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ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01
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In a Noetherian local domain, the intersection of the powers of the maximal ideal is zero

Example

Assume the Axiom of Choice.

Let (R,m) be a Noetherian local domain. Then

⋂n≥0mn=0.

Facts & Assumptions

Given: The Axiom of Choice and a Noetherian local domain (R,m).

[L1]

The Krull intersection theorem says that for a finite module M,

⋂n≥0mnM=0

when m⊆J(R) (The Krull intersection is the (1−a)-torsion submodule, and it vanishes in the Jacobson-radical case).

Verification

technique · direct
1.1

View R as a finite module over itself. Since (R,m) is local, its maximal ideal lies in the Jacobson radical. Therefore [L1] applies with M=R and gives ⋂n≥0mn=0.

L1given
2.1

This is the promised local-domain instance of Krull intersection.

algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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