Entwined Hom-modules and frobenius properties

Entwined Hom-modules were introduced by Karacuha in [13], which can be viewed as a generalization of Doi-Hom Hopf modules and entwined modules. In this paper, the sufficient and necessary conditions for the forgetful functor F : ?H(Mk)(?)CA ? ?H(Mk)A and its adjoint G : ?H(Mk)A ? ?H (Mk)(?)CA form a Frobenius pair are obtained, one is that A?C and the C*?A are isomorphic as (A;C*op#A)-bimodules, where (A,C,?) is a Hom-entwining structure. Then we can describe the isomorphism by using a generalized type of integral. As an application, a Maschke type theorem for entwined Hom-modules is given.