  We prove that if A is a finite dimensional associative H-comodule algebra
over a field F for some involutory Hopf algebra H not necessarily finite
dimensional, where either char F = 0 or char F > dim A, then the Jacobson
radical J(A) is an H-subcomodule of A. In particular, if A is a finite
dimensional associative algebra over such a field F, graded by any group, then
the Jacobson radical J(A) is a graded ideal of A. Analogous results hold for
nilpotent and solvable radicals of finite dimensional Lie algebras over a field
of characteristic 0. We use the results obtained to prove the analog of
Amitsur's conjecture for graded polynomial identities of finite dimensional
associative algebras over a field of characteristic 0, graded by any group.
