Rings with linearly ordered right annihilators
type: Item
instance of: scholarly article
title: Rings with linearly ordered right annihilators
zbMATH Open document ID: 1377.16014
author: Greg Marks
author: Ryszard Mazurek
published in: Israel Journal of Mathematics
publication date: 2017-01-11T00:00:00Z
review text: In this paper, the authors introduce the class of lineal rings, that is, rings \(R\) such that, for any subsets \(A,B\subseteq R\) we have \(\mathrm{ann}_{r}^{R}\left( A\right) \subseteq\mathrm{ann}_{r}^{R}\left( B\right) \) or \(\mathrm{ann}_{r}^{R}\left( B\right) \subseteq \mathrm{ann}_{r}^{R}\left( A\right) \), where \(\mathrm{ann}_{r}^{R}\left( A\right) \) (resp., \(\mathrm{ann}_{r}^{R}\left( B\right) \)) denotes the right annihilator of \(A\) in \(R\)(resp., the right annihilator of \(B\) in \(R\) ). \ They give many examples of lineal rings and establish some of their properties. For example, they show that there is no need to distinguish between ``right lineal'' and ``left lineal'' and the characteristic of a lineal ring is either \(0\) or a prime power. The authors study zero-divisors of lineal rings and show how linear rings are related to some important classes of rings such as reduced rings, domains, right strongly prime rings, left strongly prime rings, Armendariz rings (that is, rings \(R\) \ such that \(\left( \sum_{i}a_{i}x^{i}\right) \left( \sum_{j}b_{j}x^{j}\right) =0\) in the polynomial ring \(R\left[ x\right] \) implies that \(a_{j}b_{j}=0\) for all \(i,j\)), right distributive rings (that is, rings whose right ideals lattices are distributive) or right Bezout rings (that is, rings whose finitely generated right ideals are principal), for example. In particular, they show that in any lineal ring the right zero-divisors form a right ideal, and that this property describes linear rings among distributive rings as well as among Bezout rings. The authors prove that a polynomial ring \(R\left[ x\right] \) is lineal if and only if the ring \(R\) \ is lineal and give an example of a lineal ring that is not Armendariz. They show, however, that if a lineal ring \(R\) contains infinitely many central elements whose diferences are regular, then \(R\) is Armendariz. The authors carefully examine nilpotent elements and nilradicals of linear rings. For example,…
reviewed by: Halina France-Jackson
Mathematics Subject Classification ID: 16N40
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 6673806
zbMATH Keywords: right (resp.left) annihilators
zbMATH Keywords: prime rings
zbMATH Keywords: semiprime rings
zbMATH Keywords: reduced rings
zbMATH Keywords: right chain rings
zbMATH Keywords: right distributive rings
zbMATH Keywords: right Bezout rings
zbMATH Keywords: Armendariz rings
zbMATH Keywords: strongly prime rings
zbMATH Keywords: zero divisors
zbMATH Keywords: nilradicals
zbMATH Keywords: Kothe conjecture
MaRDI profile type: Publication
full work available at URL: s11856-016-1415-5
OpenAlex ID: W2531795407
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DOI: 10.1007/S11856-016-1415-5
description: scientific article; zbMATH DE number 6673806