Co-Frobenius Hopf Algebras: Integrals, Doi–Koppinen Modules and Injective Objects

We investigate Hopf algebras with non-zero integral from a coalgebraic point of view. Categories of Doi–Koppinen modules are studied in the special case where the defining coalgebra is left and right semiperfect, and several pairs of adjoint functors are constructed. As applications we give a very short proof for the uniqueness of the integrals and provide information about injective objects in the category of Doi–Koppinen modules.