The Cox ring of an embedded variety

We compute the Cox ring of an embedded variety X ⊆ Z within a Mori dream space, under the assumption that the pullback map induces an isomorphism at the level of divisor class groups. We show that the Cox ring of X is the intersection of finitely many localizations of a quotient image of the Cox ring of Z. As a consequence, we provide an algorithm that terminates if and only if the Cox ring of X is finitely generated, thereby generalizing previous works on the subject. We apply these results to compute the Cox ring of hypersurfaces in smooth projective toric varieties.