Suppose $f\colon R\to S$ is a ring homomorphism (and not rng homomorphism, wherein $f(1) = 1$ is not generally true). I can prove that if $a$ is invertible$^1$, then $f(a)$ also is, with its inverse being $f(a^{-1})$. However, I am not able to prove the converse, that is, if $f(a)$ is invertible, then $a$ also is, or give a counterexample.
Any help?
I've just begun studying about these things from Stillwell's Elements of Agebra, and though these are not the questions that are required to be answered in the text, these just pop up in my head, and I think that MathSE is the best place to ask them.
$^1$By inverses here, I mean multiplicative inverses.
Consider the canonical projection $\pi:\mathbb{Z}\to\mathbb{Z}/5\mathbb{Z}$. The image is a field, and so the element $\pi(2)=2+5\mathbb{Z}$ is invertible. However, $2$ is obviously not invertible in $\mathbb{Z}$.