LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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.

A separating prime for an element outside a radical

Statement

Assume the Axiom of Choice.

Let $R$ be a commutative ring, let $I ⊴ R$ be an ideal, and let $f \in R$ . If $f \notin \sqrt{I}$ , then there exists a prime ideal $p$ of $R$ such that $I \subseteq p$ and $f \notin p$ .

Facts & Assumptions

Given: A commutative ring $R$ , an ideal $I ⊴ R$ , an element $f \notin \sqrt{I}$ , and the Axiom of Choice.

[L1]

An element belongs to $\sqrt{I}$ exactly when one of its positive powers lies in $I$ (The radical of an ideal).

[L2]

If every positive power of $f$ avoids $I$ , then some prime ideal contains $I$ but avoids $f$ (Separating an element from an ideal by a prime).

Proof

technique · direct

1.1L1given

Since $f \notin \sqrt{I}$ , [L1] says that $f^{n} \notin I$ for every integer $n \geq 1$ .

2.1L2step 1.1

Applying [L2] to step 1.1 yields a prime ideal $p$ with $I \subseteq p$ and $f \notin p$ .

3.1step 2.1∎

This prime separates $f$ from the radical of $I$ .

Depends on

Used by

The radical of an ideal is the intersection of the prime ideals containing it Theorem

Dependency tree · two levels

5 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

M. Hochster, Introduction to Commutative Algebra, Math 614 notes (2020) (standard reference, not scraped)
A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., §2 Ideals (standard reference, not scraped)