the base field is required to be algebraically closed is that affine varieties automatically satisfy Hilbert's nullstellensatz: for an ideal J in where k is an algebraically closed field, Radical ideals (ideals which are their own radical) of k[V] correspond to algebraic subsets of V. Indeed, for radical ideals I and J, if and only if Hence V(I)=V(J) if and only if I=J. Furthermore, the function taking an affine algebraic set W and returning I(W), the set of all functions which also vanish on all points of W, is the inverse of the function assigning an algebraic set to a radical ideal, by the nullstellensatz. Hence the correspondence between affine algebraic sets and radical ideals is a bijection. The coordinate ring of an affine algebraic set is reduced (nilpotent-free), as an ideal I in a ring R is radical if and only if the quotient ring R/I is reduced. Prime ideals of the coordinate ring correspond to affine subvarieties. An affine algebraic set V(I) can be written as the union of two other algebraic sets if and only if I=JK for proper ideals J and K not equal to I (in which case ). This is the case if and only if I is not prime. Affine subvarieties are precisely those whose coordinate ring is an integral domain. This is because an ideal is prime if and only if the quotient of the ring by the ideal is an integral domain. Maximal ideals of k[V] correspond to points of V. If I and J are radical ideals, then if and only if As maximal ideals are radical, maximal ideals correspond to minimal algebraic sets (those which contain no proper algebraic subsets), which are points in V. If V is an affine variety with coordinate ring this correspondence becomes explicit through the map where denotes the image in the quotient algebra R of the polynomial An algebraic subset is a point if and only if the coordinate ring of the subset is a field, as the quotient of a ring by a maximal ideal is a field. The following table summarises this correspondence, for algebraic subsets of an affine variety and ideals of the corresponding coordinate ring: Products of affine varieties A product of affine varieties can be defined using the isomorphism then embedding the product in this new affine space. Let and have coordinate rings and respectively, so that their