The prime ideals intersection graph of a ring

Nikmehr, M. J., and Soleymanzadeh, B.. "The prime ideals intersection graph of a ring." Commentationes Mathematicae Universitatis Carolinae 58.2 (2017): 137-145. <http://eudml.org/doc/288184>.

@article{Nikmehr2017,
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 $G_\{p\}(R)$.},
author = {Nikmehr, M. J., Soleymanzadeh, B.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {the prime ideals intersection graph of a ring; clique number; planar graph},
language = {eng},
number = {2},
pages = {137-145},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The prime ideals intersection graph of a ring},
url = {http://eudml.org/doc/288184},
volume = {58},
year = {2017},
}

TY - JOUR
AU - Nikmehr, M. J.
AU - Soleymanzadeh, B.
TI - The prime ideals intersection graph of a ring
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2017
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 58
IS - 2
SP - 137
EP - 145
AB - 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 $G_{p}(R)$.
LA - eng
KW - the prime ideals intersection graph of a ring; clique number; planar graph
UR - http://eudml.org/doc/288184
ER -