\ds \sup \set {n \in \N: \exists p_0, \ldots, p_n \in \Spec R: \mathfrak p_0 \subsetneqq \mathfrak p_1 \subsetneqq \cdots \subsetneqq \mathfrak p_n}\)

where:

$\map {\mathrm {ht} } {\mathfrak p}$ 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.

Krull Dimension of a Topological Space

Let $T$ be a topological space.

Its Krull dimension $\map {\dim_{\mathrm {Krull} } } T$ is the supremum of lengths of chains of closed irreducible sets of $T$, ordered by the subset relation.

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

Source of Name

This entry was named for Wolfgang Krull.