Definition:Nilradical of Ring

Definition
Let $A$ be a commutative ring with unity.

Let $\operatorname{spec} \left({A}\right)$ be the set of prime ideals 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 $A$.

Alternative notation
Some sources use $N \left({A}\right)$.