ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

The ring R is a generator of R-Mod

Example

For any ring $R$ and left $R$ -module $M$ , a homomorphism $R \to M$ is determined by the image of $1$ . So the canonical coproduct of one copy of $R$ for each element of $M$ maps onto $M$ , which is the concrete generator criterion.

Facts & Assumptions

Given: A ring $R$ and a left $R$ -module $M$ .

[L1]

The canonical coproduct map criterion characterizes generators in AB3 (The cancellation and epimorphism descriptions of a generator agree).

[L2]

Module categories are Grothendieck categories, hence in particular have such a generator (Module categories are Grothendieck categories).

Verification

technique · direct

1.1L1algebra

Every module homomorphism $u : R \to M$ is determined by $u (1)$ , and every element $m \in M$ defines a homomorphism $u_{m} (r) = r m$ . Therefore the canonical map $∐_{m \in M} R \to M$ that sends the $m$ -indexed basis vector to $m$ is surjective.

2.1L1L2step 1.1∎

By [L1], this surjectivity is exactly the generator property for $R$ , and [L2] records the same conclusion abstractly at the category level.

Depends on

Used by

Nothing in the library uses this result yet.

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

Alexandre Grothendieck, Sur quelques points d'algèbre homologique, Barr translation, Section 1.9 (standard reference, not scraped)