CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Artinian local rings are Henselian

Statement

Assume the Axiom of Choice.

Every Artinian local ring is Henselian.

Facts & Assumptions

Given: A commutative Artinian local ring $(A, m)$ and the Axiom of Choice.

[L1]

In an Artinian local ring, the maximal ideal is nilpotent (An Artinian local ring has nilpotent maximal ideal, and its finite modules have finite length).

[L2]

A nilpotent ideal contained in the Jacobson radical gives a Henselian pair (Nilpotent Jacobson pairs are Henselian).

Proof

technique · use the nilpotent maximal ideal

1.1

By [L1], there exists $N \geq 1$ with $m^{N} = 0$ . Since $m$ is the maximal ideal of a local ring, it lies in the Jacobson radical.

L1given

2.1

Therefore [L2] applies to the pair $(A, m)$ , so $(A, m)$ is Henselian. Equivalently, the Artinian local ring $A$ is Henselian.

L2step 1.1∎

Depends on

Used by

A truncated polynomial local ring is Henselian Example

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

The Stacks Project, Section 10.153: Henselian local rings (standard reference, not scraped)