Maximal right nearring of quotients and semigroup generalized polynomial identity
type: Item
instance of: scholarly article
title: Maximal right nearring of quotients and semigroup generalized polynomial identity
zbMATH Open document ID: 1039.16040
DOI: 10.1007/BF03323549
published in: Results in Mathematics
publication date: 2003-10-29T00:00:00Z
review text: Quotient near-rings have been studied before by several authors, mainly using a type of Ore condition. The approach in this paper is different. The near-rings considered are right zero-symmetric near-rings. Such a near-ring \(N\) is 3-semiprime if \(xNx\neq 0\) for any non-zero element \(x\in N\). It is equiprime if for any \(a,b,c\in N\), with \(a\neq 0\) the condition \(axc= axb\) for all \(x\in N\) implies \(b= c\). The present approach derives from \textit{Y. Utumi} [Osaka Math. J. 8, 1--18 (1956; Zbl 0070.26601)] as developed in \textit{K. I. Beidar}, \textit{W. S. Martindale III} and \textit{A. V. Mikhalev} [Rings with generalized identities, Marcel Dekker, New York (1996; Zbl 0847.16001)]. The approach is to consider the set \(D(N)\) of all dense right \(N\) invariant subsets of \(N\), that is subsets \(M\) of \(N\) invariant under multiplication on the right by \(N\) such that if \(x, y\in N\) with \(x\neq 0\) there exists \(z\in N\) such that \(xz\neq 0\) and \(yz\in M\). The set of pairs \((f, J)\) is defined where \(J\in D(N)\) and \(f\) is an \(N\) homomorphism from \(J\) to \(N\), the action of \(N\) on \(J\) and \(N\) being right multiplication. By taking suitable equivalence classes of such pairs and defining addition and multiplication on them, the maximal near-ring of quotients \({\mathcal Q}_{mr}(N)\) is obtained. It contains \(N\), has a unity and is unique up to isomorphism. If \(N\) is 3-semiprime, 3-prime or equiprime, then so is \({\mathcal Q}\). The process of constructing \({\mathcal Q}(N)\) is idempotent. The near-ring \({\mathcal Q}(N)\) is von Neumann regular if and only if \(N\) is right non-singular, that is for any element \(x\in N\) there exists a non-zero right \(N\) subset \(Z_x\) such that \(xy\neq xz\) for all \(y, z\in Z_x\) with \(y\neq z\). If \(N\) is zero-symmetric, 3-prime with a minimal left \(N\)-subgroup \(L\) such that \(L= Ne\) for an idempotent \(e\), then \(G= eNe\setminus\{0\}\) is a fixed point free group of automor…
Mathematics Subject Classification ID: 16Y30
Mathematics Subject Classification ID: 16S90
Mathematics Subject Classification ID: 16R50
zbMATH DE Number: 1997667
zbMATH Keywords: maximal quotients
zbMATH Keywords: centralizer near-rings
zbMATH Keywords: generalized polynomial identity
author: K. I. Beidar
author: Y. Fong
author: Wen-Fong Ke
reviewed by: John D. P. Meldrum
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03323549
OpenAlex ID: W2030856732
description: scientific article; zbMATH DE number 1997667