On splitting of extensions of rings and topological rings. (Q605417)

The notions and results of this paper are connected with the classical Wedderburn-Mal'cev decomposition for finite-dimensional associative algebras [\textit{C. W. Curtis} and \textit{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; \textit{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, \textit{M. A. Najmark}, Normierte Algebren. Moskau: `Nauka' (1968; Zbl 0175.43702) and \textit{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 \(\pi\colon A\to R\) of topological rings is called a topological extension of \(R\). A topological extension of \(R\) splits strongly if there exists a continuous homomorphism \(\theta\colon R\to A\) such that \(\pi\circ\theta=\text{id}_R\). The author is looking for conditions under which a topological extension of \(R\) splits strongly. It is proved that if there exists an idempotent \(e\in I\) such that \(I=eI+Ie\) then the extension splits strongly. A topological extension is called singular if \((\ker\pi)^2=0\). It is proved also that if every singular topological extension of \(R\) splits strongly then every nilpotent topological extension splits strongly (a topological extension is called nilpotent if \(\ker\pi\) is nilpotent).