Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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.

Unital rings and unit-preserving ring homomorphisms form the large locally small category Ring\mathbf{Ring}

Statement

Unital rings and unit-preserving ring homomorphisms form a large locally small category Ring\mathbf{Ring}.

Facts & Assumptions

Given: Unital rings R,S,TR,S,T and unit-preserving homomorphisms f:R→Sf:R\to S and g:S→Tg:S\to T.

[L1]

A ring is an additive abelian group and multiplicative monoid with both distributive laws (Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides); a ring homomorphism preserves addition, multiplication, zero, and one (Ring homomorphism: additive, multiplicative, and required to send 11 to 11).

[L2]

The functions A→BA\to B between fixed sets form the set BAB^A (The set BAB^{A} of all functions A→BA \to B); the category and size conditions are Category, object, morphism, domain, codomain, identity, composition, and hom-collection and Small, locally small, and large categories, and the ordinals form a proper class (Burali-Forti: there is no set of all ordinals).

Proof

technique · direct
1.1

The identity of a ring preserves both operations and the unit, and g∘fg\circ f preserves them because ff and gg do; function composition supplies associativity and identity equations.

givenL1
2.1

These data form a category, and every hom-collection is a set because it is a subset of the functions between the two underlying sets.

step 1.1L2
3.1

For each ordinal α\alpha, the one-element set {α}\{\alpha\} carries the zero-ring structure with 0=1=α0=1=\alpha; these are distinct objects, so [L2] shows that Ring\mathbf{Ring} is large and locally small.

step 2.1L1L2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 36 results over 16 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