  The Chern-Galois theory is developed for corings or coalgebras over
non-commutative rings. As the first step the notion of an entwined extension as
an extension of algebras within a bijective entwining structure over a
non-commutative ring is introduced. A strong connection for an entwined
extension is defined and it is shown to be closely related to the Galois
property and to the equivariant projectivity of the extension. A generalisation
of the Doi theorem on total integrals in the framework of entwining structures
over a non-commutative ring is obtained, and the bearing of strong connections
on properties such as faithful flatness or relative injectivity is revealed. A
family of morphisms between the K0-group of the category of finitely generated
projective comodules of a coring and even relative cyclic homology groups of
the base algebra of an entwined extension with a strong connection is
constructed. This is termed a relative Chern-Galois character. Explicit
examples include the computation of a Chern-Galois character of depth 2
Frobenius split (or separable) extensions over a separable algebra R. Finitely
generated and projective modules are associated to an entwined extension with a
strong connection, the explicit form of idempotents is derived, the
corresponding (relative) Chern characters are computed, and their connection
with the relative Chern-Galois character is explained.
