Hilbert's Nullstellensatz (strong)

Hilbert's Nullstellensatz (strong): Let $k$ be an algebraically closed field and let $I\triangleleft k[x_1,\dots,x_n]$ be an ideal in the polynomial ring. Let

Then

where $\sqrt{I}$ denotes the radical of an ideal.

This identifies geometric vanishing with algebraic nilpotence modulo $I$ and implies, for instance, that varieties correspond to radical ideals and irreducible varieties correspond to prime ideals.