$\gamma: R \rightarrow S$ is a ring homomorphism and $I \lhd R$, want an example of when $\gamma(I) \ntriangleleft S$ 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!

Hint: Consider the inclusion of a domain that is not a field into its field of fractions.
