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

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The nilradical and reduced rings

Definition

Let $R$ be a commutative ring.

The nilradical of $R$ is the radical of the zero ideal,

Nil (R) = \sqrt{(0)} .

Thus $x \in Nil (R)$ exactly when $x^{n} = 0$ for some integer $n \geq 1$ .

The ring $R$ is reduced when $Nil (R) = (0)$ , equivalently when the only nilpotent element of $R$ is $0$ .

Depends on

The radical of an ideal

Used by

A ring is reduced exactly when zero is an intersection of primes Corollary
Passing to the reduced quotient does not change the prime spectrum Corollary
The nilradical is the intersection of all prime ideals Corollary
The reduced quotient by the nilradical Corollary
A non-Noetherian nilradical need not be nilpotent Example
A Noetherian ring has finitely many minimal prime ideals Theorem
The nilradical of a Noetherian ring is nilpotent Theorem

Dependency tree · two levels

3 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)