\begin{definition}[Definition:Radical of Ideal of Ring/Definition 2]
Let $A$ be a commutative ring with unity.
Let $I$ be an ideal of $A$.
Let $A / I$ be the quotient ring.
Let $\Nil {A / I}$ be its nilradical.
Let $\pi: A \to A / I$ be the quotient mapping.
The '''radical''' of $I$ is the preimage of $\Nil {A / I}$ under $\pi$:
:$\map \Rad I = \pi^{-1} \sqbrk {\Nil {A / I} }$
\end{definition}