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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 [{\it C. W. Curtis} and {\it 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; {\it 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, {\it M. A. Najmark}, Normierte Algebren. Moskau: `Nauka’ (1968; Zbl 0175.43702) and {\it 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).
16W80Topological and ordered associative rings and modules
16S70Extensions of associative rings by ideals
46H05General theory of topological algebras
54H13Topological fields, rings, etc. (topological aspects)
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