DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 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 $a\mid b$ , if $b=ac$ for some $c\in R$ .

Elements $a,b\in R$ are associates if $a=ub$ 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

Irreducible and prime elements of an integral domain Definition
Unique factorisation domain Definition

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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