The principal small intersection graph of a commutative ring


  • Soheila Khojasteh Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran



Let $R$ be a commutative ring with non-zero identity. The small intersection graph of $R$, denoted by $G(R)$, is a graph with the vertex set $V(G(R))$, where $V(G(R))$ is the set of all proper non-small ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I \cap J$ is not small in $R$. In this paper, we introduce a certain subgraph $PG(R)$ of $G(R)$, called the principal small intersection graph of $R$. It is the subgraph of $G(R)$ induced by the set of all proper principal non-small ideals of $R$. We study the diameter, the girth, the clique number, the independence number and the domination number of $PG(R)$. Moreover, we present some results on the complement of the principal small intersection graph.


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