TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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

Statement

Assume the Axiom of Choice.

Let $R$ be a commutative ring and let $I ⊴ R$ be an ideal. Then

\sqrt{I} = ⋂_{\begin{array}{c} p \in Spec R \\ I \subseteq p \end{array}} p,

where the intersection is taken to be $R$ if no prime ideal contains $I$ .

Facts & Assumptions

Given: A commutative ring $R$ , an ideal $I ⊴ R$ , and the Axiom of Choice.

[L1]

Every prime ideal containing $I$ also contains $\sqrt{I}$ (Primes containing an ideal contain its radical).

[L2]

Every element outside $\sqrt{I}$ is omitted by some prime ideal containing $I$ (A separating prime for an element outside a radical).

Proof

technique · direct

1.1

Let $x \in \sqrt{I}$ . By [L1], every prime ideal containing $I$ also contains $x$ . Therefore $x$ belongs to the displayed intersection.

L1given

1.2

Let $x \notin \sqrt{I}$ . By [L2], there is a prime ideal $p$ containing $I$ with $x \notin p$ . Hence $x$ does not belong to the displayed intersection.

L2given

2.1

Steps 1.1 and 1.2 prove that an element belongs to $\sqrt{I}$ exactly when it belongs to every prime ideal containing $I$ , which is the claimed equality.

step 1.1step 1.2∎

Depends on

Used by

The nilradical is the intersection of all prime ideals Corollary
Computing sqrt((x²,xy)) from its containing primes Example
Vanishing sets detect radicals Lemma

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

J. S. Milne, A Primer of Commutative Algebra, v4.03, §2 Ideals (standard reference, not scraped)
A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., §2 Ideals (standard reference, not scraped)