DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-27

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The radical of an ideal

Definition

Let $R$ be a commutative ring and let $I ⊴ R$ be an ideal. The radical of $I$ is

\sqrt{I} = {x \in R : x^{n} \in I for some integer n \geq 1} .

The ideal $I$ is radical when $I = \sqrt{I}$ .

Depends on

Used by

The nilradical and reduced rings Definition
A separating prime for an element outside a radical Lemma
Primes containing an ideal contain its radical Lemma
Radical membership via positive powers Lemma
Radicals and quotient correspondence Lemma
Radicals commute with localization Lemma
The radical of an ideal is an ideal 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)