In https://en.wikipedia.org/wiki/Zariski_topology it says that $V(I+J) = V(I) \cap V(J)$. Two questions:
1- Is my solution for this claim correct and rigorous: If $x$ is a zero of some $f \in I$ and some $g \in J$ then it is a zero of any linear combination of them over the ring $R$, so $V(I) \cap V(J) \subset V(I+J)$. If $x \in V(I+J)$ then $x \in V(I)$ and $\in V(J)$, since $I+J$ contains $I$ and $J$ as special cases?
2- How to generalize $V(I+J) = V(I) \cap V(J)$ to $V(\sum_i J_i) = \cap_i V(J_i)$, where sum and intersection is taken over an arbitrary set of indices?