  Work of Jean Renault shows that, for topologically principal \'etale
groupoids, a diagonal-preserving isomorphism of reduced $C^*$-algebras yields
an isomorphism of groupoids. Several authors have proved analogues of this
result for ample groupoid algebras over integral domains under suitable
hypotheses. In this paper, we extend the known results by allowing more general
coefficient rings and by weakening the hypotheses on the groupoids. Our
approach has the additional feature that we only need to impose conditions on
one of the two groupoids. Applications are given to Leavitt path algebras.
