On a new radical in a topological ring

Abstract

The radical which is referred to in this paper was treated extensively by Wright in the case of topological 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.