There are many questions about Examples of a non-Noetherian rings on this site, but this is a bit different.
Let $f:R\to S$ be a homomorphism and R be Noetherian. If $f$ is surjective, then $S$ is also Noetherian. But there are examples (on the site) that $f$ is 1-1, while $S$ is non-Noetherian.
What if $f$ is not surjective and not 1-1? is there examples that $S$ is non-Noetherian?
Thank you.