Affine -variety is a locus of zeros of a set of polynomials in the affine -dimensional space . Usually is taken to be a field.
Given a field , an affine -variety is a maximal spectrum (= set of maximal ideals) of a finitely generated noetherian (commutative unital) -commutative algebra without nilpotents, equipped with the Zariski topology; the algebra can be recovered as the coordinate ring of the variety; this correspondence is an equivalence of categories, if the morphisms are properly defined.
The converse requires in addition some finiteness condition. (Ballico 08).