Actions of monoidal categories and generalized Hopf smash products

Let R be a k-algebra, and a monoidal category. Assume given the structure of a -category on the category of left R-modules; that is, the monoidal category is assumed to act on the category by a coherently associative bifunctor . We assume that this bifunctor is right exact in its right argument. In this setup we show that every algebra A (respectively coalgebra C) in gives rise to an R-ring A R (respectively an R-coring C R) whose modules (respectively comodules) are the A-modules (respectively C-comodules) within the category . We show that this very general scheme for constructing (co)associative (co)rings gives conceptual explanations for the double of a quasi-Hopf algebra as well as certain doubles of Hopf algebras in braided categories, each time avoiding ad hoc computations showing associativity.