  A definition of a Doi-Koppinen datum over a noncommutative algebra is
proposed. The idea is to replace a bialgebra in a standard Doi-Koppinen datum
with a bialgebroid. The corresponding category of Doi-Koppinen modules over a
noncommutative algebra is introduced. A weak Doi-Koppinen datum and module of
[G. Bohm. Comm. Algebra, 28:4687--4698, 2000] are shown to be examples of a
Doi-Koppinen datum and module over an algebra. A coring associated to a
Doi-Koppinen datum over an algebra is constructed and various properties of
induction and forgetful functors for Doi-Koppinen modules over an algebra are
deduced from the properies of corresponding functors in the category of
comodules of a coring.
