Sum of ideals-Intersection of algebraic sets

Question

In my lecture notes I have the following:

$$ \begin{array}{ccl} \text{Sum of ideals} & & \text{Intersection of algebraic sets} \\[4pt] I+J & \longrightarrow & V(I+J)=V(I)\cap V(J) \\ \operatorname{Rad}(I(V)+I(W)) & \longleftarrow & V\cap W \end{array} $$

I understand the first one, that the sum of two ideals is the intersection of the algebraic sets of these two ideals.

But I don't understand the second one. Can you explain it to me?

Answer 1

Hint: $V \cap W$ is the largest algebraic set contained in both $V$ and $W.$ Hence $I(V \cap W)$ is the smallest radical ideal containing both $I(V)$ and $I(W).$ Thus $I(V \cap W)= \text{Rad}(I(V)+I(W)).$ (The correspondence between largest algebraic set and smallest radical ideal follows from HILBERT NULLSTELLENSATZ)