False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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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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FALSE: elementary divisors determine the free rank

Statement

False claim. The elementary divisors of a finitely generated module over a PID determine its free rank.

Facts & Assumptions

Given: Free rank (The free rank of a finitely generated module over a PID) and the convention that elementary divisors record only torsion cyclic summands (Invariant factors and elementary divisors of a finitely generated module over a PID).

[L1]

Every nonzero commutative unital ring has invariant basis number for finite bases: if $R^{m} ≅ R^{n}$ , then $m = n$ (Every nonzero commutative ring has invariant basis number for finite bases).

Refutation

technique · direct

1.1

Let $R$ be any PID. The free modules $R$ and $R^{2}$ are torsion-free, so both have empty elementary-divisor lists. The zero module $R^{0}$ has the same empty torsion data as well.

givenalgebra

2.1

By [L1], their free ranks are respectively $1$ , $2$ , and $0$ and are genuinely distinct. Thus identical elementary-divisor data does not determine the free rank.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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

K. Conrad, Modules over a PID, structure-theorem rank data (standard reference, not scraped)
M. Brussel, Finitely Generated Modules over a PID, structure theorem (standard reference, not scraped)