The classical affine dictionary forgets nilpotents and non-rational points

For an algebraically closed field $k$, the classical affine dictionary keeps only reduced affine $k$-algebras and only their $k$-rational points.

By Affine algebraic sets and reduced affine k-algebras at the object level, an affine algebraic set recovers exactly its reduced coordinate ring. Nilpotent structure is invisible to the point set: if one replaces a ring by its reduction, the same classical algebraic set is obtained. By Points of an affine algebraic set correspond to maximal ideals of its coordinate ring, the visible points are precisely maximal ideals with residue field $k$, so closed points with larger residue fields and nonmaximal prime ideals do not appear in this register.

These are the two losses scheme theory repairs later: nilpotents return through nonreduced affine schemes, and non-rational or generic points return through the full prime spectrum.