On Andrunakievich's chain and Koethe's problem.
type: Item
instance of: scholarly article
title: On Andrunakievich's chain and Koethe's problem.
zbMATH Open document ID: 1216.16009
published in: Israel Journal of Mathematics
publication date: 2011-01-10T00:00:00Z
full work available at URL: 82.pdf
review text: Köthe's famous open problem asks whether the nil radical \(\text{Nil}(R)\) of any ring \(R\) contains all nil left (or equivalently, right) ideals of \(R\) [\textit{G. Köthe}, M. Z. 32, 161-186 (1930; JFM 56.0143.01)]. Recently it was shown that this is equivalent to ask whether \(A(R/A(R))=0\) for every ring \(R\) where, for a ring \(S\), \(A(S)\) denotes the ideal \(A(S)=\sum(L\mid L\) is a nil left ideal of \(S)\) [\textit{E. R. Puczyłowski}, in Algebra and its applications. Proc. int. conf., Athens, OH 2005. Contemp. Math. 419, 269-283 (2006; Zbl 1133.16018)]. In pursuit of an answer to these problems, the authors define and study a chain of ideals \(\{A_\alpha(R)\}\), \(\alpha\) an ordinal number, for a ring \(R\) called the `Andrunakievich chain' as follows: \(A_0(R)=0\). For \(\alpha>0\), suppose \(A_\beta(R)\) has been defined for all \(\beta<\alpha\). If \(\alpha\) is a limit ordinal, let \(A_\alpha(R)=\bigcup_{\beta<\alpha}A_\beta(R)\). If \(\alpha\) is not a limit ordinal, then \(A_\alpha(R)\) is the ideal of \(R\) containing \(A_{\alpha-1}(R)\) for which \(A(R/A_{\alpha-1}(R))=A_\alpha(R)/A_{\alpha-1}(R)\). One may then ask about the existence of an \(\alpha\) such that for every ring \(R\), the Andrunakievich chain of \(R\) stabilizes at an ordinal \(\leq\alpha\). The authors show that this question is also equivalent to Köthe's problem. In particular, it is shown that if Köthe's problem has a negative solution, there is a field \(F\) such that for every ordinal number \(\alpha\) there is an \(F\)-algebra \(R_\alpha\) such that \(A_\alpha(R_\alpha)=R_\alpha\) and \(R_\alpha/A_\beta(R_\alpha)\) is not nil for all \(\beta<\alpha\) (i.e. \(A_\beta(R_\alpha)\neq R_\alpha\) for all \(\beta<\alpha\)). This construction is achieved by considering a certain matrix algebra over the field \(F\).
reviewed by: Stefan Veldsman
Mathematics Subject Classification ID: 16N40
zbMATH DE Number: 5834337
zbMATH Keywords: Köthe problem
zbMATH Keywords: nil radical
zbMATH Keywords: Andrunakievich chains
zbMATH Keywords: nil rings
zbMATH Keywords: radicals of rings
zbMATH Keywords: Köthe conjecture
author: M. A. Chebotar
author: P.-H. Lee
author: E. R. Puczyłowski
MaRDI profile type: Publication
OpenAlex ID: W1997400812
cites work: Q3140660
cites work: Q3435582
DOI: 10.1007/S11856-010-0096-8
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description: scientific article; zbMATH DE number 5834337