ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

A ring viewed as a one-object preadditive category with its matrices

Example

For a ring $R$ , the one-object preadditive category of A one-object preadditive category is the same thing as a ring becomes, after adjoining finite biproducts, the matrix category ${M a t}_{R}$ . In particular, the object $2$ of ${M a t}_{R}$ is the biproduct of two copies of the unique object of the ring category.

Facts & Assumptions

Given: A ring $R$ .

[L1]

A one-object preadditive category is the same thing as a ring (A one-object preadditive category is the same thing as a ring).

[L2]

In ${M a t}_{R}$ , a morphism $n \to m$ is an $m \times n$ matrix (The matrix category over a ring).

Verification

technique · direct

1.1

By [L1], the unique object $*$ of the ring category has endomorphism ring $R$ . In ${M a t}_{R}$ , the object $1$ recovers that same one-object category, because endomorphisms $1 \to 1$ are $1 \times 1$ matrices, hence elements of $R$ , by [L2].

L1L2

2.1

The object $2$ has the standard column injections $1 \to 2$ and row projections $2 \to 1$ , namely $(\begin{array}{c} 1 \\ 0 \end{array})$ , $(\begin{array}{c} 0 \\ 1 \end{array})$ , $(\begin{array}{cc} 1 & 0 \end{array})$ , and $(\begin{array}{cc} 0 & 1 \end{array})$ . Their block identities are exactly the biproduct equations, so $2 ≅ 1 \oplus 1$ .

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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

Kiran S. Kedlaya, Solid modules over an ordinary ring, Example 1.2.2 (standard reference, not scraped)
Gabriele Lobbia, Wojciech Rozowski, Ralph Sarkis, and Fabio Zanasi, Quantitative Monoidal Algebra, Definition 25 (standard reference, not scraped)