[Math] Does a surjective ring homomorphism have to be surjective on the unit groups

Question

I know ring homomorphisms map units to units, which made me curious about the following. Suppose $f:R\to R'$ is a surjective ring homomorphism, mapping $1$ to $1'$. Is it necessarily surjective from $U(R)\to U(R')$?

I know if $f(u)$ is a unit in $R'$ with $f(w)$ its inverse, then $f(uw)=f(wu)=1'$, but I see no reason to conclude $uw=wu=1$ in $R$ without assuming $f^{-1}(1')=\{1\}$. But I can't find a counterexample, so I'm not sure whether it's true or not.

Answer 1

Look at the quotient map $\mathbf Z \to \mathbf Z/5\mathbf Z$. The residue class of $2$ is a unit in the target, but the only units of $\mathbf Z$ are $\pm 1$.