2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69395The relations between the radical of crossed product $R #_σH$ 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 $R#_σH$ is semiprime.24pagesQuantum AlgebraRings and Algebras16A30text