On the regular radical of a group ring

If R is a commutative ring with identity and G a group, it is shown that th e regular radical of the group ring RG is trivial if the regular radical of R is trivial. This answers in the affirmative the commutative case of a qu estion in [4]. Further, the regular radical of RG is computed when G is an Abelian group or when G is locally finite and the order of each element of G is a unit in R.