Let $H$ be a bialgebra, let $A$ be an algebra and a left $H$-comodule coalgebra, let $B$ be an algebra and a right $H$-comodule coalgebra. Also let $f : H \otimes H \rightarrow A \otimes H, R : H \otimes A \rightarrow A \otimes H$, and $T : B \otimes H \rightarrow H \otimes B$ be linear maps.
We present necessary and sufficient conditions for the one-sided Brzezi´nski’s crossed product algebra $A\#^f_RH_T\#B$ and the two-sided smash coproduct coalgebra $A \times H \times B$ to form a bialgebra, which generalizes the main results from [On
Ranford biproduct // Communs Algebra. – 2015. – 43, № 9. – P. 3946 – 3966]. It is clear that both Majid’s double biproduct
[Double-bosonization of braided groups and the construction of $U_q(g)$ // Math. Proc. Cambridge Phil. Soc. – 1999. – 125,
№ 1. – P. 151 – 192] and the Wang – Jiao – Zhao’s crossed product [Hopf algebra structures on crossed products // Communs
Algebra. – 1998. – 26. – P. 1293 – 1303] are obtained as special cases.
