Morita theory for corings and cleft entwining structures

Using the theory of corings, we generalize and unify Morita contexts introduced by Chase and Sweedler [Hopf Algebras and Galois Theory, in: Lecture Notes in Math., vol. 97, Springer-Verlag, Berlin, 1969], Doi [Generalized smash products and Morita contexts for arbitrary Hopf algebras, in: J. Bergen, S. Montgomery (Eds.), Advances in Hopf algebras, in: Lecture Notes in Pure and Appl. Math., vol. 158, Dekker, New York, 1994], and Cohen, Fischman, and Montgomery [J. Algebra 133 (1990) 351]. We discuss when the contexts are strict. We apply our theory to corings arising from entwining structures, and this leads us to the notion of cleft entwining structure.