On rings which are sums of two subrings
type: Item
instance of: scholarly article
title: On rings which are sums of two subrings
zbMATH Open document ID: 0645.16005
DOI: 10.1007/BF01194866
author: Miguel Ferrero
author: E. R. Puczyłowski
published in: Archiv der Mathematik
publication date: 1989-01-01T00:00:00Z
review text: A result proved by \textit{O. H. Kegel} [Math. Ann. 149, 258-260 (1963; Zbl 0106.254)] states that a ring \(R\) is nilpotent if it is a sum of two nilpotent subrings \(R_ 1\) and \(R_ 2\). In this paper we study the case when one of the subrings is nilpotent (\(T\)-nilpotent) and the other is S- radical, for several radicals \(S\), including the most popular radicals. We get, for example, that if \(R_ 1\) is nilpotent and \(R_ 2\) is S- radical, where \(S\) is an \(N\)-radical, then \(R\) is \(S\)-radical. Also if \(R_ 1\) is right \(T\)-nilpotent and \(R_ 2\) is Jacobson radical (locally nilpotent), then \(R\) is Jacobson radical (locally nilpotent). The analogous results for the nil radical are equivalent to the Koethe conjecture. Several other results of this type and examples are included in the paper. Finally, it is an open question whether \(R\) is prime radical if \(R_ 1\) is \(T\)-nilpotent and \(R_ 2\) is prime radical.
Mathematics Subject Classification ID: 16N40
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16Nxx
zbMATH DE Number: 4051819
zbMATH Keywords: sums of two subrings
zbMATH Keywords: nilpotent subrings
zbMATH Keywords: right T-nilpotent
zbMATH Keywords: Jacobson radical
zbMATH Keywords: locally nilpotent
zbMATH Keywords: nil radical
zbMATH Keywords: Koethe conjecture
zbMATH Keywords: prime radical
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01194866
OpenAlex ID: W2000082237
description: scientific article; zbMATH DE number 4051819