Completely simple topological commutative rings

The paper considers a generalization to topological algebras of the concept of algebraical simplicity (see, definitions 1 and 1$'$ below). Such topological algebras are called completely simple. Completely simple topological commutative rings and Abelian groups are described. As an appendix, a new proof is obtained for Kowalsky's theorem on fields with topologies that cannot be weakened.