Rings which are sums of two subrings
type: Item
instance of: scholarly article
title: Rings which are sums of two subrings
zbMATH Open document ID: 0929.16017
DOI: 10.1016/S0022-4049(97)00191-6
published in: Journal of Pure and Applied Algebra
publication date: 2000-01-18T00:00:00Z
review text: Suppose that an associative ring \(R\) has the form \(R=R_1+R_2\), where \(R_1\) and \(R_2\) are subrings of \(R\). One can ask what properties of \(R\) can be derived if \(R_1\) and \(R_2\) satisfy certain conditions. The main studies of rings which are sums of two subrings concern radicals and polynomial identities. The authors also concentrate on these topics in this paper. Given a ring \(A\), define the left hyperannihilator \(l(A)\) of \(A\) to be the union \(\bigcup_{\alpha\geq 0}l_\alpha(A)\) where \(l_0(A)=0\) and for \(\alpha\geq 1\) \[ l_\alpha(A)=\bigl\{x\in A\mid xA\subseteq\bigcup_{\beta<\alpha}l_\beta(A)\bigr\}. \] Similarly define \(r(A)\), the right hyperannihilator of \(A\). The intersection \(N(A)=l(A)\cap r(A)\) is called the hyperannihilator of \(A\). In a paper written in 1964, O. H. Kegel asked whether at least one of \(N(R_1)\) or \(N(R_2)\) is contained in \(\beta(R)\), the prime radical of \(R\). The authors give an example which answers Kegel's question in the negative. Their example also shows that there are no direct relations between radicals of \(R\) and those of \(R_1\) and \(R_2\). It has been previously noted that the field of rational numbers is a sum of two Jacobson radical subrings. In the second part of this paper the authors characterize all fields of that type. They prove that for a field \(F\) the following are equivalent: 1. \(F=A+B\) for some Jacobson radical rings \(A\) and \(B\); 2. \(F=A+B\) for some proper subrings \(A\) and \(B\) of \(F\); 3. \(\text{ch }F=0\) or \(\text{tr}_{F_0}F>0\), where \(F_0\) is the prime subfield of \(F\). A descending chain \(I_1\supseteq I_2\supseteq\cdots\) of ideals of a ring \(A\) is said to absorb \(A\) if for every natural number \(n\) one has \(AI_n+I_nA\subseteq I_{n+1}\). O. H. Kegel asked whether the existence of properly descending absorbing chains of ideals of \(R_1\) and \(R_2\) is sufficient to force the ring \(R\) to be non-simple. The authors show that the answer is negative. T…
Mathematics Subject Classification ID: 16N20
Mathematics Subject Classification ID: 16R40
Mathematics Subject Classification ID: 16W99
Mathematics Subject Classification ID: 12F99
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 1308264
zbMATH Keywords: sums of subrings
zbMATH Keywords: radicals
zbMATH Keywords: polynomial identities
zbMATH Keywords: hyperannihilators
zbMATH Keywords: prime radical
zbMATH Keywords: Jacobson radical subrings
zbMATH Keywords: descending absorbing chains of ideals
zbMATH Keywords: nil rings of bounded index
zbMATH Keywords: simple nil rings
author: Marek Kępczyk
author: E. R. Puczyłowski
MaRDI profile type: Publication
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full work available at URL: s0022-4049(97)00191-6
OpenAlex ID: W1986024693
description: scientific article; zbMATH DE number 1308264