On radicals of module coalgebras

We introduce the notion of an idempotent radical class of module coalgebras over a bialgebra BB. We prove that if RR is an idempotent radical class of BB-module coalgebras, then every BB-module coalgebra contains a unique maximal BB-submodule coalgebra in RR. Moreover, a BB-module coalgebra CC is a member of RR if, and only if, DBDB is in RR for every simple subcoalgebra DD of CC. The collection of BB-cocleft coalgebras and the collection of HH-projective module coalgebras over a Hopf algebra HH 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.