# Is the image of a ring homomorphism an ideal?

Let $\phi : R \rightarrow S$ be a ring homomorphism, then is $im(\phi)$ an ideal in $S$? I ask this because I am studying about modules and in that we say that for a given $R$-module homomorphism the image is a submodule, which makes me wonder whether the image is an ideal or not?

I don't see how it would be since it would need to be closed under multiplication by elements in $S$ and for that we need the map to be surjective?