When Is a Smash Product Semiprime ?

Miriam Cohen raised the 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 selfinjective ring whenever A is H-semiprime and H has a bijective antipode. For a semiprime Goldie PI H-module algebra A with central invariants we show that A#H 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. This allows us to conclude that Cohen’s question has a positive answer for all ‘known’ simple semisimple Hopf algebras. Applying some recent results of Etinghof and Gelaki we show that every semisimple cosemisimple triangular Hopf algebra over a field is strongly semisimple.