DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-03

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.
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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.

Unique factorisation domain

Definition

An integral domain $R$ is a unique factorisation domain (UFD) when every nonzero nonunit is a finite product of irreducible elements, and whenever

$a = p_{1} \dots p_{m} = q_{1} \dots q_{n}$

are products of irreducibles, then $m = n$ and, after a permutation, each $p_{i}$ is associate to the corresponding $q_{i}$ . Thus uniqueness is only up to order and multiplication of factors by units.

Depends on

Used by

Gauss lemma over a UFD Lemma
Every principal ideal domain is a unique factorisation domain Theorem
For every field F, F[x] is a unique factorisation domain Theorem
The ring of holomorphic germs is a UFD Theorem

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

Sharifi, Abstract Algebra, Advanced Ring Theory (standard reference, not scraped)