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On splitting of extensions of rings and topological rings. (English) Zbl 1208.16034
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 \(\pi\colon A\to R\) 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).
16W80 Topological and ordered rings and modules
16S70 Extensions of associative rings by ideals
46H05 General theory of topological algebras
54H13 Topological fields, rings, etc. (topological aspects)