Definition:Radical of Ideal of Ring/Definition 2

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

Let $I$ be an ideal of $A$. Let $A/I$ be the quotient ring.

Let $\operatorname{Nil}(A/I)$ be its nilradical.

Let $\pi : A \to A/I$ be the quotient mapping.

The radical of $I$ is the preimage of $\operatorname{Nil}(A/I)$ under $\pi$:
 * $\operatorname{rad}(I) = \pi^{-1}(\operatorname{Nil}(A/I))$

Also see

 * Equivalence of Definitions of Radical of Ideal of Ring