Zero-multiplication rng ($xy=0\text{ for all }x,y$)

Any abelian group becomes a nonunital ring by declaring every product zero. Associativity and the distributive property hold because both sides are zero. When the group has more than one element, no multiplicative identity element exists: an identity would have to satisfy $ex=x$, whereas every such product is zero. This differs from the zero ring, which has just one element.