Let $R$ be a finite ring such that the group of units of $R$, $U(R)$, has $34$ elements. I would like to find the characteristic of $R$.
Let $k:= \mathrm{Char}(R)$. If $\varphi$ denotes the Euler totient, then $\varphi(k)$ divides $34$, hence $k\in \{2,3,4,6\}$. I think it is possible to rule out some of these values, but I'm not sure how. And if we could rule them all out, even better, then there is no such ring. Any idea is appreciated.