Rings which are generated by their units: a graph theoretical approach

The ring Z2 × Z2, having only one unit, cannot be generated by its units. It turns out, in the general theory of rings, that this is essentially the only example. In this note, we give an elementary proof of “A finite commutative ring with nonzero identity is generated by its units if and only if it cannot have Z2 ×Z2 as a quotient.” The proof uses graph theory, and offers, as a byproduct, that in this case, every element is the sum of at most three units.