Zero-multiplication rng
= Zero-multiplication rng
{title2=$xy=0\text{ for all }x,y$}
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.