The prime ideals intersection graph of a ring

Nikmehr, M. J. ; Soleymanzadeh, B.

Commentationes Mathematicae Universitatis Carolinae, Tome 58 (2017) no. 2, pp. 137-145.

Résumé

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 $G_{p}(R)$.

Détail

DOI : 10.14712/1213-7243.2015.205

Classification : 05C40, 05C69, 13A15
Keywords: the prime ideals intersection graph of a ring; clique number; planar graph