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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).
MSC:
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)
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