Ring of regular functions over union of open sets

Question

In what generality does the ring of regular functions of a scheme $X$ over a finite union of open sets equal the intersection of the regular functions over respective open set? That is what properties of $X, U, V$ with open sets $U, V \subset X$ give $$\mathscr{O}_X(U \cup V) = \mathscr{O}_X(U) \cap \mathscr{O}_X(V)?$$