Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-28
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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.

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Module categories have enough projectives

Statement

Assume the Axiom of Choice. For every ring R, the abelian category R-Mod has enough projectives.

Facts & Assumptions

Given: A ring R.

[L1]

Every left R-module is a quotient of a free left R-module (Every module is a quotient of a free module).

[L2]

Under the Axiom of Choice, free modules are projective (Equivalent characterizations of projective modules).

[L3]

Having enough projectives means admitting a projective epimorphism onto every object (A category with enough projectives and with enough injectives).

Proof

technique · direct
1.1L1

Let M be a left R-module. By [L1], the canonical free module R(M) admits a surjection R(M)↠M.

2.1L2step 1.1

Under the Axiom of Choice, [L2] makes R(M) projective. So M admits a projective epimorphism from step 1.1.

3.1L3step 2.1∎

Since M was arbitrary, [L3] shows that R-Mod has enough projectives.

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