On directed zero-divisor graphs of finite rings

Abstract

For an artinian ring $R$, the directed zero-divisor graph $Γ(R)$ is connected if and only if there is no proper one-sided identity element in $R$. Sinks and sources are characterized and clarified for finite ring $R$, especially, it is proved that for {\it any} ring $R$, if there exists a source $b$ in $Γ(R)$ with $b^2=0$, then $|R|=4$ and $R=\{0,a,b,c\}$, where $a$ and $c$ are left identity elements and $ba=0=bc$. Such a ring $R$ is also the only ring such that $Γ(R) $ has exactly one source. This shows that $Γ(R)$ can not be a network for any ring $R$.