A graph over the commutative rings
type: Item
instance of: scholarly article
title: A graph over the commutative rings
zbMATH Open document ID: 1390.13041
author: Leila Hamedi-Mobarra
author: Dawood Hassanzadeh Lelekaami
author: H. Roshan-Shekalgourabi
published in: Boletín de la Sociedad Matemática Mexicana. Third Series
publication date: 2018-02-06T00:00:00Z
review text: Let \(R\) be a commutative ring with identity, and \(M\) be an \(R\)-module. After several graph constructions from commutative rings, there are studies which associate graphs with module. More specifically zero-divisor graph of modules are studied in the recent past. The comaximal graph of a commutative ring \(R\) is the simple undirected graph with vertex set \(R\) and two distinct vertices \(a\) and \(b\) are adjacent if \(Ra+Rb=R\). Motivated by this, the authors of this paper, associate a graph \(G(R,M)\) as the graph with vertex set \(R\) where two distinct elements \(a\) and \(b\) are adjacent if and only if \(aM+bM=M.\) Obviously, \(G(R,R)\) is the comaximal graph of \(R\) and \(G(\{0\},M)\) is the complete graph. Having introduced the graph \(G(R,M),\) the authors considered the relationship between the completeness of subgraphs of \(G(R,M)\) and algebraic properties of \(R\) and \(M\). In addition, they studied the bipartite and complete bipartite subgraphs of the associated graph. Also they have characterized the diameter of certain subgraphs of \(G(R,M)\).
reviewed by: T. Tamizh Chelvam
Mathematics Subject Classification ID: 13C99
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 05C10
zbMATH DE Number: 6835187
zbMATH Keywords: associated graph
zbMATH Keywords: connected graph
zbMATH Keywords: complete bipartite graph
MaRDI profile type: Publication
full work available at URL: s40590-016-0132-8
OpenAlex ID: W2430163061
cites work: When a zero-divisor graph is planar or a complete \(r\)-partite graph
cites work: Zero-divisor graphs of non-commutative rings.
cites work: On the Zero-Divisor Graph of a Ring
cites work: The zero-divisor graph of a commutative ring
cites work: On the diameter and girth of a zero-divisor graph
cites work: Q4023148
cites work: Zero-Divisor Graphs of Polynomials and Power Series Over Commutative Rings
cites work: Zero-divisor graph of \(C(X)\)
cites work: Coloring of commutative rings
cites work: Q4380359
cites work: Complemented zero-divisor graphs and Boolean rings
cites work: Q3695368
cites work: Q5297727
cites work: Comaximal graph of commutative rings
cites work: A note on graphical representation of rings
cites work: Graphs associated to co-maximal ideals of commutative rings
cites work: Q4002547
cites work: Koatomare Moduln
DOI: 10.1007/S40590-016-0132-8
description: scientific article; zbMATH DE number 6835187