The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. Let us consider a Hopf algebra A such that its Jacobson radical J is a nilpotent Hopf ideal and H := A/J is a semisimple algebra. We prove that the canonical projection of A on H has a section which is an H-colinear algebra map. Furthermore, if H is cosemisimple too, then we can choose this section to be an (H,H)-bicolinear algebra morphism. This fact allows us to describe A as a ‘generalized bosonization’ of a certain algebra R in the category of Yetter-Drinfeld modules over H. As an application we give a categorical proof of Radford’s result about Hopf algebras with projections. We also consider the dual situation. Let A be a bialgebra such that its coradical is a Hopf sub-bialgebra with antipode. Then there is a retraction of the canonical injection of H into A which is an H-linear coalgebra morphism. Furthermore, if H is semisimple too, then we can choose this retraction to be an (H,H)-bilinear coalgebra morphism. Then, also in this case, we can describe A as a 'generalized bosonization' of a certain coalgebra R in the category of Yetter-Drinfeld modules over H.
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