Can a finite ring have a characteristic $0$? I am working on studying Abstract Algebra and doing one of the review problems for the exam.  I do not think a finite ring can have characteristic zero because when I think of, for example, Integers mod $4$, this ring has a characteristic greater than $0$. I also remember learning about infinite rings and I almost naturally associate an infinite ring with having a characteristic of $0$.  Could anyone help me prove the above statement? I am thinking of starting off with a contradiction: let R be a finite ring with characteristic $0$...
I'm not sure whether your ring has unity, so here is a general argument.

If a ring $R$ has characteristic zero, then, for each $n&gt;0$, there is some $a\in R$ with additive order $&gt;n$.  This means the set $\{0,a,2a,\ldots ,na\}$ consists of distinct elements (this requires the existence of additive inverses), so that $|R|&gt;n$.

Since $n$ was arbitrary, $R$ is infinite.

This argument is a little simpler if we can assume that $R$ has unity.  In that case, $R$ must actually contain $\mathbb{Z}$ as an additive subgroup.
No it can't be zero because the characteristic of a finite ring, say (R,+,.), divides the order of (R,+). It's a trivial proof.