Equality of ideals and their vareties.

Question

Let $I_1 $,$I_2 $ $\in \mathbb{C}[x_1,x_2,...,x_n] $ be two polynomial ideals.

If their affine varieties, $\mathbb{V}(I_1)=\mathbb{V}(I_2)$ are equal then is $I_1=I_2$ always?

Answer 1

As pointed out in the comments, this is false in general. However, $\sqrt{I_1} = \sqrt{I_2}$ by Hilbert's Nullstellensatz.