Let A = $k[x_1, x_2, \ldots, x_n]$ and let $I_{\lambda}$ be an ideal of A.
Let J = $\sum_{\lambda \in \Lambda} I_{\lambda}$ be a finite sum. Show that $V(J) = \cap_{\lambda \in \Lambda} V(I_{\lambda})$.
It seems quite obvious that $V(\sum I_{\lambda}) \subseteq \cap_{\lambda \in \Lambda} V(I_{\lambda}) $.
However, I am not at all sure where to start for showing the other inclusion. Ideas?