TheoremStatement: 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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A submodule of an arbitrary-rank free module over a PID is free

Statement

Assume the Axiom of Choice. If $R$ is a PID, $F$ is a free $R$ -module, and $N \subseteq F$ , then $N$ is free (with no finite-rank assumption on $F$ ).

Proof

Given: a basis of $F$ , a submodule $N \subseteq F$ , and the Axiom of Choice. By the well-ordering theorem, index the basis as $(e_{α})_{α < κ}$ .

1.1

Put $F_{α} = ⟨ e_{β} : β < α ⟩$ and $N_{α} = N \cap F_{α}$ . The image of $N_{α + 1}$ in $F_{α + 1} / F_{α} ≅ R$ is an ideal $I_{α}$ of $R$ , hence is either zero or free of rank one. Thus $0 \to N_{α} \to N_{α + 1} \to I_{α} \to 0$ splits.

givenalgebra

2.1

At each successor with $I_{α} \neq 0$ , choose a generator and a lift $x_{α} \in N_{α + 1}$ ; the splitting gives $N_{α + 1} = N_{α} \oplus R x_{α}$ . At a limit $λ$ , every element has finite support, so $N_{λ} = ⋃_{α < λ} N_{α}$ and the nested union of the earlier bases is a basis. Transfinite induction through the terminal stage $κ$ therefore gives a basis of $N_{κ} = N$ . For $κ = 0$ , this is the empty basis of $N = 0$ .

step 1.1construct∎

Depends on

Used by

Boundaries and cycles in a free complex over a PID are free Lemma

Dependency tree · two levels

17 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

Weibel, An Introduction to Homological Algebra (standard reference, not scraped)