CorollaryStatement: Literature-sourcedProof: AI-adaptedverified 2026-09-24 (gpt-6-sol)

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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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Height is bounded by the minimal number of local generators

Statement

Assume the Axiom of Choice. Let $R$ be a Noetherian commutative ring and let $p \in Spec (R)$ . Then $ht (p) \leq μ_{R_{p}} (p R_{p}),$ where $μ$ denotes the minimal number of generators.

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice), a Noetherian commutative ring $R$ , and a prime ideal $p$ .

[L1]

The localization $R_{p}$ is a Noetherian local ring with maximal ideal $p R_{p}$ (Every quotient and every localisation of a Noetherian ring is Noetherian, $R_{p}$ is local with unique maximal ideal $p R_{p}$ ).

[L2]

Under Choice, a prime minimal over an ideal generated by $n \geq 1$ elements has height at most $n$ (Krull's height theorem).

[L3]

The height of $p$ is the dimension of $R_{p}$ (The height of a prime ideal).

Proof

technique · direct

1.1L1L2given

By [L1], the maximal ideal of the local ring $R_{p}$ is $p R_{p}$ . Let $n$ be its minimal number of generators, finite by Noetherianity. If $n = 0$ , the maximal ideal is zero and the local ring is a field, so its dimension is zero. If $n \geq 1$ , the maximal ideal is minimal over the ideal generated by those same $n$ elements. Applying the Choice-qualified [L2] inside $R_{p}$ gives $\dim R_{p} \leq n$ .

2.1L3step 1.1

By [L3], $ht (p) = \dim R_{p}$ , so step 1.1 says $ht (p) \leq μ_{R_{p}} (p R_{p})$ .

3.1step 2.1∎

This is the claimed local-generator bound on height.

Depends on

Used by

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Dependency tree · two levels

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Sources

J. S. Milne, A Primer of Commutative Algebra, v4.03, §21 (standard reference, not scraped)
Melvin Hochster, Dimension theory and systems of parameters (standard reference, not scraped)