Co-Frobenius hopf algebras: Integrals, Doi-Koppinen modules and injective objects

We investigate Hopf algebras with non-zero integral from a coalgebraic poin t of view. Categories of Doi-Koppinen modules are studied in the special ca se where the defining coalgebra is left and right semiperfect, and several pairs of adjoint functors are constructed. As applications we give a very s hort proof for the uniqueness of the integrals and provide information abou t injective objects in the category of Doi-Koppinen modules. (C) 1999 Acade mic Press.