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 a
nd 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 Schneiders Theorem II [H.-J. Schneider, Israel J. Math., 72 (1990), 167--195], is proven. This is then applied to the $C$-ring associated to a weak entwining structure and a structure theorem for a weak $A$-Galois coextension is derived. The theory of matrix ring contexts for a firm coalgebra (or {em infinite matrix ring contexts}) is outlined. A Galois connection associated to a matrix $C$-ring is constructed.
Weak coalgebra-Galois extensions are studied. A notion of an invertible weak entwining structure is introduced. It is proven that, within an invertible weak entwining structure, the surjectivity of the canonical map implies bijectivity provided the s
tructure coalgebra C is either coseparable or projective as a C-comodule.
