Table of Contents
ISRN Algebra
Volume 2012, Article ID 282054, 13 pages
http://dx.doi.org/10.5402/2012/282054
Research Article

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

Department of Mathematics, Saurashtra University, Rajkot 360 005, India

Received 12 March 2012; Accepted 3 April 2012

Academic Editors: D. Anderson, A. V. Kelarev, and C. Munuera

Copyright © 2012 S. Visweswaran. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝑅 be a commutative ring with identity which has at least two nonzero zero-divisors. Suppose that the complement of the zero-divisor graph of 𝑅 has at least one edge. Under the above assumptions on 𝑅 , it is shown in this paper that the complement of the zero-divisor graph of 𝑅 is complemented if and only if 𝑅 is isomorphic to 𝐙 / 3 𝐙 Γ— 𝐙 / 3 𝐙 as rings. Moreover, if 𝑅 is not isomorphic to 𝐙 / 3 𝐙 Γ— 𝐙 / 3 𝐙 as rings, then, it is shown that in the complement of the zero-divisor graph of 𝑅 , either no vertex admits a complement or there are exactly two vertices which admit a complement.