Morita Theory for corings and cleft entwining structures

The Galois theory of matrix $C$-rings

A theory of monoids in the category of bicomodules of a coalgebra $C$ or $C$-rings is developed. This can be viewed as a dual version of the coring theory. The notion of a matrix ring context consisting of two bicomodules and two maps is introduced and the corresponding example of a $C$-ring (termed a {\em matrix $C$-ring}) is constructed. It is shown that a matrix ring context can be associated to any bicomodule which is a one-sided quasi-finite injector. Based on this, the notion of a {\em Galois module} is introduced and the structure theorem, generalising Schneider's Theorem II [H.-J. Schn…