DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-27

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Krull dimension of a nonzero ring

Definition

Let $R$ be a nonzero commutative ring. A strict chain of prime ideals of length $n$ is a sequence

p_{0} ⊊ p_{1} ⊊ \dots ⊊ p_{n}

of prime ideals of $R$ .

The Krull dimension of $R$ is the supremum of all integers $n \geq 0$ for which such a chain exists. This supremum is allowed to be infinite.

On this page the zero ring is left outside the definition so that later chain statements do not hide that degenerate boundary.

Depends on

Prime ideals and maximal ideals in a commutative ring

Used by

Dimension of a quotient via chains above an ideal Corollary
The height of a prime ideal Definition
Height equals local dimension Lemma

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, A Primer of Commutative Algebra, v4.03, Definition 3.14 (standard reference, not scraped)
The Stacks Project, Section 10.60: Dimension of rings (standard reference, not scraped)