Definition:Nilradical of Ring

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:

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

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

Generalizations

• Definition:Radical of Ideal of Ring