The radical which is
referred to in this paper was treated extensively by Wright in the case of
topologicaI groups. The present course of attack here is threefold: (1) to show the
proximity of large powers of topologically nilpotent elements to the radical in a
topological ring, (2) to determine a nilpotence condition on the radical and (3) to
characterize the radical of all locally compact simple rings without divisors of
zero. For a topological group, the radical possesses little, if any, algebraic
structure aside from being a subgroup of the group. Viewed as an additive
subgroup of a topological ring R, it is shown that the radical is an ideal of R.
Relative to the nilpotence of the radical, the additive group structure of
locally compact connected Jacobson semi simple rings is established to within
topological isomorphism. In the final section the theorem on nilpotence is used
to characterize the radical of locally compact simple rings having no zero
divisors.
