LemmaStatement: 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.
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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The defining ideal of a Henselian pair lies in the Jacobson radical

Statement

If $(A, I)$ is a Henselian pair, then $I \subseteq J (A)$ .

Facts & Assumptions

Given: A Henselian pair $(A, I)$ .

[L1]

A Henselian pair is defined by the Jacobson-radical condition together with unique lifting of coprime monic factorizations (Henselian pairs and Henselian local rings).

Proof

technique · unpack the definition

1.1

By [L1], one clause in the definition of a Henselian pair is exactly the containment $I \subseteq J (A)$ .

L1given

2.1

Therefore the defining ideal of a Henselian pair lies in the Jacobson radical.

step 1.1∎

Depends on

Henselian pairs and Henselian local rings

Used by

Lifted coprime factorisations are unique Proposition

Dependency tree · two levels

2 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)