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.

Left modules over a fixed ring and module homomorphisms form the large locally small category $R - M o d$

Statement

For a fixed ring $R$ , left $R$ -modules and module homomorphisms form a large locally small category $R - M o d$ .

Facts & Assumptions

Given: Left $R$ -modules $M, N, P$ and module homomorphisms $f : M \to N$ , $g : N \to P$ .

[L1]

Left modules satisfy the axioms in Unital left and right modules over a ring; unqualified module means left module, and module homomorphisms preserve addition and scalar multiplication (Module homomorphism and isomorphism, kernel, image and cokernel).

[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 notions 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

Identity maps are module homomorphisms, and $g \circ f$ preserves addition and scalar multiplication by the two homomorphism laws; function composition is associative and unital.

2.1step 1.1L2

Thus left $R$ -modules and their homomorphisms form a category whose hom-collections are sets.

3.1step 2.1L1L2∎

For every ordinal $α$ , the singleton ${α}$ carries a transported zero $R$ -module structure, producing a proper class of distinct objects by [L2]; hence $R - M o d$ is large and locally small.

Depends on

Dependency tree · 2 levels

Sources

Emily Riehl, Category Theory in Context, Chapter 1

Left modules over a fixed ring and module homomorphisms form the large locally small category R-ModR-Mod