On directed zero-divisor graphs of finite rings Original Research Article

For an artinian ring R, the directed zero-divisor graph Γ(R)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 a finite ring R. Especially, it is proved that for any ring R, if there exists a source y in Γ(R)Γ(R) with y2=0y2=0, then |R|=4|R|=4 and R={0,x,y,z}R={0,x,y,z}, where x and z are left identity elements and yx=0=yzyx=0=yz. Such a ring R is also the only ring such that Γ(R)Γ(R) has exactly one source. This shows that Γ(R)Γ(R) cannot be a network for any finite or infinite ring R.