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CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04
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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.
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Quotients of Henselian local rings are Henselian

Statement

Let (A,m) be a Henselian local ring and let J⊊A be a proper ideal. Then A/J is a Henselian local ring.

Facts & Assumptions

Given: A Henselian local ring (A,m) and a proper ideal J⊊A.

[L1]

Quotient pairs inherit the coprime monic factor-lifting property (Henselian factor lifting descends to quotients).

Proof

technique · identify the maximal ideal of the quotient and apply the quotient-pair lemma
1.1

The quotient A/J is local with maximal ideal (m+J)/J.

givenalgebra
1.2

Since (A,m) is Henselian, the pair (A,m) is Henselian. Applying [L1] with I=m shows that the quotient pair (A/J,(m+J)/J) has the Henselian factor-lifting property.

L1given
2.1

Together with step 1.1, this is exactly the definition of a Henselian local ring for A/J.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources