  It is an open question whether the smash product of a semisimple Hopf algebra
and a semiprime module algebra is semiprime. In this paper we show that the
smash product of a commutative semiprime module algebra over a semisimple
cosemisimple Hopf algebra is semiprime. In particular we show that the central
$H$-invariant elements of the Martindale ring of quotients of a module algebra
form a von Neumann regular and self-injective ring whenever $A$ is semiprime.
For a semiprime Goldie PI $H$-module algebra $A$ with central invariants we
show that $\AH$ is semiprime if and only if the $H$-action can be extended to
the classical ring of quotients of $A$ if and only if every non-trivial
$H$-stable ideal of $A$ contains a non-zero $H$-invariant element. In the last
section we show that the class of strongly semisimple Hopf algebras is closed
under taking Drinfeld twists. Applying some recent results of Etingof and
Gelaki we conclude that every semisimple cosemisimple triangular Hopf algebra
over an algebraically closed field is strongly semisimple.
