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TheoremStatement: Literature-sourcedProof: AI-adaptedverified 2026-09-24 (gpt-6-sol)
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Krull's principal ideal theorem

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let R be a Noetherian commutative ring, let x∈R, and let p be a prime ideal minimal over (x). Then ht⁡(p)≤1.

Facts & Assumptions

Given: The Axiom of Choice, a Noetherian commutative ring R, an element x∈R, and a prime ideal p minimal over (x).

[L1]

The principal-ideal bound reduces to the case of a Noetherian local domain whose maximal ideal is minimal over one nonzero element (Reduce the principal ideal theorem to a Noetherian local domain).

[L2]

Under the assumed AC, in that reduced local-domain situation every prime properly below the maximal ideal is zero, so the maximal ideal has height at most 1 (The symbolic-power step inside the principal ideal theorem).

Proof

technique · direct
1.1L1givenchoose

By [L1], choose a minimal prime q⊆p and pass to the Noetherian local domain A=(R/q)p/q whose maximal ideal is minimal over the image of x.

2.1L2step 1.1

Fact [L2] applies to A, so its maximal ideal has height at most 1.

3.1L1step 2.1∎

The reduction packaged in [L1] identifies this with the desired bound ht⁡(p)≤1 upstairs in R.

Depends on

Dependency tree · 2 levels

Sources