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ON THE REGULAR DIGRAPH OF IDEALS OF COMMUTATIVE RINGS

Part of: Chain conditions, growth conditions, and other forms of finiteness Chain conditions, finiteness conditions Graph theory

Published online by Cambridge University Press:  16 October 2012

M. AFKHAMI ,
M. KARIMI  and
K. KHASHYARMANESH
M. AFKHAMI*
Affiliation:
Department of Mathematics, University of Neyshabur, PO Box 91136-899, Neyshabur, Iran (email: [email protected])
M. KARIMI
Affiliation:
Department of Pure Mathematics, Ferdowsi University of Mashhad, PO Box 1159-91775, Mashhad, Iran (email: [email protected])
K. KHASHYARMANESH
Affiliation:
Department of Pure Mathematics, Ferdowsi University of Mashhad, PO Box 1159-91775, Mashhad, Iran (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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Let $R$ be a commutative ring. The regular digraph of ideals of $R$, denoted by $\Gamma (R)$, is a digraph whose vertex set is the set of all nontrivial ideals of $R$ and, for every two distinct vertices $I$ and $J$, there is an arc from $I$ to $J$ whenever $I$ contains a nonzero divisor on $J$. In this paper, we study the connectedness of $\Gamma (R)$. We also completely characterise the diameter of this graph and determine the number of edges in $\Gamma (R)$, whenever $R$ is a finite direct product of fields. Among other things, we prove that $R$ has a finite number of ideals if and only if $\mathrm {N}_{\Gamma (R)}(I)$ is finite, for all vertices $I$ in $\Gamma (R)$, where $\mathrm {N}_{\Gamma (R)}(I)$ is the set of all adjacent vertices to $I$ in $\Gamma (R)$.

Keywords

regular digraphconnectednessdiameter

MSC classification

Secondary: 05C20: Directed graphs (digraphs), tournaments 13E05: Noetherian rings and modules 16P20: Artinian rings and modules
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 2 , October 2013 , pp. 177 - 189
Copyright
Copyright © 2012 Australian Mathematical Publishing Association Inc. 

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