When Is the Complement of the Comaximal Graph of a Commutative Ring Planar?

S Visweswaran,Jaydeep Parejiya

doi:10.1155/2014/736043

Abstract

Let R be a commutative ring with identity. In this paper we classify rings R such that the complement of comaximal graph of R is planar. We also consider the subgraph of the complement of comaximal graph of R induced on the set S of all nonunits of R with the property that each element of S is not in the Jacobson radical of R and classify rings R such that this subgraph is planar.

Highlights

All rings considered in this paper are commutative with identity 1 ≠ 0
Sharma and Bhatwadekar in [1] introduced a graph Ω(R) on a commutative ring R, whose vertices are the elements of R and two distinct vertices x and y are adjacent if and only if Rx + Ry = R
In [1, Theorem 2.3], the authors showed that χ(Ω(R)) < ∞ if and only if the ring R is finite and in this case χ(Ω(R)) = ω(Ω(R)) = t + l, where t and l denote, respectively, the number of maximal ideals of R and number of units of R

Summary

Introduction

All rings considered in this paper are commutative with identity 1 ≠ 0. In [6], Gaur and Sharma studied the graph whose vertices are the elements of a ring R and two distinct vertices x and y are adjacent if and only if there exists a maximal ideal of R containing both x and y. If R is a ring with exactly three maximal ideals, it is shown in Theorem 24 that (G(R))c satisfies (G(R))c

Results

Conclusion