For any nontrivial idempotent $e$ in any commutative ring $R$, there is an idempotent $f = 1 - e$ such that $ef = 0$ and $e + f = 1$. This leads to the decomposition $R \cong eR \times fR$.
Is the same true of commutative semirings (with multiplicative neutral and additive neutral which is also a multiplicative zero)? That is, in such a semiring $S$, for any nontrivial idempotent $e$, does there exist an idempotent $f$ such that $ef = 0$ and $e + f = 1$? Suppose $S$ is finite also? If not, can someone give a counter example?