The natural numbers are the initial commutative semiring. Thus, for any commutative semiring $R$, there is a unique semiring map $\mathbb{N}\to R$.

For which $R$ is this map an epimorphism?

Some examples where it is:

- Obviously, if $R=\mathbb{N}$.
- If it is surjective, e.g. $R=\mathbb{Z}/n$.
- If it adjoins additive inverses, e.g. $R=\mathbb{Z}$.
- If it adjoins multiplicative inverses, e.g. $R=\mathbb{Q}_{\ge 0}$ (or smaller localizations of $\mathbb{N}$).
- If it does both, e.g. $R=\mathbb{Q}$ (or smaller localizations of $\mathbb{Z}$).

Are these the only possibilities?