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.
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.

Divisibility and associates in an integral domain

Definition

Let $R$ be an integral domain. For $a, b \in R$ , say that $a$ divides $b$ , written $a ∣ b$ , if $b = a c$ for some $c \in R$ .

Elements $a, b \in R$ are associates if $a = u b$ for a unit $u \in R^{\times}$ . Thus associates differ only by multiplication by an invertible factor. This relation is used to state uniqueness of factorisations without artificially choosing representatives of unit classes.

Depends on

Used by

Invariant factors and elementary divisors of a finitely generated module over a PID Definition
Irreducible and prime elements of an integral domain Definition
Unique factorisation domain Definition
Normalization of the cusp semigroup ring Example
Finite-variable polynomial algebras over fields are integrally closed Lemma
Gauss lemma over a UFD Lemma
Irreducible holomorphic germs are prime Lemma
Noetherian domains are atomic Lemma
Under AC, Cartier and Weil divisors agree on a locally factorial Noetherian integral scheme Theorem

Dependency tree · two levels

8 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)