The Radicals of Crossed Products

The relations between the radical of crossed product R #_\sigma H and algebra $R$ are obtained. Using this theory, the author shows that if $H$ is a finite-dimensional semisimple, cosemisimle, and either commutative or cocommutative Hopf algebra, then $R$ is $H$-semiprime iff $R$ is semiprime iff R#_{\sigma}H is semiprime.