Solving some linear equations over alternative rings
type: Item
instance of: scholarly article
title: Solving some linear equations over alternative rings
zbMATH Open document ID: 1201.17019
author: Yongge Tian
published in: Results in Mathematics
publication date: 2010-11-17T00:00:00Z
review text: A (not necessarily associative) ring \(R\) is said to be alternative if the multiplication in \(R\) satisfies the ``alternative'' laws: \(a(ab)=(a^2)b;~(ba)a=b(a^2)\) for all \(a,b \in R\). Let \(R\) be an alternative ring. The author of the paper under review considers the solvability of the linear equations \(ax=b,xa=b\) and \(axa=b\) for all \(a,b \in R\). Necessary and sufficient conditions for the existence of each of the equations is given. These results are also supplemented with the expression for the most general solution and conditions guaranteeing uniqueness of solutions. Each of these conditions involve the von Neumann regularity of the element \(a\). (\(a \in R\) is said to be von Neumann regular if there exists \(x \in R\) such that \(axa=a\)). Further, the author derive conditions under which each of the equations \(ax-ya=b,ax-xa=b\) and \(x-axa=b\) have solutions in \(R\), where \(a\) is assumed to be either idempotent or involutory.
reviewed by: K. C. Sivakumar
Mathematics Subject Classification ID: 17D05
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 17A30
zbMATH DE Number: 5816642
zbMATH Keywords: alternative ring
zbMATH Keywords: von Neumann regularity
zbMATH Keywords: linear equations
MaRDI profile type: Publication
full work available at URL: s00025-010-0041-5
OpenAlex ID: W2024071370
cites work: Solutions to \(x-axb=w\)
cites work: Equations \(ax = c\) and \(xb = d\) in rings and rings with involution with applications to Hilbert space operators
cites work: Q5522244
cites work: Alternative regular rings without nilpotent elements
DOI: 10.1007/S00025-010-0041-5
description: scientific article; zbMATH DE number 5816642