PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck 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 $R i n g$

Statement

Unital rings and unit-preserving ring homomorphisms form a large locally small category $R i n g$ .

Facts & Assumptions

Given: Unital rings $R, S, T$ and unit-preserving homomorphisms $f : R \to S$ and $g : S \to T$ .

[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 $1$ to $1$ ).

[L2]

The functions $A \to B$ between fixed sets form the set $B^{A}$ (The set $B^{A}$ of all functions $A \to B$ ); 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

1.1givenL1

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

2.1step 1.1L2

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.

3.1step 2.1L1L2∎

For each ordinal $α$ , the one-element set ${α}$ carries the zero-ring structure with $0 = 1 = α$ ; these are distinct objects, so [L2] shows that $R i n g$ is large and locally small.

Depends on

Dependency tree · 2 levels

Sources

Emily Riehl, Category Theory in Context, Chapter 1

Unital rings and unit-preserving ring homomorphisms form the large locally small category RingRing