The intersection property in the theory of radicals of topological algebras

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 is a radical of classK, then the following conditions are equivalent: 1) is a topological special radical; 2) the -semisimple algebras are topological subdirect products of prime -semisimple algebras ofK;This result is a corollary of a general result that establishes necessary and sufficient conditions for the radical to have the intersection property with respect to a class of prime algebras. Bibliography: 17 titles.Translated fromTrudy Seminara imeni I. G. Petrovskogo, No. 15, pp. 178–188.