About this Journal Submit a Manuscript Table of Contents
ISRN Discrete Mathematics
Volume 2011 (2011), Article ID 459547, 7 pages
http://dx.doi.org/10.5402/2011/459547
Research Article

Zero-Divisor Graphs with respect to Ideals in Noncommutative Rings

1Department of Mathematics, University of Guilan, P.O. Box 1914, Rasht, Iran
2Department of Mathematics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil 56199-11367, Iran

Received 15 July 2011; Accepted 25 August 2011

Academic Editor: W. F. Klostermeyer

Copyright © 2011 Shahabaddin Ebrahimi Atani and Ahmad Yousefian Darani. 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 and 𝐼 an ideal of 𝑅 . The zero-divisor graph of 𝑅 with respect to 𝐼 , denoted Ξ“ 𝐼 ( 𝑅 ), is the undirected graph whose vertex set is { π‘₯ ∈ 𝑅 ⧡ 𝐼 | π‘₯ 𝑦 ∈ 𝐼 for some 𝑦 ∈ 𝑅 ⧡ 𝐼 } with two distinct vertices π‘₯ and 𝑦 joined by an edge when π‘₯ 𝑦 ∈ 𝐼 . In this paper, we extend the definition of the ideal-based zero-divisor graph to noncommutative rings.