ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-27

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.

Additive functors out of the matrix category

Example

An additive functor $F : {M a t}_{R} \to A b$ is determined by the left $R$ -module $M : = F (1)$ , and then $F (n) ≅ M^{\oplus n}$ for every $n$ .

Facts & Assumptions

Given: An additive functor $F : {M a t}_{R} \to A b$ .

[L1]

Additive functors out of the one-object ring category are exactly left $R$ -modules (Additive functors from a ring to abelian groups are left modules).

[L2]

The matrix category is equivalent to the finitely generated free modules (The matrix category is fully faithful in modules and, with chosen bases, equivalent to finite free modules).

[L3]

Between additive categories, additivity is equivalent to preserving finite biproducts (A functor between additive categories is additive exactly when it preserves finite biproducts).

Verification

technique · direct

1.1

Restrict $F$ to the full one-object subcategory on $1 \in {M a t}_{R}$ . By [L1], this restriction is the same thing as a left $R$ -module, namely $M : = F (1)$ .

2.1

The object $n$ of ${M a t}_{R}$ is the $n$ -fold biproduct of $1$ . Since $F$ is additive, [L3] says it preserves those finite biproducts, so $F (n) ≅ M^{\oplus n}$ . This is the matrix-category form of the finite-free-module equivalence in [L2].

L2L3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

Kiran S. Kedlaya, Solid modules over an ordinary ring, Example 1.2.2 (standard reference, not scraped)
Kiran S. Kedlaya, Solid modules over an ordinary ring, Example 1.2.6 (standard reference, not scraped)