= Rng with two elements
= Rngs with two elements
{synonym}
Up to an operation-preserving <bijection>, there are two <rngs with two elements>. Write the elements as $0,u$. The additive <abelian group> must satisfy $u+u=0$. The <distributive property> forces all products involving zero to vanish, leaving only the choice $u^2=0$ or $u^2=u$. The first gives a <zero-multiplication rng>; the second gives the <prime field> $\mathbb F_2$, with identity $u$. Both satisfy the axioms, and they are not isomorphic because an isomorphism must send the unique nonzero element to the unique nonzero element and preserve its square. In particular, \b[there is just one unital <ring> with two elements].
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