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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 $\pi :A\to R$ of topological rings is called a topological extension of $R$. A topological extension of $R$ splits strongly if there exists a continuous homomorphism $\theta :R\to A$ 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 ${\left(ker\pi \right)}^{2}=0$. 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).

##### MSC:
 16W80 Topological and ordered associative rings and modules 16S70 Extensions of associative rings by ideals 46H05 General theory of topological algebras 54H13 Topological fields, rings, etc. (topological aspects)