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 $A$.

Also see

 * Intersection of Ring Ideals is Ideal
 * Equivalence of Definitions of Nilradical of Ring