DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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.

Divisible modules over an integral domain

Let $R$ be an integral domain and $D$ a left $R$ -module (Unital left and right modules over a ring; unqualified module means left module, Zero divisor, and integral domain: a commutative ring with $1 \neq 0$ and no zero divisors). The module $D$ is divisible if for every $0 \neq r \in R$ and every $d \in D$ , there exists $x \in D$ with $r x = d$ . Equivalently, multiplication by every nonzero $r$ is surjective on $D$ .

The zero module is divisible. For $R = Z$ , divisible $R$ -modules are precisely divisible abelian groups.

Depends on