The topological Jacobson radical of rings. I. (Q1930198)

The authors give a new definition for a topological Jacobson radical of a topological ring \(R\) as the intersection of annihilators of all topologically irreducible left \(R\)-modules. They show that this definition coincides with the classical definition. Several properties of the topological Jacobson radical are further studied. They also show that any topological ring can be embedded as a closed ideal into a unital ring so that the topological Jacobson radicals of these two rings coincide. It is proved that the topological Jacobson radical of a direct product of topological rings coincides with the direct product of topological Jacobson radicals of the rings involved. Also some properties of topological Jacobson radicals connected with the ring of continuous functions are established.