The notions and results of this paper are connected with the classical Wedderburn-Mal’cev decomposition for finite-dimensional associative algebras [C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras. Pure Appl. Math. 11. New York-London: Interscience Publishers (1962; Zbl 0131.25601), Chapter X; N. Jacobson, The theory of rings. Mathematical Survey 1. New York: AMS (1943; Zbl 0060.07302), Chapter V]. There is an extensive literature on this topic in the case of topological algebras and rings [see, for instance, M. A. Najmark, Normierte Algebren. Moskau: ‘Nauka’ (1968; Zbl 0175.43702) and K. Numakura, Proc. Japan Acad. 35, 313-315 (1959; Zbl 0090.02802)].
A ring with topology in which the addition is continuous and the multiplication is separately continuous is called a topological ring. A continuous surjective homomorphism of topological rings is called a topological extension of . A topological extension of splits strongly if there exists a continuous homomorphism such that .
The author is looking for conditions under which a topological extension of splits strongly. It is proved that if there exists an idempotent such that then the extension splits strongly. A topological extension is called singular if . It is proved also that if every singular topological extension of splits strongly then every nilpotent topological extension splits strongly (a topological extension is called nilpotent if is nilpotent).