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?


I think that the characteristic is related to (or the same as) the order of the group $(,+)$

Not really unless $R$ is finite, when by Lagrange's theorem the maximum order of an element must divide $|R|$. $|R|$ can easily be infinite while having positive characteristic, for example.  The connection is not strong.

i.e., every ring with prime characteristic is either $ℤ_$ or something related to $ℤ_.$

The only relationship is that a ring (with identity) with characteristic $p$ contains a copy of $\mathbb Z_p$. This is true for $p$ not prime as well.  It is just the subring generated by the identity $1$.
Even when $R$ is finite and has positive characteristic, it is completely possible for it to be noncommutative: a small example would be the upper triangular matrices over the field of two elements. It has eight elements, is noncommutative, and characteristic $2$.
