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

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

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

Also see

 * Equivalence of Definitions of Radical of Ideal of Ring