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

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Nilpotent Jacobson pairs are Henselian

Statement

Let $A$ be a commutative ring and let $I \subseteq J (A)$ be a nilpotent ideal. Then $(A, I)$ is a Henselian pair.

Facts & Assumptions

Given: A commutative ring $A$ and a nilpotent ideal $I \subseteq J (A)$ .

[L1]

Every complete separated adic pair is Henselian (Complete separated adic pairs are Henselian).

Proof

technique · nilpotence makes the adic tower stationary

1.1

Choose $N \geq 1$ with $I^{N} = 0$ . Then for every $r \geq N$ , the quotients $A / I^{r}$ stabilize at $A$ , so $A$ is automatically complete and separated for the $I$ -adic topology.

givenchoosealgebra

2.1

Applying [L1] to the ideal $I$ shows that $(A, I)$ is Henselian.

L1step 1.1∎

Depends on

Complete separated adic pairs are Henselian

Used by

Artinian local rings are Henselian Corollary

Dependency tree · two levels

9 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 15.11: Henselian pairs (standard reference, not scraped)