LetK be a class of associative topological algebras that is closed under subalgebras with the induced topology, direct products, quotients, and semidirect topological products with respect to continuous homomorphisms. If α is a radical of classK, then the following conditions are equivalent: 1) α is a topological special radical; 2) the α-semisimple algebras are topological subdirect products of prime α-semisimple algebras ofK;

In this article semisimple classes of topological algebras are characterized in a series of varieties of rings (in particular, all subvarieties of the varieties of alternative and Jordan algebras). Characterizations of semisimple classes of hereditary radicals are obtained, and the heredity of semisimple classes of radicals is demonstrated. It is proved that the construction of lower radicals stabilizes on the first infinite ordinal in the varieties of topological algebras considered.