Additive identity in a semiring need not be multiplicative annihilator.

Question

We know that an additive identity in a ring is always a multiplicative annihilator. But this doesn't need always be true in case of a semiring. Consider that $e$ is the additive identity of a semiring $S$, then for any $a\in S$, we see that $a.e=a.(e+e)=a.e+a.e \implies a.e=e \iff -a.e \in S.$ Looking for a better suggestion. Thanks

Answer 1

But this doesn't need always be true in case of a semiring

I’m not sure what definition you’re using, but semirings are usually defined to require that the additive identity is absorbing.

If you’d like an example of something slightly less than a semiring which has a non absorbing zero, see Examples for almost-semirings without absorbing zero.