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

Dependency tree · 2 levels

Sources

Sharifi, Abstract Algebra, Advanced Ring Theory