Extensions of zip rings.
type: Item
instance of: scholarly article
title: Extensions of zip rings.
zbMATH Open document ID: 1071.16020
author: Chan Yong Hong
author: Nam Kyun Kim
author: Tai Keun Kwak
author: Yang Lee
published in: Journal of Pure and Applied Algebra
publication date: 2005-02-22T00:00:00Z
review text: Throughout \(R\) is an associative ring with identity. Denote by \(UTM_n(R)\) the \(n\times n\) upper triangular matrix ring over a ring \(R\) and by \(aUTM_n(R)\) (\(n\geq 2\)) the ring having as elements upper triangular matrices with entries in \(R\) and all the entries on the main diagonal equal. A ring \(R\) is called right zip provided that if the right annihilator \(r_R(X)\) of a subset \(X\) of \(R\) is zero, then \(r_R(Y)=0\) for a finite subset \(Y\subseteq X\). \textit{C. Faith} raised [in Commun. Algebra 19, No. 7, 1867-1892 (1991; Zbl 0729.16015)] a number of questions on zip rings, in connection with which the authors establish a series of results. It is proved that a ring \(R\) is a right (left) zip ring if and only if \(aUTM_n(R)\) is a right (left) zip ring. An Armendariz ring \(R\) is a right zip ring if and only if \(R[x]\) is a right zip ring. Given a monoid \(G\) and a ring \(R\), denote by \(R[G]\) the monoid ring of \(G\) over \(R\). A monoid \(G\) is called a u.p. monoid (unique product monoid) if given any two non-empty finite subsets \(A\) and \(B\) of \(G\) there exists at least one \(c\in G\) that has a unique representation in the form \(c=ab\) with \(a\in A\) and \(b\in B\). The authors show that, for a commutative ring \(R\) and a u.p. monoid \(G\) that contains an infinite cyclic submonoid, \(R\) is a zip ring if and only if \(R[G]\) is a zip ring.
reviewed by: Septimiu Crivei
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20M25
zbMATH DE Number: 2136954
zbMATH Keywords: right zip rings
zbMATH Keywords: Armendariz rings
zbMATH Keywords: u.p. monoids
zbMATH Keywords: upper triangular matrix rings
zbMATH Keywords: right annihilators
zbMATH Keywords: monoid rings
MaRDI profile type: Publication
full work available at URL: j.jpaa.2004.08.025
OpenAlex ID: W2086686051
cites work: Armendariz rings and gaussian rings
cites work: Rings whose faithful left ideals are cofaithful
cites work: Zip rings and mal'cev domains
cites work: Rings with zero intersection property on annihilators: zip rings
cites work: On annihilator ideals of a polynomial ring over a noncommutative ring
cites work: Armendariz rings and reduced rings
cites work: Q4113971
cites work: Armendariz rings
cites work: The Finite Intersection Property on Annihilator Right Ideals
DOI: 10.1016/J.JPAA.2004.08.025
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description: scientific article; zbMATH DE number 2136954