LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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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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Subgroups of free abelian groups are free

Statement

Assume the Axiom of Choice. Every subgroup of a free abelian group is free abelian.

Facts & Assumptions

Given: A free abelian group $F$ with a basis, a subgroup $H \leq F$ , and the Axiom of Choice. By the well-ordering theorem, index the basis as $(e_{α})_{α < κ}$ .

Proof

technique · direct

1.1

Put $F_{α} = ⟨ e_{β} : β < α ⟩$ and $H_{α} = H \cap F_{α}$ . At a successor stage, $H_{α + 1} / H_{α}$ is a subgroup of $F_{α + 1} / F_{α} ≅ Z$ , hence is free; the finite-free PID result A submodule of a free module of finite rank over a PID is free of no larger rank supplies the required one-generator extension.

givenconstruct

2.1

If $H_{α + 1} / H_{α}$ is nonzero, choose a lift $x_{α}$ of its positive generator. Since the quotient is free, the short exact sequence splits and a basis of $H_{α}$ extends by $x_{α}$ to one of $H_{α + 1}$ . At a limit stage $H_{λ} = ⋃_{α < λ} H_{α}$ , and the nested union of the previously chosen bases is a basis. Transfinite induction through the terminal stage $κ$ therefore gives a basis of $H_{κ} = H \cap F_{κ} = H$ . This also covers $κ = 0$ , when $H = 0$ has the empty basis.

step 1.1algebra∎

Depends on

Used by

The integers have global dimension one Theorem

Dependency tree · two levels

6 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

Charles A. Weibel, An Introduction to Homological Algebra, Chapters 3–4 (standard reference, not scraped)