When Is the Complement of the Zero-Divisor Graph of a Commutative Ring Complemented?

S Visweswaran

doi:10.5402/2012/282054

Abstract

Let be a commutative ring with identity which has at least two nonzero zero-divisors. Suppose that the complement of the zero-divisor graph of has at least one edge. Under the above assumptions on , it is shown in this paper that the complement of the zero-divisor graph of is complemented if and only if is isomorphic to as rings. Moreover, if is not isomorphic to as rings, then, it is shown that in the complement of the zero-divisor graph of , either no vertex admits a complement or there are exactly two vertices which admit a complement.

Highlights

The rings considered in this paper are commutative rings with identity satisfying the further condition that there exist two distinct zero-divisors whose product is nonzero
Let R be a commutative ring with identity
Recall from 1 that the zero-divisor graph of R denoted by Γ R is an undirected graph whose vertex set is the set of all nonzero zero-divisors of R and two distinct nonzero zero-divisors x, y of R are joined by an edge in this graph if and only if xy 0

Summary

Introduction

The rings considered in this paper are commutative rings with identity satisfying the further condition that there exist two distinct zero-divisors whose product is nonzero. Let R be a commutative ring with identity. Several researchers investigated the properties of zero-divisor graphs of commutative rings with identity. The following survey article 2 gives a very clear account of the problems solved in the area of zero-divisor graphs of commutative rings along with necessary history of the problems attempted in this area. Recall from 3, 4 that two distinct vertices a, b of G are said to be orthogonal, written a ⊥ b if a and b are adjacent in G and there is no vertex c of G which is adjacent to both a and b in G; that is, the edge a − b of G is not a part of any triangle in G.

ISRN Algebra

As Z Since