The coordinate ring of a plane and a disjoint line

A disjoint union of closed subsets of \(\AA^n\) has coordinate ring the product of their coordinate rings. In \(\AA^3_\CC\) with coordinates \(x, y, z\), let \(X_1 = V(x, z-1)\) be the line through \((0,0,1)\) parallel to the \(y\)-axis, let \(X_2 = V(z)\) be the \(xy\)-plane, and let \(X = X_1 \cup X_2\).

Then \[I(X) = (x, z-1) \cdot (z) = (xz, z^2 - z)\] and the coordinate ring decomposes as \[\CC[X] = \frac{\CC[x,y,z]}{(xz, z^2-z)} \cong \frac{\CC[x,y,z]}{(x, z-1)} \times \frac{\CC[x,y,z]}{(z)} .\]