On endomorphism-regularity of zero-divisor graphs
type: Item
instance of: scholarly article
title: On endomorphism-regularity of zero-divisor graphs
zbMATH Open document ID: 1156.05025
author: Dancheng Lu
author: Tongsuo Wu
published in: Discrete Mathematics
publication date: 2008-09-04T00:00:00Z
review text: Let \(G\) be a graph and \(\text{End}(G)\) the semigroup consisting of all the endomorphisms of \(G\). An element \(a\) of a semigroup \(S\) is called regular if \(a=aba\) for some \(b\in S\), and \(S\) is called regular if every element in \(S\) is regular. A graph \(G\) is called end-regular if \(\text{End}(G)\) is regular. The zero-divisor graph of a ring \(R\), denoted by \(\Gamma(R)\), is a graph whose vertex set \(Z(R)^\ast\) consisting of all non-zero zero-divisors of \(R\) and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(xy=0\). Let \(R\) be a ring containing nontrivial idempotents. Then the main result of the present paper is to prove that \(\Gamma(R)\) is end-regular if and only if \(R\) is isomorphic to one of the following rings: \(\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2\); \(\mathbb{Z}_2\times\mathbb{Z}_4\); \(\mathbb{Z}_2\times(\mathbb{Z}_2[x]/(x^2))\); \(F_1\times F_2\), where \(F_1\), \(F_2\) are fields.
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 5321355
zbMATH Keywords: graph
zbMATH Keywords: ring
zbMATH Keywords: zero-divisor graph
zbMATH Keywords: endomorphisms
MaRDI profile type: Publication
full work available at URL: j.disc.2007.08.057
OpenAlex ID: W1984864135
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DOI: 10.1016/J.DISC.2007.08.057
description: scientific article; zbMATH DE number 5321355