Let $R$ be the polynomial ring of $n$ variables over $\mathbb C$. It is known that a radical ideal $I (\ne R)$ defines a non-empty set $\mathbf V(I) \subset \mathbb C^n$.
I am looking for a counterexample. Can we have a non-radical ideal $I (\ne R)$ such that $\mathbf V(I)=\varnothing$?