Some results on the annihilator graph of a commutative ring

Afkhami, Mojgan ; Khashyarmanesh, Kazem ; Rajabi, Zohreh

Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 151-169.

Résumé

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 $n \geq 1$.

Détail

DOI : 10.21136/CMJ.2017.0436-15

Classification : 05C75, 05C99, 13A99
Keywords: annihilator graph; zero-divisor graph; outerplanar; ring-graph; cut-vertex; clique number; weakly perfect; chromatic number; polynomial ring; ring of fractions