The Von Neumann Regular Radical and Jacobson Radical of Crossed Products
Abstract
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⊗F)=
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 J.R.Fisher asked.