Abstract: We develop the Radford’s biproduct theorem which plays an important role in giving a negative answer to a conjecture of I Kaplansky. Let B, H be two Hopf algebras with H acting weakly on B and α, β : B → H be two linear maps verifying suitable conditions. We consider in this paper a twisted Hopf crossed coproduct B^#_×βαH and derive a necessary and sufficient condition for B^#_×βαH with a Hopf smash product structure to be a bialgebra which generalizes in [14, Theorem 1.1] and the well-known Radford biproduct theorem [10, Theorem 1].

Key words: Hopf algebra, twisted Hopf crossed coproduct, Hopf smash product