CorollaryStatement: Literature-sourcedProof: Literature-sourcedprecheck passaudited 2026-09-01

How statement and proof provenance work

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Height is bounded by the minimal number of local generators

Statement

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: 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]

A prime minimal over an ideal generated by $n$ 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

1.1

By [L1], the maximal ideal of the local ring $R_{p}$ is $p R_{p}$ . If it is minimally generated by $n$ elements, then it is certainly minimal over the ideal generated by those same $n$ elements. Applying [L2] inside $R_{p}$ gives $\dim R_{p} \leq n$ .

L1L2given

2.1

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

L3step 1.1

3.1

This is the claimed local-generator bound on height.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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

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)