Czechoslovak Mathematical Journal, Vol. 67, No. 1, pp. 151-169, 2017

Some results on the annihilator graph of a commutative ring

Mojgan Afkhami, Kazem Khashyarmanesh, Zohreh Rajabi

Received August 13, 2015.   First published February 24, 2017.

Mojgan Afkhami, Department of Mathematics, University of Neyshabur, P. O. Box 91136-899, Neyshabur, Iran, e-mail: mojgan.afkhami@yahoo.com; Kazem Khashyarmanesh, Zohreh Rajabi, Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran, e-mail: Khashyar@ipm.ir, rajabi261@yahoo.com

Abstract: Let $R$ be a commutative ring. The annihilator graph of $R$, denoted by ${\rm AG}(R)$, is the undirected graph with all nonzero zero-divisors of $R$ as vertex set, and two distinct vertices $x$ and $y$ are adjacent if and only if ${\rm ann}_R(xy) \neq{\rm ann}_R(x)\cup{\rm ann}_R(y)$, where for $z \in R$, ${\rm ann}_R(z) = \lbrace r \in R \colon rz = 0\rbrace$. In this paper, we characterize all finite commutative rings $R$ with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings $R$ whose annihilator graphs have clique number $1$, $2$ or $3$. Also, we investigate some properties of the annihilator graph under the extension of $R$ to polynomial rings and rings of fractions. For instance, we show that the graphs ${\rm AG}(R)$ and ${\rm AG}(T(R))$ are isomorphic, where $T(R)$ is the total quotient ring of $R$. Moreover, we investigate some properties of the annihilator graph of the ring of integers modulo $n$, where $n \geq1$.

Keywords: annihilator graph; zero-divisor graph; outerplanar; ring-graph; cut-vertex; clique number; weakly perfect; chromatic number; polynomial ring; ring of fractions

Classification (MSC 2010): 05C75, 13A99, 05C99

DOI: 10.21136/CMJ.2017.0436-15

Full text available as PDF.


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