Definition:Krull Dimension
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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.