If \(R\) is a (non-commutative possibly) ring let \(\Gamma(R)\) be the digraph with vertices \(x\in R\) a nonzero zero-divisor and arrows \(x\to y\) (\(x\neq y\)) iff \(xy=0\). In the commutative case \(x\to y\) means \(y\to x\) also and \(x-y\) can be taken to be an undirected edge. This not being the only definition, nor perhaps the standard one, it is nevertheless a simple and natural way to look at the graphical representation of the zero-divisor structure.
Main theorems extend results or prove new ones. Thus if \(R\) does not contain nonzero nilpotent elements \(xy=0\) (\(x\to y\)) implies \(yx=0\) (\(y\to x\)) for \(|\Gamma(R)|\geq 2\). Hence \(\Gamma(R)\) is an undirected graph and, e.g., not a tournament. Similarly, if \(R\) is finite then \(\Gamma(R)\) is not a network. Some examples of digraphs which cannot be zero-divisor graphs of rings (in the sense of this paper) are provided and it is shown that for \(R\) finite, \(\Gamma(R)\) has an even number of edges as well.