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Keywords:
the prime ideals intersection graph of a ring; clique number; planar graph
the prime ideals intersection graph of a ring; clique number; planar graph
Summary:
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 $G_{p}(R)$.
References:
[1] Akbari S., Nikandish R., Nikmehr M.J.: Some results on the intersection graphs of ideals of rings. J. Algebra Appl. 12 (2013), no. 4, 1250200. DOI 10.1142/S0219498812502003 | MR 3037274 | Zbl 1264.05056
[2] Anderson D.F., Badawi A.: The generalized total graph of a commutative ring. J. Algebra Appl. 12 (2013), no. 5, 1250212. DOI 10.1142/S021949881250212X | MR 3055569 | Zbl 1272.13003
[3] Anderson D.F., Badawi A.: On the total graph of a commutative ring without the zero element. J. Algebra Appl. 11 (2012), no. 4, 1250074. DOI 10.1142/S0219498812500740 | MR 2959423 | Zbl 1247.13004
[4] Anderson D.F., Badawi A.: The total graph of a commutative ring. J. Algebra 320 (2008), no. 7, 2706–2719. DOI 10.1016/j.jalgebra.2008.06.028 | MR 2441996 | Zbl 1247.13004
[5] Anderson D.F., Livingston P.: The zero-divisor graph of a commutative ring. J. Algebra 217 (1999), no. 2, 434–447. DOI 10.1006/jabr.1998.7840 | MR 1700509 | Zbl 1035.13004
[6] Atiyah M.F., Macdonald I.G.: Introduction to Commutative Algebra. Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802 | Zbl 0238.13001
[7] Beck I.: Coloring of commutative rings. J. Algebra 116 (1988), no. 1, 208–226. DOI 10.1016/0021-8693(88)90202-5 | MR 0944156 | Zbl 0654.13001
[8] Corbas B., Williams G.D.: Rings of order $p^5$, I. Nonlocal rings. J. Algebra 231 (2000), 677-690. DOI 10.1006/jabr.2000.8349 | MR 1778165 | Zbl 1017.16014
[9] Raghavendran R.: Finite associative rings. Compositio Math. 21 (1969), 195–229. MR 0246905 | Zbl 0179.33602
[10] Syslo M.M.: Characterizations of outerplanar graphs. Discrete Math. 26 (1979), no. 1, 47–53. DOI 10.1016/0012-365X(79)90060-8 | MR 0535083 | Zbl 0401.05040
[11] West D.B.: Introduction to Graph Theory. second edition, Prentice Hall, Upper Saddle River, 2001. MR 1367739 | Zbl 1121.05304
[12] Lu D., Wu T.: On bipartite zero-divisor graphs. Discrete Math. 309 (2009), no. 4, 755–762. DOI 10.1016/j.disc.2008.01.044 | MR 2502185
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