Consider algebras over a given commutative ring \(R\), and let \(A\) be a Hopf algebra which is a finitely generated and projective module over \(R\). Suppose \(B\) is an \(A\)-Galois extension of a subalgebra \(C\) and there is a homomorphism \(\alpha\) of \(B\) into an algebra \(E\). Then \(E\) is a bimodule over \(B\), a bimodule over \(C\) by restriction of \(\alpha\), and the opposite \(\Omega^{\text{op}}\) of the ring \(\Omega\) of endomorphisms of the right \(C\)-module \(E\) is a left \(A\)-module algebra. Right multiplication by an element of the centralizer \(V\) of \(\alpha(C)\) in \(E\) is an endomorphism of the right \(C\)-module \(E\); and thus \(V\) can be identified with a subring of \(\Omega^{\text{op}}\) which is a left \(A\)-module subalgebra, and the action of \(A\) on \(V\) is the Miyashita-Ulbrich action. On the other hand, let \(E\) be a ring extension of an algebra \(C\) such that the right \(C\)-module \(E\) is a progenerator. If \(E\) is an Azumaya algebra; then the ring \(\Omega\) of endomorphisms of the right \(C\)-module \(E\) is isomorphic to \(E\otimes V^{\text{op}}\) where \(V\) is the centralizer of \(C\) in \(E\), and \textit{Y. Doi} and \textit{M. Takeuchi} [J. Algebra 121, No. 2, 488-516 (1989; Zbl 0675.16004)] used Morita duality to show that any action of \(A\) on \(V\) by which \(V\) becomes a left \(A\)-module algebra arises from an \(A\)-Galois extension of \(C\). In this paper, the author shows that \(\Omega\) is isomorphic to \(E\otimes_ZV^{\text{op}}\), where \(Z\) is the center of \(E\), whenever \(E\) is an \(H\)-separable extension of \(C\) and uses the same methods of Morita duality to obtain a very satisfactory generalization of the results of Doi and Takeuchi.