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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?

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You are correct. It isn't in general an ideal. An easy way to see this is that if your rings are unital, then the image of a homomorphism must contain 1, so it could only be an ideal if the homomorphism were surjective.

Alternatively pick any subring that is not an ideal and let your homomorphism be the inclusion.

Another alternative way to see this is what you mentioned in your post.

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