Structure of \(H\)-semiprime Artinian algebras. (Q431840)

Let \(H\) be a Hopf algebra over the ground field \(k\). It is shown in this paper that every \(H\)-semiprime right Artinian left \(H\)-module algebra \(A\) is actually quasi-Frobenius and \(H\)-semisimple. If \(H\) grows slower than exponentially, then all \(H\)-equivariant \(A\)-modules are \(A\)-projective. Moreover, with the additional assumption that \(H\) is cosemisimple it is proved that the Jacobson radical of any right Artinian left \(H\)-module algebra is stable under the action of \(H\). In particular, \(A\) is semisimple whenever \(A\) is \(H\)-semiprime.