This paper examines crossed
products R ∗ H where the Hopf algebra H acts weakly on the algebra R and is
twisted by a Hopf cocycle t. Invertible cocycles are discussed and a related sort of
weak action which we call “fully invertible” is introduced. This condition allows us to
undo the action of H in a useful way and allows reasonable behavior of ideals in
crossed products. Many crossed products of interest are of this type, including
crossed products of cocommutative Hopf algebras with invertible cocycles,
crossed products of irreducible Hopf algebras, and all smash products with
bijective antipode. We construct the quotient ring Q of an H-prime ring
and discuss actions which become inner when extended to Q. This is then
applied to describe prime ideals in crossed products over H-prime rings with
extended inner actions and it is shown that some of these crossed products are
semiprime.
