The total graph of a commutative ring
type: Item
instance of: scholarly article
title: The total graph of a commutative ring
zbMATH Open document ID: 1158.13001
author: David F. Anderson
author: Ayman Badawi
published in: Journal of Algebra
publication date: 2008-12-02T00:00:00Z
review text: Let \(R\) be a commutative ring with \(Nil(R)\) its ideal of nilpotent elements, \(Z(R)\) its set of zero-divisors, and \(Reg(R)\) its set regular elements. Let \(T(\Gamma(R))\) be the total graph of \(R\), \(Reg(\Gamma(R))\) be the subgraph of \(T(\Gamma(R))\) with vertices \(Reg(R)\), \(Z(\Gamma(R))\) be the subgraph of \(T(\Gamma(R))\) with vertices \(Z(R)\) and \(Nil(\Gamma(R))\) be the subgraph of \(T(\Gamma(R))\) with vertices \(Nil(R)\). The main goal of the first section to provide general structure theorem for \(Reg(\Gamma(R))\), \(Z(\Gamma (R))\) and \(Nil(\Gamma(R))\), in case where \(Z(R)\) is an ideal of \(R\), \(Z(\Gamma(R))\) is a complete subgraph of \(T(\Gamma(R))\) and is disjoint from \(Reg(\Gamma(R))\). In the first section, the authors complete the diameter and the girth of \(Reg(\Gamma(R))\), and show that \(Z(\Gamma(R))\) is always connected and \(T(\Gamma(R))\) is never connected if \(Z(R)\) is an ideal. Finally, the basic of properties of \(Nil(\Gamma(R))\) are given below: Let \(R\) be a commutative ring: (1) \(Nil(\Gamma(R))\) is a complete (induced) subgraph of \(Z(\Gamma(R))\); (2) Each vertex of \(Nil(\Gamma(R))\) is adjacent to each distinct vertex; (3) \(Nil(\Gamma(R))\) is disjoint from \(Reg(\Gamma(R))\); (4) If \(\{0\}\neq Nil(R)\subseteq Z(R)\), then \(gr(Z(\Gamma(R)))=3\). Section 2 traits the second case of \(Z(R)\), i.e., when \(Z(R)\) is not an ideal of \(R\). It shows that \(Z(\Gamma(R))\) is always connected but never complete, \(Z(\Gamma(R))\) and \(Reg(\Gamma(R))\) are never disjoint subgraphs of \(T(\Gamma(R))\) and \(|Z(R)|>3\). The authors conclude this section by the investigation of the girth of \(Z(\Gamma(R))\), \(Reg(\Gamma(R))\) and \(T(\Gamma(R))\) when \(Z(R)\) is not an ideal of \(R\).
reviewed by: Driss Karim
Mathematics Subject Classification ID: 13A99
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13B10
Mathematics Subject Classification ID: 13C99
zbMATH DE Number: 5376932
zbMATH Keywords: Commutative rings
zbMATH Keywords: Zero-divisor graph
zbMATH Keywords: Regular elements
zbMATH Keywords: Total graph
zbMATH Keywords: Zero-divisors.
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.06.028
OpenAlex ID: W2044882319
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DOI: 10.1016/J.JALGEBRA.2008.06.028
description: scientific article; zbMATH DE number 5376932