Is there a noncommutative ring with prime characteristic? Is there a noncommutative ring with prime characteristic?
My first answer is “no”, because $(R, +)$ is an abelian group, and I think that the characteristic is related to (or the same as) the order of the group $(R, +)$, i.e., every ring with prime characteristic is either $\mathbb{Z}_p$ or something related to $\mathbb{Z}_p$. But I'm not able to prove it, or give a counterexample.
Where am I wrong?

$M_2(\mathbb{F}_p).\phantom{--------}$
