Let $\phi: (R, +, \cdot)\rightarrow (S, \oplus, \star)$ be a semiring map. Then $\operatorname{im}\phi$ is a sub semiring of $S$. In general, it is not an ideal. I look for an example in which $\operatorname{im}\phi$ is an ideal of $S$. The following is my attempt :
Let $\phi$ be an onto map. Let $ \phi(m)\in \operatorname{im}\phi$ and $\phi(r)\in S$, then $\phi(m)\star \phi(r)=\phi(m\cdot r)\in \operatorname{im}\phi$(since $m\cdot r \in R$ and $\phi$ is onto). Similarly, $\phi(r)\star \phi(m)\in \operatorname{im}\phi$. Hence $\operatorname{im}\phi$ is an ideal of $S$ if $\phi$ is onto.
Is this attempt correct? Any other appropriate examples (if any) are appreciated.
The image of a semiring hom will contain $1$. Any ideal that contains $1$ is the whole ring/semiring. So the image is an ideal iff the map is surjective.