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On directed zero-divisor graphs of finite rings

Tongsuo Wu
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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 $R$.

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Tongsuo Wu. 2004-03-13. On directed zero-divisor graphs of finite rings. https://arxiv.org/abs/math/0403062

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