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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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.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#RfHT#B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ A{\#}_R^f{H}_T\#B $$\end{document} 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. [ABSTRACT FROM AUTHOR]