An affine scheme is a locally ringed space isomorphic to the spectrum Spec(R) of a commutative ring R. A scheme is a locally ringed space X admitting a covering by open sets U_i, such that each U_i (as a locally ringed space) is an affine scheme. In particular, X comes with a sheaf O_X, which assigns to every open subset U a commutative ring O_X(U) called the ring of regular functions on U.