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# nilradical

Let $R$ be a commutative ring. An element $x\in R$ is said to be nilpotent if $x^{n}=0$ for some positive integer $n$. The set of all nilpotent elements of $R$ is an ideal of $R$, called the nilradical of $R$ and denoted $\operatorname{Nil}(R)$. The nilradical is so named because it is the radical of the zero ideal.

The nilradical of $R$ equals the prime radical of $R$, although proving that the two are equivalent requires the axiom of choice.

Defines:

nilpotent

Related:

PrimeRadical, JacobsonRadical

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

13A10*no label found*

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