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 of $A$ by $I$.

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

Let $\pi: A \to A / I$ be the quotient epimorphism from $A$ onto $A / I$.

The radical of $I$ is the preimage of $\Nil {A / I}$ under $\pi$:
 * $\map \Rad I = \pi^{-1} \sqbrk {\Nil {A / I} }$

Also see

 * Equivalence of Definitions of Radical of Ideal of Ring