The intersection of the powers of the topological Jacobson radical and topological Krull dimension

In this paper, it is proved that a certain power of the topological Jacobson radical for a ring annihilates a left module having topological Krull dimension over this ring. The estimation of this power depends on the topological Krull dimension and the dual topological Krull dimension. A similar estimation for discrete Jacobson radical holds true. Levitzky's theorem is generalized for topological rings.