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 $R$.

19 pages, 2 figures