False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

False statement: every Artinian module is Noetherian

Statement

False claim: every Artinian module is Noetherian. See The Prüfer $p$ -group is Artinian but not Noetherian.

Facts & Assumptions

Given: The hypotheses and objects in the false claim.

[L1]

For every prime $p$ , the Prüfer group $Z (p^{\infty}) \leq Q / Z$ is Artinian but not Noetherian as a $Z$ -module. (The Prüfer $p$ -group is Artinian but not Noetherian).

Refutation

technique · direct

1.1L1givenalgebra

We use the preceding Prüfer $p$ -group, quoting its complete subgroup classification to establish DCC and its strict ascending chain of finite cyclic subgroups to refute Noetherianity.

2.1step 1.1L1givenalgebra∎

One witness refutes a universally quantified claim, and $Z (p^{\infty})$ is one: it is Artinian and not Noetherian by [L1], so "every Artinian module is Noetherian" is false. This proves the stated claim.

Depends on

The Prüfer $p$-group is Artinian but not Noetherian

Used by

Nothing in the library uses this result yet.

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

Arvind Nair, Algebra I, Lecture 5 (standard reference, not scraped)