In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space.

More formally, an affine algebraic set is the set of the common zeros over an algebraically closed field k of some family of polynomials in the polynomial ring ${\displaystyle k[x_{1},\ldots ,x_{n}].}$ An affine variety is an affine algebraic set which is not the union of two smaller algebraic sets; algebraically, this means that (the radical of) the ideal generated by the defining polynomials is prime. One-dimensional affine varieties are called affine algebraic curves, while two-dimensional ones are affine algebraic surfaces.

Some texts use the term variety for any algebraic set, and irreducible variety an algebraic set whose defining ideal is prime (affine variety in the above sense).

In some contexts (see, for example, Hilbert's Nullstellensatz), it is useful to distinguish the field k in which the coefficients are considered, from the algebraically closed field K (containing k) over which the common zeros are considered (that is, the points of the affine algebraic set are in Kⁿ). In this case, the variety is said defined over k, and the points of the variety that belong to kⁿ are said k-rational or rational over k. In the common case where k is the field of real numbers, a k-rational point is called a real point. When the field k is not specified, a rational point is a point that is rational over the rational numbers. For example, Fermat's Last Theorem asserts that the affine algebraic variety (it is a curve) defined by xⁿ + yⁿ − 1 = 0 has no rational points for any integer n greater than two.

An affine algebraic set is the set of solutions in an algebraically closed field k of a system of polynomial equations with coefficients in k. More precisely, if ${\displaystyle f_{1},\ldots ,f_{m}}$ are polynomials with coefficients in k, they define an affine algebraic set

An affine (algebraic) variety is an affine algebraic set that is not the union of two proper affine algebraic subsets. Such an affine algebraic set is often said to be irreducible.

If X is an affine algebraic set, and ${\displaystyle \mathrm {I} (X)}$ is the ideal of all polynomials that are zero on X, then the quotient ring ${\displaystyle A(X)=k[x_{1},\ldots ,x_{n}]/\mathrm {I} (X)}$ (also denoted ${\displaystyle \Gamma (X)}$ or ${\displaystyle k[X]}$, although the latter may be mistaken for the polynomial ring in one indeterminate) is called the coordinate ring of X. The ideal ${\displaystyle \mathrm {I} (X)}$ is radical, so the coordinate ring is a reduced ring, and, if X is an (irreducible) affine variety, then ${\displaystyle \mathrm {I} (X)}$ is prime, so the coordinate ring is an integral domain. The elements of the coordinate ring ${\displaystyle A(X)}$ can be thought of as polynomial functions on X and are also called the regular functions or the polynomial functions on the variety. They form the ring of regular functions on the variety, or, simply, the ring of the variety; in more technical terms (see § Structure sheaf), it is the space of global sections of the structure sheaf of X.

The dimension of a variety is an integer associated to every variety, and even to every algebraic set, whose importance relies on the large number of its equivalent definitions (see Dimension of an algebraic variety).