Let H be a bialgebra, let A be an algebra and a left H-comodule coalgebra, and let B be an algebra and a right H-comodule coalgebra. Also let f : H ⨂ H → A ⨂ H, R : H ⨂ A → A ⨂ H, and T : B ⨂ H → H ⨂ B be linear maps. We present necessary and sufficient conditions for the onesided Brzeziński’s crossed product algebra \( A{\#}_R^f{H}_T\#B \) and the two-sided smash coproduct coalgebra A × H × B to form a bialgebra, which generalizes the main results from [“On Ranford biproduct,” Comm. Algebra, 43, No. 9, 3946–3966 (2015)]. It is clear that both the Majid double biproduct [“Doublebosonization of braided groups and the construction of Uq(g), ” Math. Proc. Cambr. Philos. Soc., 125, No. 1, 151–192 (1999)] and the Wang–Jiao–Zhao crossed product [“Hopf algebra structures on crossed products,” Comm. Algebra, 26, 1293–1303 (1998)] are obtained as special cases.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 11, pp. 1533–1540, November, 2018.
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Ma, T.S., Li, H.Y. & Dong, L.H. A Class of Double Crossed Biproducts. Ukr Math J 70, 1767–1776 (2019). https://doi.org/10.1007/s11253-019-01621-y
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DOI: https://doi.org/10.1007/s11253-019-01621-y