Page "Zariski topology" Paragraph 29

Another way, perhaps more similar to the original, to interpret the modern definition is to realize that the elements of A can actually be thought of as functions on the prime ideals of A ; namely, as functions on Spec A.

Simply, any prime ideal P has a corresponding residue field, which is the field of fractions of the quotient A / P, and any element of A has a reflection in this residue field.

Furthermore, the elements that are actually in P are precisely those whose reflection vanishes at P. So if we think of the map, associated to any element a of A: