Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01
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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.
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Krull's principal ideal theorem

Statement

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: 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]

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
1.1

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.

L1givenchoose
2.1

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

L2step 1.1
3.1

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

L1step 2.1∎

Depends on

Used by

Dependency tree · two levels

16 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