The structure of bialgebra with a Hopf module

Let H be a Hopf algebra, B a bialgebra, and (B, ◃, ρ) a right H-Hopf module. Assume that (B, ρ) is a right H-comodule algebra, (B, ◃) is a right H-module coalgebra, and let A = B co H = {a ∈ B | ρ(a) = a ⊗ 1}. Then we prove that B has a factorization of A□ ◁ (the underlying space is A ⊗ H) as a bialgebra, which generalizes Radford’s factorization of bialgebras with projection [12].