Properties of annihilator graph of a commutative semigroup

Yahya Talebi ; Sahar Akbarzadeh

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2018), pp. 22-31

Résumé

Let $S$ be a commutative semigroup with zero. Let $Z(S)$ be the set of all zero-divisors of $S$. We define the annihilator graph of $S$, denoted by $ANN_{G}(S)$, as the undirected graph whose set of vertices is $Z(S)^{\ast}=Z(S)-\{0\}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $ann_{S}(xy)\neq ann_{S}(x)\cap ann_{S}(y)$. In this paper, we study some basic properties of $ANN_{G}(S)$ by means of $\Gamma(S)$. We also show that if $Z(S)\neq S$, then $ANN_{G}(S)$ is a subgraph of $\Gamma(S)$. Moreover, we investigate some properties of the annihilator graph $ANN_{G}(S)$ related to minimal prime ideals of $S$. We also study some connections between the domination numbers of annihilator graphs and zero-divisor graphs.