A Hopf module is an $A$-module for an algebra $A$ as well as
a $C$-comodule for a coalgebra $C$, satisfying a suitable
compatibility condition between the module and comodule
structures. To formulate the compatibility condition one
needs some kind of interaction between $A$ and $C$. The
most classical case arises when $A=C=:H$ is a bialgebra.
Many subsequent variants of this were unified independently
by Doi and
Koppinen; in their version an auxiliary bialgebra $H$,
over which $A$ is a comodule algebra and $C$ a module
coalgebra, governs the compatibility. Another very general
type of interaction between $A$ and $C$ is an entwining
map as studied by Brzezi\'nski --- without an
auxiliary bialgebra.
Every Doi-Koppinen datum induces
an entwining structure, so Brzezi\'nski's notion of
an entwined module includes that of
a Doi-Koppinen Hopf module. This paper investigates whether the inclusion is
proper.
By work of Tambara, every entwining structure
can be obtained from a suitable Doi-Koppinen datum whenever the
algebra under consideration is finite
dimensional.
We show by examples that this need not be true in general.
