Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28
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.

The zero module has empty support and no associated primes

Example

For the zero R-module, Ass⁡R(0)=∅andSupp⁡R(0)=∅. Also the primary-submodule definition does not apply to 0⊆0, because it requires a proper submodule.

Facts & Assumptions

Given: A commutative ring R and the zero R-module.

[L1]

Associated primes are prime annihilators of elements (Associated primes of a module).

[L2]

Support consists of the primes whose localization is nonzero (Support of a module).

[L3]

Primaryity is defined only for proper submodules (Primary submodules and primary ideals).

Verification

technique · direct
1.1

The only element of the zero module is 0, and its annihilator is the whole ring R, which is not a prime ideal of itself. Therefore [L1] gives Ass⁡R(0)=∅.

L1givenalgebra
1.2

Every localization of the zero module is again the zero module, so [L2] gives Supp⁡R(0)=∅.

L2given
1.3

The inclusion 0⊆0 is not proper, so [L3] shows that it is not a candidate for the primary-submodule definition. This is the exact boundary that prevents a vacuous use of primaryity at the zero module.

L3given
2.1

Thus the zero module sits outside associated-prime and primaryity claims in exactly the stated way.

step 1.1step 1.2step 1.3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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