Let $R$ be a ring with $10$ elements, show that $R$ is commutative.

Question

Let $R$ be a ring with $10$ elements, show that $R$ is commutative.

$R$ is a ring which contains $10$ elements and doesn't have to include $1$.

Answer 1

The additive group is cyclic, let $g$ be a generator. Then for arbitrary elements $a=ng$, $b=mg$, we have $$a\cdot b = ng\cdot mg=nm(g\cdot g)= mn(g\cdot g)=mg\cdot ng=b\cdot a.$$