Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck 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∞)≤Q/Z is Artinian but not Noetherian as a Z-module. (The Prüfer p-group is Artinian but not Noetherian).

Refutation

technique · direct
1.1

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.

L1givenalgebra
2.1

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

step 1.1L1givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 18 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources