It is a famous open question whether the smash product of a semisimple Hopf algebra and a semiprime module algebra is semiprime. The aim of this paper is to answer this question partially. The main result is that the smash product of a commutative semiprime module algebra over a semisimple cosemisimple Hopf algebra is semiprime. On the other hand, it is proved that for a semiprime Goldie PI -module algebra with central invariants, is semiprime if and only if the -action can be extended to the classical ring of quotients of , and if and only if every non-trivial -stable ideal of contains a non-zero -invariant element.
In the last section, it is shown that the class of strongly semisimple Hopf algebras is closed under taking Drinfeld twists. Applying some recent results of P. Etingof and S. Gelaki [see Int. Math. Res. Not. 1998, No. 16, 851-861 (1998; Zbl 0918.16027); ibid. 2000, No. 5, 223-234 (2000; Zbl 0957.16029)], it is concluded that every semisimple cosemisimple triangular Hopf algebra over a field is strongly semisimple.