# Definition:Nilradical of Ring/Definition 2

## Definition

Let $A$ be a commutative ring with unity.

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 $A$.