Let I an ideal of a conmutative ring A. We define the radical of I

$\displaystyle \sqrt{I}=\{ x \in A | \exists n such.that. x^n \in I \}$

A) Prove that the radical of I is an ideal

B) Describe the radical of some ideal of $\displaystyle \mathbb{Z}$

C) If $\displaystyle A=K[X_1, \dots, X_n]$ prove that $\displaystyle \sqrt{I}\subset \mathcal{I}(\mathcal{V}(I))$