Suppose that $V$ is a nonempty affine algebraic set. If $k[V]$ is the direct sum of two non-zero ideals, prove that V is disconnected in the Zariski topology.
I am preparing for the midterm, and I stuck on this problem.
If $k[V]=\frac{k[A^n]}{I(V)}=A+B$ sum of ideals then how can I connect that with connectedness of set $V$?
Thank you!