ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge 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.

The standard basis and a universal map from $R^{3}$

Example

For a unital ring $R$ , the module $R^{3}$ has standard basis $e_{1} = (1, 0, 0), e_{2} = (0, 1, 0), e_{3} = (0, 0, 1) .$ Given $m_{1}, m_{2}, m_{3}$ in a left $R$ -module $M$ , the unique homomorphism sending $e_{i}$ to $m_{i}$ is $T (r_{1}, r_{2}, r_{3}) = r_{1} m_{1} + r_{2} m_{2} + r_{3} m_{3} .$

Facts & Assumptions

Given: A unital ring $R$ , a left $R$ -module $M$ , and elements $m_{1}, m_{2}, m_{3} \in M$ .

[F1]

$R^{3}$ is free on its three standard basis vectors, and every vector has a unique coordinate expression (The free module on a set and its standard basis).

[L1]

A map on a basis extends uniquely to a module homomorphism (Universal property of the free module on a set).

Verification

technique · direct

1.1

The displayed formula is additive and $R$ -linear by the module axioms, and it sends $e_{i}$ to $m_{i}$ .

givenF1algebra

2.1

If $S : R^{3} \to M$ is another homomorphism with $S (e_{i}) = m_{i}$ , then [F1] gives $(r_{1}, r_{2}, r_{3}) = r_{1} e_{1} + r_{2} e_{2} + r_{3} e_{3}$ , so linearity gives $S (r_{1}, r_{2}, r_{3}) = r_{1} m_{1} + r_{2} m_{2} + r_{3} m_{3} = T (r_{1}, r_{2}, r_{3})$ .

step 1.1F1

3.1

Therefore $T$ is the unique universal extension asserted by [L1].

step 1.1step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 10 results over 6 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

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)

The standard basis and a universal map from R3R3