*(English)*Zbl 1076.16038

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 $H$-module algebra $A$ with central invariants, $A\#H$ is semiprime if and only if the $H$-action can be extended to the classical ring of quotients of $A$, and if and only if every non-trivial $H$-stable ideal of $A$ contains a non-zero $H$-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.

##### MSC:

16W30 | Hopf algebras (assoc. rings and algebras) (MSC2000) |

16S40 | Smash products of general Hopf actions |