DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

Krull dimension of a nonzero ring

Definition

Depends on

Dependency tree · 2 levels

Sources