Let $R$ be a ring without unity. Suppose that $char R=n$. Does there must exists an element $r\in R$ with $|r|=n$? Let $R$ be a ring without unity (not necessarily commutative). Suppose that $char R=n$. Does there must exists an element $r\in R$ with $|r|=n$?
$|r|$ denote the additive order of $r$, 
that is, 
the smallest positive integer $n$ such that $nr=0$.
I already know that if it is false, 
then the counterexample must be infinite.
But I can't find a counterexample.

This is true, and the multiplicative structure is irrelevant for this. An abelian group $G$ with exponent $n$ has an element of this order.
To see this, take $a\in G$ with $o(a)$ maximal. If $o(a)<n$, then there is a $b\in G$ with $o(b)$ not a divisor of $o(a)$ (because otherwise $\exp(G)=o(a)$), so $p^k| o(b)$ but $p^k\nmid o(a)$ for some prime $p$.
Take suitable multiples $c=sa$, $d=tb$ so that $o(d)=p^k$, $o(c)=o(a)/p^m$, where $p^m$ is the highest power of $p$ that divides $o(a)$. Then $o(c), o(d)$ are relatively prime, so $o(c+d)=p^ko(c)=p^{k-m}o(a)>o(a)$, but this contradicts our choice of $a$.
