  The notions of Hom-coring, Hom-entwining structure and associated entwined
Hom-module are introduced. A theorem regarding base ring extension of a
Hom-coring is proven and then is used to acquire the Hom-version of Sweedler
coring. Motivated by the work of T. Brzezinski, a Hom-coring associated to an
entwining Hom-structure is constructed and an identification of entwined
Hom-modules with Hom-comodules of this Hom-coring is shown. The dual algebra of
this Hom-coring is proven to be a $\psi$-twisted convolution algebra. By a
construction, it is shown that a Hom-Doi-Koppinen datum comes from a
Hom-entwining structure and that the Doi-Koppinen Hom-Hopf modules are the same
as the associated entwined Hom-modules. A similar construction regarding an
alternative Hom-Doi-Koppinen datum is also given. A collection of Hom-Hopf-type
modules are gathered as special examples of Hom-entwining structures and
corresponding entwined Hom-modules, and structures of all relevant Hom-corings
are also considered.
