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 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.

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.