On a generalization of right duo rings
type: Item
instance of: scholarly article
zbMATH Open document ID: 1354.16003
author: Nam Kyun Kim
author: Tai Keun Kwak
author: Yang Lee
published in: Bulletin of the Korean Mathematical Society
publication date: 2016-06-16T00:00:00Z
full work available at URL: http://www.mathnet.or.kr/mathnet/kms_content.php
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16N20
Mathematics Subject Classification ID: 16N40
Mathematics Subject Classification ID: 16S36
zbMATH DE Number: 6594411
zbMATH Keywords: right \(\pi\)-duo ring
zbMATH Keywords: (weakly) right duo ring
zbMATH Keywords: (strongly) \(\pi\)-regular ring
zbMATH Keywords: polynomial ring
zbMATH Keywords: matrix ring
MaRDI profile type: Publication
OpenAlex ID: W2422144985
DOI: 10.4134/BKMS.B150441
title: On a generalization of right duo rings
review text: A ring \(R\) is \textit{right duo} when every right ideal is a two-sided ideal. Equivalently, for every \(a\in R\) the containment \(Ra\subseteq aR\) holds. Similarly a ring is \textit{right quasi-duo} when every maximal right ideal is two-sided. These are important weakenings of commutativity that have found multiple applications in the literature.NEWLINENEWLINEStill following the literature, a ring \(R\) is called \textit{weakly right duo} if for every \(a\in R\) there exists some \(n\geq 1\) such that \(Ra^n\subseteq a^nR\). The paper under review generalizes this condition even further by introducing the following new definition: a ring \(R\) is \textit{right \(\pi\)-duo} if for every \(a\in R\) there exists some \(n\geq 1\) such that \(Ra^n\subseteq aR\). Clearly, weakly right duo rings are right \(\pi\)-duo. Much of the paper revolves around (re-)proving facts about weakly right duo rings for the more general class of right \(\pi\)-duo rings. For instance, the authors show that right \(\pi\)-duo rings are always abelian, right Ore, and right quasi-duo.NEWLINENEWLINEThe authors further show that the right \(\pi\)-duo property passes to factor rings, certain restricted upper-triangular matrix rings, and central localizations by non-zero-divisors, but not to Dorroh extensions or polynomial rings. There are many classes of rings where the abelian, right quasi-duo rings coincide with the weakly right duo rings, and thus also coincide with the right \(\pi\)-duo rings. A few examples and results are of this flavor, requiring the reader to forage through the literature for proofs.NEWLINENEWLINEIn this review, I have given a slightly different definition of right \(\pi\)-regularity than the authors. However, the conditions are equivalent, even for non-unital rings, by slightly modifying the proof of their Proposition 1.1.NEWLINENEWLINEFinally, I should mention there seems to be a hole in Example 1.2(1). First, since iterated Laurent series do not commute, the ring \(V_…
reviewed by: Pace P. Nielsen
description: scientific article; zbMATH DE number 6594411