Page "Zariski topology" Paragraph 33

More generally, V ( I ) for any ideal I is the common set on which all the " functions " in I vanish, which is formally similar to the classical definition.

In fact, they agree in the sense that when A is the ring of polynomials over some algebraically closed field k, the maximal ideals of A are ( as discussed in the previous paragraph ) identified with n-tuples of elements of k, their residue fields are just k, and the " evaluation " maps are actually evaluation of polynomials at the corresponding n-tuples.

Since as shown above, the classical definition is essentially the modern definition with only maximal ideals considered, this shows that the interpretation of the modern definition as " zero sets of functions " agrees with the classical definition where they both make sense.