  We introduce the notion of 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, DB 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 bijective antipode.
