Does every idempotent semiring has characteristic zero?

Question

Let $(S, +, \cdot, 0, 1)$ be a semiring. If for all $s\in S$ and non- negative integer $m$, $ms=s+s+...+s(m$times) $=0$, then $S$ is said to be of characteristic $m$. If no such $m$ exists then $S$ is said to be of characteristic $0$. In my opinion, every idempotent (or, additively idempptent) semiring will have characteristic $0$ (since $s+s+...+s=s$) for all $s\in S$. Please do correct me if i am wrong.