$I(V_1)+I(V_2) \neq I(V_1 \cap V_2)$?

Question

Let $V_1,V_2 \subset \mathbb{A}^n(k)$ affine varieties ($k$ field). I've proved $I(V_1)+I(V_2) \subset I(V_1 \cap V_2)$, but I don't know how to prove $\supset$. I think that's maybe because that inclusion it's false. Could someone help me to prove it, or to give a counterexample if it's false, please? Thanks

Answer 1

In the classical language this is indeed false: $I_1 + I_2$ cuts out the right variety, but may not be radical. Here's a standard example: intersect the parabola $y=x^2$ and the line $y=0$. The ring is rightfully trying to keep track of the tangency, and scheme-theoretically it would be allowed to do so!