## M. J. Nikmehr, B. SoleymanzadehThe prime ideals intersection graph of a ring

Comment.Math.Univ.Carolin. 58,2 (2017) 137-145.

Abstract:Let $R$ be a commutative ring with unity and $U(R)$ be the set of unit elements of $R$. In this paper, we introduce and investigate some properties of a new kind of graph on the ring $R$, namely, the prime ideals intersection graph of~$R$, denoted by $G_{p}(R)$. The $G_{p}(R)$ is a graph with vertex set $R^*-U(R)$ and two distinct vertices $a$ and $b$ are adjacent if and only if there exists a prime ideal $\mathfrak{p}$ of $R$ such that $a,b\in \mathfrak{p}$. We obtain necessary and sufficient conditions on $R$ such that $G_{p}(R)$ is disconnected. We find the diameter and girth of $G_{p}(R)$. We also determine all rings whose prime ideals intersection graph is a star, path, or cycle. At the end of this paper, we study the planarity and outerplanarity of~$G_{p}(R)$.

Keywords: the prime ideals intersection graph of a ring; clique number; planar graph

DOI: DOI 10.14712/1213-7243.2015.205
AMS Subject Classification: 05C40 05C69 13A15

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