# Definition:Nilradical of Ring

## Contents

## Definition

Let $A$ be a commutative ring with unity.

### Definition 1

The **nilradical** of $A$ is the subset consisting of all nilpotent elements of $A$.

### Definition 2

Let $\Spec A$ denote the prime spectrum of $A$.

The **nilradical** of $A$ is:

- $\displaystyle \Nil A = \bigcap_{\mathfrak p \mathop \in \Spec A} \mathfrak p$

That is, it is the intersection of all prime ideals of $A$.

## Also denoted as

Some sources use $N \left({A}\right)$ or $\mathfrak N$.

## Also see

- Equivalence of Definitions of Nilradical of Ring (which uses the axiom of choice)
- Definition:Jacobson Radical