Abstract
We introduce the notion of an idempotent radical class of module coalgebras over a bialgebra B . We prove that if R is an idempotent radical class of B -module coalgebras, then every B -module coalgebra contains a unique maximal B -submodule coalgebra in R . Moreover, a B -module coalgebra C is a member of R if, and only if, D B is in R for every simple subcoalgebra D of C . The collection of B -cocleft coalgebras and the collection of H -projective module coalgebras over a Hopf algebra H are idempotent radical classes. As applications, we use these idempotent radical classes to give another proofs for a projectivity theorem and a normal basis theorem of Schneider without assuming a bijective antipode.
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