Intermediate rings between matrix rings and Ornstein dual pairs

We show that isomorphism of intermediate rings between row and column finit e matrix rings and row finite matrix rings implies Morita equivalence of th e coefficient rings and equality of the cardinality of the set of indices. Among the applications we extend the Isomorphism Theorem for Dual Pairs ove r Division Rings to Ornstein dual pairs over any class of rings for which M orita equivalence implies isomorphism.