Let $f: R_1 \rightarrow R_2$ be a ring homomorphism. Does $r \in R_1$ being non-invertible mean $f(r)$ is non-invertible? Let $f: R_1 \rightarrow R_2$ be a ring homomorphism. Does $r \in R_1$ being non-invertible mean $f(r)$ is non-invertible?
This seems incorrect, but I find it difficult finding counter-example... 
Thanks for any assistance! 

Hint: Consider the inclusion $\Bbb Z \to \Bbb Q$.
