  Let X be a hypersurface of a Mori dream space Z. We provide necessary and
sufficient conditions for the Cox ring R(X) of X to be isomorphic to R(Z)/(f),
where R(Z) is the Cox ring of Z and f is a defining section for X. We apply our
results to Calabi-Yau hypersurfaces of toric Fano fourfolds. Our second
application is to general degree d hypersurfaces in P^n containing a linear
subspace of dimension n-2.
