Intersection in a Sheaf of Rings

Question

Suppose $\mathcal{F}$ is a sheaf of rings on a topological space $X$, with open subsets $U,V$, with $U\cap V\ne\emptyset$.

If $\mathcal{F}(U)=A$ and $\mathcal{F}(V)=B$, what can be said about $\mathcal{F}(U\cap V)$? Is it isomorphic to $A\cap B$?