Definition:Nilradical of Ring/Definition 2

Definition
Let $A$ be a commutative ring with unity. Let $\operatorname{Spec} \left({A}\right)$ be the prime spectrum of $A$.

Then the Nilradical of $A$ is:
 * $\displaystyle \operatorname{Nil} \left({A}\right) = \bigcap_{\mathfrak p \in  \operatorname{Spec} \left({A}\right)}\mathfrak p$

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

Also see

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