Actions of commutative hopf algebras

We show that actions of finite-dimensional semisimple commutative Hopf algebras H on H-module algebras A are essentially group-gradings. Moreover we show that the centralizer of H in the smash product A # H equals A^H ⊗ H. Using these we invoke results about group graded algebras and results about centralizers of separable subalgebras to give connections between the ideal structure of A, A^H and A # H. Examples of the above occur naturally when one considers: (1) finite abelian groups G of automorphisms of an algebra A with G ⁻¹ ɛ A; (2) G-graded algebras, for finite groups G; (3) finite-dimensional restricted Lie algebras L, with semisimple restricted enveloping algebra u(L), acting as derivations on an algebra A.