Weakly periodic and subweakly periodic rings.
type: Item
instance of: scholarly article
title: Weakly periodic and subweakly periodic rings.
zbMATH Open document ID: 1038.16023
DOI: 10.1155/S0161171203210589
author: Amber Rosin
author: Adil Yaqub
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2003-06-18T00:00:00Z
full work available at URL: https://eudml.org/doc/50770
review text: Let \(R\) be a ring, \(N\) the set of nilpotent elements, \(J\) the Jacobson radical, and \(P= \{x\in R\mid\) there exists an integer \(n= n(x)> 1\) such that \(x^n= x\}\). The ring \(R\) is called periodic if for each \(x\in R\), there exist distinct positive integers \(m\), \(n\) for which \(x^n= x^m\); and \(R\) is called weakly periodic if \(R= P+ N\). It is known that every periodic ring is weakly periodic; whether the converse holds is an open question. The authors generalize further, defining \(R\) to be subweakly periodic if \(R\setminus J\subseteq P+ N\); and they prove commutativity theorems for subweakly periodic rings. Their principal result establishes commutativity of subweakly periodic rings satisfying the following two conditions: (i) \(J\) is commutative; (ii) for each \(x, y\in R\), there exist words \(w(x, y)\) and \(w'(x,y)\) in \(x\) and \(y\), for which \(w(x, y)[xy, yx]= 0= [xy, yx]w'(x, y)\).
Mathematics Subject Classification ID: 16U80
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16N40
zbMATH DE Number: 1930093
zbMATH Keywords: weakly periodic rings
zbMATH Keywords: subweakly periodic rings
zbMATH Keywords: commutativity theorems
zbMATH Keywords: nilpotent elements
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1930093