I just proved that if $\gamma: R \rightarrow S$ is a surjective ring homomorphism and $I \lhd R$ then $\gamma(I) \lhd S$. I'm now looking for an example of where this fails if $\gamma$ is not surjective. I've been looking for one for awhile but alas, i have not come up with anyone. Looking for a new perspective, any advice is appreciated!
2026-03-29 11:50:05.1774785005
$\gamma: R \rightarrow S$ is a ring homomorphism and $I \lhd R$, want an example of when $\gamma(I) \ntriangleleft S$
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Hint: Consider the inclusion of a domain that is not a field into its field of fractions.