Strong primeness in matrix rings.
type: Item
instance of: scholarly article
title: Strong primeness in matrix rings.
zbMATH Open document ID: 1306.16011
author: John E. van den Berg
author: Henry (Maya) Robert Thackeray
published in: Journal of Algebra
publication date: 2014-10-01T00:00:00Z
review text: A ring \(R\) is right strongly prime if for every nonzero \(a\in R\), there exists a nonempty finite subset \(S\) of \(R\) (called a right insulator for \(a\)) such that the set \(aS\) has a trivial right annihilator. A ring \(R\) is right strongly prime of bound \(n\) if there exists a positive integer \(n\) such that every nonzero element in \(R\) has a right insulator of size \(n\) and no smaller such \(n\) exists. Left strongly prime rings and left insulators are defined in a dual manner. A ring \(R\) is uniformly strongly prime if \(R\) contains a finite subset \(S\) (called a uniform insulator for \(R\)) that is a right insulator for every nonzero \(a\in R\). A ring \(R\) is uniformly strongly prime of bound \(n\) if \(n\) is the smallest positive integer for which \(R\) has a uniform insulator of size \(n\). The bound of uniform strong primeness of the ring \(\mathbb M_n(R)\) of all \(n\) by \(n\) matrices over a unitary ring \(R\) is denoted by \(m_n(R)\). In this paper the authors develop a method for the calculation of \(m_n(R)\) that involves reduction to a system of bilinear equations over \(R\) and, using this method, they obtain some new results concerning bounds of uniform strong primeness in matrix rings. In particular, the authors investigate the bound \(m_2(D)\) for a division ring \(D\) and show that \(m_2(\mathbb H)=3\), where \(\mathbb H\) denotes the ring of quaternions. They define a division pseudoalgebra over a division ring \(D\) to be a \(D\)-bimodule \(_DM_D\) with a multiplication \(\odot\colon M^2\to M\) which is left linear in its first argument, right linear in its second and such that for \(y\in M\setminus\{0\}\), \(x\in M\), each of the equations \(y\odot w=x\) and \(w\odot y=x\) has exactly one solution \(w\in M\). They show that for every natural number \(n\) and every division ring \(D\), there exists a division pseudoalgebra over \(D\) which is isomorphic to \(D^n\) as a \(D\)-module precisely when \(m_n(D)=n\). The authors prov…
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16K40
zbMATH DE Number: 6350929
zbMATH Keywords: uniformly strongly prime rings
zbMATH Keywords: matrix rings
zbMATH Keywords: division rings
zbMATH Keywords: right insulators
zbMATH Keywords: right annihilators
zbMATH Keywords: division pseudoalgebras
reviewed by: Halina France-Jackson
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2014.07.032
OpenAlex ID: W2064354239
cites work: Strongly Prime Rings
cites work: Q4285304
cites work: On uniform bounds of primeness in matrix rings
cites work: Q4235664
cites work: Solution theory for complete bilinear systems of equations
cites work: Q3755592
cites work: The “fundamental theorem of algebra” for quaternions
cites work: Some consequences of a theorem of Bott
cites work: Q4819371
DOI: 10.1016/J.JALGEBRA.2014.07.032
description: scientific article; zbMATH DE number 6350929