in is not an algebraic variety (not even algebraic set). Note that is not a polynomial in (although a polynomial in real variables .) On the other hand, the complement of the origin in is an algebraic (affine) variety, since the origin is the zero-locus of . This may be explained as follows: the affine line has dimension one and so any subvariety of it other than itself must have strictly less dimension; namely, zero. For similar reasons, a unitary group (over the complex numbers) is not an algebraic variety, while the special linear group is a closed subvariety of , the zero-locus of . (Over a different base field, a unitary group can however be given a structure of a variety.) Basic results An affine algebraic set V is a variety if and only if I(V) is a prime ideal; equivalently, V is a variety if and only if its coordinate ring is an Every nonempty affine algebraic set may be written uniquely as a finite union of algebraic varieties (where none of the varieties in the decomposition is a subvariety of any other). The dimension of a variety may be defined in various equivalent ways. See Dimension of an algebraic variety for details. A product of finitely many algebraic varieties (over an algebraically closed field) is an algebraic variety. A finite product of affine varieties is affine and a finite product of projective varieties is projective. Isomorphism of algebraic varieties Let be algebraic varieties. We say and are isomorphic, and write , if there are regular maps and such that the compositions and are the identity maps on and respectively. Discussion and generalizations The basic definitions and facts above enable one to do classical algebraic geometry. To be able to do more — for example, to deal with varieties over fields that are not algebraically closed — some foundational changes are required. The modern notion of a variety is considerably more abstract than the one above, though equivalent in the case of varieties over algebraically closed fields. An abstract algebraic variety is a particular kind of scheme; the generalization to schemes on the geometric side enables an extension of the correspondence described above to a wider class of rings. A scheme is a locally ringed space such that every point has a neighbourhood that, as a locally ringed space, is isomorphic to a spectrum of a ring. Basically,