Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2017; 54(1): 331-342

Online first article November 3, 2016      Printed January 31, 2017

https://doi.org/10.4134/BKMS.b160135

Copyright © The Korean Mathematical Society.

On the annihilator graph of group rings

Mojgan Afkhami, Kazem Khashyarmanesh, and Sepideh Salehifar

University of Neyshabur, Ferdowsi University of Mashhad, Ferdowsi University of Mashhad

Abstract

Let $R$ be a commutative ring with nonzero identity and $G$ be a nontrivial finite group. Also, let $Z(R)$ be the set of zero-divisors of $R$ and, for $a\in Z(R)$, let $\T{ann}(a) = \lbrace r\in R\ \vert \ ra=0\rbrace$. The annihilator graph of the group ring $RG$ is defined as the graph $AG(RG)$, whose vertex set consists of the set of nonzero zero-divisors, and two distinct vertices $x$ and $y$ are adjacent if and only if $\T{ann}(xy)\neq \T{ann}(x) \cup \T{ann}(y)$. In this paper, we study the annihilator graph associated to a group ring $RG$.

Keywords: zero-divisor graph, annihilator graph, bipartite graph, planar graph, line graph

MSC numbers: 05C69, 05C75, 13A15