Prime intersection graph of ideals of a ring
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Prime intersection graph of ideals of a ring KUKIL KALPA RAJKHOWA1,∗ and HELEN K SAIKIA2 1 Department of Mathematics, Cotton University, Guwahati 781 001, India 2 Department of Mathematics, Gauhati University, Guwahati 781 014, India
*Corresponding author. E-mail: [email protected]; [email protected]
MS received 8 November 2018; revised 29 March 2019; accepted 26 July 2019 Abstract. Let R be a ring. The prime intersection graph of ideals of R, denoted by G P (R), is the graph whose vertex set is the collection of all non-trivial (left) ideals of R with two distinct vertices I and J are adjacent if and only if I ∩ J = 0 and either one of I or J is a prime ideal of R. We discuss connectedness in G P (R). The concepts of bipartition, planarity and colorability are interpreted. Finally, we introduce the idea of traversability in G P (Zn ). The core part of this paper is observed in the ring Zn . Keywords.
Prime intersection graph; ring; prime ideal; connected graph.
2010 Mathematics Subject Classification.
05C25, 13C99.
1. Introduction In 2008, Chakrabarty et al. [8] introduced the motivating insight of graphical aspect of algebraic structures, namely intersection graphs of ideals of rings. In that intersection graph, the vertex set is the collection of non-trivial ideals of a ring and any two vertices are adjacent if their intersection is non-zero. They observed almost all fundamental concepts of the intersection graphs of ideals of rings. The central part of their interpretation depended upon the ring Zn . After that introduction, Akbari et al. [1] studied some more results of intersection graph of ideals of rings. They interestingly noticed characteristics between the graph-theoretic properties of this graph and some algebraic properties of rings. In [2], Akbari et al. extended the concept of intersection graphs of ideals of rings into intersection graphs of submodules of modules. Akbari et al. [3] also discussed on the complement of the intersection graph of submodules of a module. Rajkhowa and Saikia [18] generalized the intersection graph introduced by Chakrabarty et al. [8] and investigated the features of the center in that graph. Some more discussions of intersection graphs can be found in [4,9,14,19]. In this paper, we introduce the prime intersection graph of ideals of rings. Let R be a ring. Then the prime intersection graph of R, denoted by G P (R), is an undirected graph with vertex set as the collection of non-trivial (left) ideals of R and any two vertices I, J are adjacent if and only if I ∩ J = 0 and one of I and J is a prime ideal of R. The vital part of this discussion is based on the ring Zn and continues for generalized interpreta© Indian Academy of Sciences 0123456789().: V,-vol
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Proc. Indian Acad. Sci. (Math. Sci.)
(2020) 130:17
tion. We first study the aspects of connectedness in the prime intersection graph G P (Zn ) and then proceed for the same for an arbitrary ring. We also find the diameter, completeness character for the prime intersection graph G P (Zn
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