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

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Left modules over a fixed ring and module homomorphisms form the large locally small category R-ModR\text{-}\mathbf{Mod}

Statement

For a fixed ring RR, left RR-modules and module homomorphisms form a large locally small category R-ModR\text{-}\mathbf{Mod}.

Facts & Assumptions

Given: Left RR-modules M,N,PM,N,P and module homomorphisms f:M→Nf:M\to N, g:N→Pg: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→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 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
1.1

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

givenL1
2.1

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

step 1.1L2
3.1

For every ordinal α\alpha, the singleton {α}\{\alpha\} carries a transported zero RR-module structure, producing a proper class of distinct objects by [L2]; hence R-ModR\text{-}\mathbf{Mod} 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: 41 results over 19 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