Let P be the set of prime ideals of a commutative ring A. Then an affine scheme is a technical mathematical object defined as the ring spectrum σ(A) of P, regarded as a local-ringed space with a structure sheaf. A local-ringed space that is locally isomorphic to an affine scheme is called a scheme (Itô 1986, p. 69). An affine scheme is a generalization of the notion of affine variety, where the coordinate ring is replaced by any commutative unit ring, and the variety with the Zariski topology is replaced by any topological space.

affine variety | prime ideal | ring spectrum | scheme