The Von Neumann Regular Radical and Jacobson Radical of Crossed Products

We construct the H-von Neumann regular radical for H-module algebras and show that it is an H-radical property. We obtain that the Jacobson radical of a twisted graded algebra is a graded ideal. For a twisted H-module algebra R, we also show that r _j (R# H) = r _Hj (R)# H and the Jacobson radical of R is stable, when k is an algebraically closed field or there exists an algebraic closure F of k such that r _j (RF) = r _j (R) F, where H is a finite-dimensional, semisimple, cosemisimple, commutative or cocommutative Hopf algebra over k. In particular, we answer two questions of J. R. Fisher.