Definition:Krull Dimension

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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.