On directed zero-divisor graphs of finite rings

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 any ring R, if there exists a source b in Γ(R) with b = 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.