TheoremStatement: Literature-sourcedProof: 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.

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

Statement

To give a one-object preadditive category is exactly to give a ring. More precisely, if $C$ is preadditive with one object $*$ , then ${End}_{C} (*)$ is a ring under hom-group addition and composition as multiplication. Conversely, every ring $R$ defines a one-object preadditive category with endomorphism ring $R$ . Under this correspondence, additive functors are exactly unital ring homomorphisms.

Facts & Assumptions

Given: A preadditive category with one object $*$ , or a ring $R$ .

[L1]

In a preadditive category every hom-set is an abelian group and composition is bilinear (Preadditive category).

[L2]

A ring is an abelian group under addition, a monoid under multiplication, and a ring homomorphism preserves addition, multiplication, and $1$ (Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides, Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$ ).

[F1]

A functor preserves identities and composition (Covariant functor, identity functor, composite functor, and contravariant functor).

Proof

technique · direct

1.1

Suppose $C$ has one object $*$ . Put $R : = {End}_{C} (*)$ . By [L1], $R$ is an abelian group under its hom-group addition. The category identity $1_{*}$ is a multiplicative identity for composition, composition is associative, and the two displayed bilinearity laws in [L1] are exactly the distributive laws. Hence $R$ is a ring.

L1L2

1.2

Conversely, let $R$ be a ring. Form a category with one object $*$ and $End (*) = R$ , taking composition to be ring multiplication. The ring multiplication is associative with identity $1_{R}$ , and [L2] gives bilinearity with respect to the additive group law. So this is a preadditive category.

2.1

If $F$ is an additive functor between one-object preadditive categories, then its action on the unique hom-set preserves addition because $F$ is additive, preserves multiplication because functors preserve composition, and preserves $1$ because functors preserve identities by [F1]. Thus $F$ is a unital ring homomorphism by [L2]. The converse is the same verification in the opposite direction. Steps 1.1 and 1.2 are inverse constructions, so the two notions are the same data.

L2F1step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

14 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)
Mike Prest, Modules as exact functors, Rings as categories (standard reference, not scraped)