ON THE REGULAR DIGRAPH OF IDEALS OF COMMUTATIVE RINGS

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...

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Bibliographic Details
Published inBulletin of the Australian Mathematical Society Vol. 88; no. 2; pp. 177 - 189
Main Authors AFKHAMI, M., KARIMI, M., KHASHYARMANESH, K.
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.10.2013
Subjects
Apexes
Commutativity
Graph theory
Rings (mathematics)
Vertex sets
13E05
connectedness
diameter
regular digraph
16P20
primary 05C20; secondary 05C69
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Summary: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 $\Gamma (R)$.
ISSN:0004-9727
1755-1633
DOI:10.1017/S0004972712000792