Noncommutativity and noncentral zero divisors
type: Item
instance of: scholarly article
title: Noncommutativity and noncentral zero divisors
zbMATH Open document ID: 0921.16012
DOI: 10.1155/S0161171299220674
author: Howard E. Bell
author: A. Klein
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1999-09-28T00:00:00Z
full work available at URL: https://eudml.org/doc/48229
review text: For \(R\) any associative ring, let \(Z\) denote the center of \(R\) and \(D\) the set of all zero divisors in \(R\). The authors prove that if \(1\in R\) is a finite noncommutative ring, \(R\cong S_1\oplus\cdots\oplus S_k\) for indecomposable rings \(S_j\), and \(n\geq 1\) is the number of noncommutative \(S_j\) then \((4^n-1)| D\cap Z|\leq| D-Z|\). When \(D\) is replaced with the set \(N\) of all nilpotent elements in \(R\) the conclusion is that \((p^n-1)| N\cap Z|\leq| N-Z|\) where \(p\) is the smallest prime divisor of \(| R|\). If \(S\) is the set of elements in \(R\) with a nonzero two sided annihilator, then for \(R\) infinite, \(| S-Z|=| D-Z|\) and \(| S-Z|\) is infinite or empty. Other similar results in the paper involve periodic, weakly periodic, or quasiperiodic rings.
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16N40
Mathematics Subject Classification ID: 16U70
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 1275558
zbMATH Keywords: centers
zbMATH Keywords: zero divisors
zbMATH Keywords: finite noncommutative rings
zbMATH Keywords: indecomposable rings
zbMATH Keywords: nilpotent elements
zbMATH Keywords: quasi-periodic rings
MaRDI profile type: Publication
OpenAlex ID: W1981337690
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description: scientific article; zbMATH DE number 1275558