Affine gluing along closed subschemes ($R=A{\times }_{C}B$)

Suppose a scheme is the scheme-theoretic union of affine closed subschemes $Y=\mathrm{Spec}A$ and $Z=\mathrm{Spec}B$, meaning their ideal sheaves have zero intersection. The intersection $W$ is closed in both, hence affine with coordinate ring $C$ and surjections $A\to C$, $B\to C$. Then the whole scheme is $\mathrm{Spec}(A{\times }_{C}B)$. Both projections of this fiber product are surjective; their kernels have zero intersection and product, and their quotient by the sum is $C$. Thus its spectrum has the same closed pieces, intersection and topology. The structure sheaf is the kernel of $i_{\ast }{\mathcal{O}}_Y\oplus j_{\ast }{\mathcal{O}}_Z\to k_{\ast }{\mathcal{O}}_W$ on both spaces, identifying their locally ringed structures. A reduced Noetherian scheme has finitely many irreducible components, so induction proves it is affine if and only if its reduced irreducible components are affine. The intersection $W$ need not be reduced.