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

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A module over a Noetherian ring has no associated primes exactly when it is zero

Statement

Let R be a Noetherian commutative ring and let M be a left R-module. Then Ass⁡R(M)=∅⟺M=0.

Facts & Assumptions

Given: A Noetherian commutative ring R and a left R-module M.

[L1]

Every nonzero R-module has an associated prime (A nonzero module over a Noetherian ring has an associated prime).

Proof

technique · direct
1.1

If M=0, then the only element of M is 0, whose annihilator is the whole ring R. Since R is not a prime ideal of itself, no associated prime can occur, so Ass⁡R(M)=∅.

givenalgebra
1.2

If M≠0, then [L1] gives Ass⁡R(M)≠∅.

L1given
2.1

Steps 1.1 and 1.2 prove the equivalence.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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