I'm trying to prove that, if $V$ is an affine algebraic set, then $V$ is connected in the Zariski topology iff $k[V]$ is not the direct sum of two ideals. Note that $k$ is algebraically closed here.
So, first I started by assuming that $k[V]=I_1\oplus I_2$. Then, using the Chinese Remainder Theorem and basic properties of the correspondence between ideals and algebraic sets, I was able to write $V=\mathcal Z(I_1)\sqcup\mathcal Z(I_2)$, but of course, to show that the set is not connected, I need to write $V$ as a disjoint union of OPEN sets, not closed sets. I tried showing that they were open by doing some tricks with their complements (which are NOT necessarily each other, since $V$ may not be the whole space).
How do I finish this argument? Am I on the right track?