Affine$k$-variety is a locus of zeros of a set of polynomials in the affine $n$-dimensional space $\mathbf{A}^n_k$. Usually $k$ is taken to be a field.

Definition

Given a field $k$, an affine $k$-variety is a maximal spectrum (= set of maximal ideals) of a finitely generated noetherian (commutative unital) $k$-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.

Affine varieties can be embedded as closed subvarieties into an affine space (in the sense of algebraic geometry). As topological spaces affine varieties are noetherian.