When are associates unit multiples?
type: Item
instance of: scholarly article
title: When are associates unit multiples?
zbMATH Open document ID: 1092.13002
author: S. J. Forman
author: Joe Stickles
author: D. D. Anderson
author: Michael Axtell
published in: Rocky Mountain Journal of Mathematics
publication date: 2005-03-11T00:00:00Z
review text: Let \(R\) be a commutative ring with identity. Two elements \(a,b\in R\) are called: (i) associates, denoted \(a\sim b\), if \(a| b\) and \(b| a\); (ii) strong associates, denoted \(a\approx b\), if \(a=ub\) for some unit \(u\) of \(R\); (iii) very strong associates, denoted \(a\cong b\), if \(a\sim b\) and the conditions \(a\neq 0\) and \(a=rb\) imply that \(r\) is a unit. One always has \(a\cong b\Rightarrow a\approx b\Rightarrow a\sim b\). The ring \(R\) is called: (i) strongly associate if \(a,b\in R\), \(a\sim b\) implies \(a\approx b\); (ii) présimplifiable if \(a,b\in R\), \(a\sim b\) (or \(a\approx b\)) implies \(a\cong b\). It is known that every présimplifiable ring is strongly associate. The authors establish various properties of these kinds of rings.
reviewed by: Septimiu Crivei
Mathematics Subject Classification ID: 13A05
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13F99
zbMATH DE Number: 2144255
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181069828
OpenAlex ID: W2053399991
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DOI: 10.1216/RMJM/1181069828
description: scientific article; zbMATH DE number 2144255