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

Definition

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

Used by

Dependency tree · two levels

6 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

A. Kleshchev, Lectures on Abstract Algebra for Graduate Students, sections 3.6, 3.14, and 3.15 (standard reference, not scraped)
The Stacks Project, Algebra (standard reference, not scraped)
P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)