Annihilator Digraphs and Extended Zero-Divisor Digraphs of Semigroups and Rings

Abstract

Let $S$ be a semigroup with zero. This paper studies the zero-divisor digraph $\overrightarrow{\Gamma}(S)$ and the extended zero-divisor digraph $\overrightarrow{\Gamma}_{\!E}$, and introduces the annihilator digraph $\overrightarrow{\mathrm{AG}}(S)$ via left and right annihilators. The diameter bound when every zero-divisor is nilpotent, and the sinks and the sources of $\overrightarrow{\Gamma}_{\!E}(S)$ being identical to those of $\overrightarrow{\Gamma}(S)$ is demonstrated. The conditions in which $\overrightarrow{\Gamma}(S)=\overrightarrow{\Gamma}_{\!E}(S)=\overrightarrow{\mathrm{AG}}(S)$ holds are established, and the connectedness, diameter, girth, and vertex degrees of $\overrightarrow{\mathrm{AG}}(S)$ are bounded when every zero-divisor is nilpotent or two-sided. The extended zero-divisor digraph is connected if and only if the zero-divisor digraph is connected, and it contains a directed cycle if and only if the zero-divisor digraph does. The knit degrees of $\overrightarrow{\Gamma}(S)$, $\overrightarrow{\Gamma}_{\!E}(S)$, and $\overrightarrow{\mathrm{AG}}(S)$ are computed. For a unital ring $R$, the equality $\overrightarrow{\Gamma}_{\!E}(R)=\overrightarrow{\Gamma}(R)$ is characterized by nilpotency indices and one-sided annihilator conditions; it holds for the full matrix ring $M_{n}(F)$ over a field $F$ if and only if $n=2$, and $\overrightarrow{\mathrm{AG}}(M_{n}(F))$ is connected and contains a directed cycle. Moreover, for an artinian noncommutative ring $R$, it was proved that $\overrightarrow{\Gamma}(R)$ is connected if and only if $\overrightarrow{\Gamma}_{\!E}(R)$ is connected if and only if every one-sided identity element of $R$ is a two-sided identity of $R$.