On splitting of extensions of rings and topological rings.
type: Item
instance of: scholarly article
title: On splitting of extensions of rings and topological rings.
zbMATH Open document ID: 1208.16034
author: Mart Abel
published in: Annals of Functional Analysis
publication date: 2010-11-23T00:00:00Z
full work available at URL: https://eudml.org/doc/224914
full work available at URL: AFA-tex v1 n1 a12.pdf
review text: 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).
reviewed by: Mihail I. Ursul
Mathematics Subject Classification ID: 16W80
Mathematics Subject Classification ID: 16S70
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 54H13
zbMATH DE Number: 5818628
zbMATH Keywords: topological extensions
zbMATH Keywords: topological rings
zbMATH Keywords: splitting extensions
MaRDI profile type: Publication
OpenAlex ID: W1982136310
DOI: 10.15352/AFA/1399900998
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description: scientific article; zbMATH DE number 5818628