  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
twisted graded algebra is a graded ideal. For twisted $H$-module algebra $R$,
we also show that $r_{j}(R#_\sigma H)= r_{Hj}(R)#_\sigma H$ 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 $r_j(R \otimes F) = r_j(R)
\otimes F$, where $H$ is a finite-dimensional, semisimple, cosemisimple,
commutative or cocommutative Hopf algebra over $k$. In particular, we answer
two questions J.R.Fisher asked.
