  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. 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.
