Structural Properties of the Essential Ideal Graph of &lt;img src=image/13424240_01.gif&gt;

P Jamsheena,A V Chithra

doi:10.13189/ms.2021.090521

Abstract

Let <img src=image/13424240_02.gif> be a commutative ring with unity. The essential ideal graph of <img src=image/13424240_02.gif>, denoted by <img src=image/13424240_03.gif>, is a graph with vertex set consisting of all nonzero proper ideals of A and two vertices <img src=image/13424240_04.gif> and <img src=image/13424240_05.gif> are adjacent whenever <img src=image/13424240_06.gif> is an essential ideal. An essential ideal <img src=image/13424240_04.gif> of a ring <img src=image/13424240_02.gif> is an ideal <img src=image/13424240_04.gif> of <img src=image/13424240_02.gif> (<img src=image/13424240_07.gif>), having nonzero intersection with every other ideal of <img src=image/13424240_02.gif>. The set <img src=image/13424240_08.gif> contains all the maximal ideals of <img src=image/13424240_02.gif>. The Jacobson radical of <img src=image/13424240_02.gif>, <img src=image/13424240_09.gif>, is the set of intersection of all maximal ideals of <img src=image/13424240_02.gif>. The comaximal ideal graph of <img src=image/13424240_02.gif>, denoted by <img src=image/13424240_11.gif>, is a simple graph with vertices as proper ideals of A not contained in <img src=image/13424240_09.gif> and the vertices <img src=image/13424240_04.gif> and <img src=image/13424240_05.gif> are associated with an edge whenever <img src=image/13424240_10.gif>. In this paper, we study the structural properties of the graph <img src=image/13424240_03.gif> by using the ring theoretic concepts. We obtain a characterization for <img src=image/13424240_03.gif> to be isomorphic to the comaximal ideal graph <img src=image/13424240_11.gif>. Moreover, we derive the structure theorem of <img src=image/13424240_12.gif> and determine graph parameters like clique number, chromatic number and independence number. Also, we characterize the perfectness of <img src=image/13424240_12.gif> and determine the values of <img src=image/13424240_13.gif> for which <img src=image/13424240_12.gif> is split and claw-free, Eulerian and Hamiltonian. In addition, we show that the finite essential ideal graph of any non-local ring is isomorphic to <img src=image/13424240_12.gif> for some <img src=image/13424240_13.gif>.

Highlights

In 1998, Beck initiated the concept of identifying a graph structure from a commutative ring structure and studied a graph formed out of zero divisors of a commutative ring[8]
The comaximal ideal graph of S, denoted by C(S), is a simple graph with vertices as proper ideals of A not contained in J(S) and the vertices P and Q are associated with an edge whenever P + Q = S
Wu[18] set out to explore the comaximal ideal graph of a commutative ring C(S) in 2012. It is a simple graph with vertex set consisting of the proper ideals of A not contained in J(S) and an edge is made between the vertices P and Q if and only if they are comaximal

Summary

Introduction

In 1998, Beck initiated the concept of identifying a graph structure from a commutative ring structure and studied a graph formed out of zero divisors of a commutative ring[8]. Wu[18] set out to explore the comaximal ideal graph of a commutative ring C(S) in 2012 It is a simple graph with vertex set consisting of the proper ideals of A not contained in J(S) and an edge is made between the vertices P and Q if and only if they are comaximal. This motivates the study of the essential ideal graph of a commutative ring with unity. We compute the matching number, covering number, clique number, chromatic number and independence number of the essential ideal graph of the commutative ring S = F1 × F2 × · · · × Fn, n ≥ 2, where each Fi is a field for i = 1, 2, · · · , n.

Preliminaries

Properties of the Essential Ideal Graph

Isomorphism properties of ES

Conclusion