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:

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

\(=\)

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

\(\ds \)

\(=\)

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

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.

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.

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