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Published
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On the module intersection graph of ideals of rings
Volume 63, no. 1
(2022),
pp. 93–107
https://doi.org/10.33044/revuma.1936
Abstract
Let $R$ be a commutative ring and $M$ an $R$-module. The $M$-intersection
graph of ideals of $R$ is an undirected simple graph, denoted by $G_{M}(R)$,
whose vertices are non-zero proper ideals of $R$ and two distinct vertices are
adjacent if and only if $IM\cap JM\neq 0$. In this article, we focus on how
certain graph theoretic parameters of $G_M(R)$ depend on the properties of
both $R$ and $M$. Specifically, we derive a necessary and sufficient condition
for $R$ and $M$ such that the $M$-intersection graph $G_M(R)$ is either
connected or complete.
Also, we classify all $R$-modules according to the diameter value of $G_M(R)$.
Further, we characterize rings $R$ for which $G_M(R)$ is perfect or
Hamiltonian or pancyclic or planar. Moreover, we show that the graph
$G_{M}(R)$ is weakly perfect and cograph.
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Published by the Unión Matemática Argentina |
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