Let $f:R\rightarrow S$ be a ring homomorphism. Prove or disprove: if $f$ is one-to-one and $R$ is a field, then $S$ is a field.
My attempt: I believe this statement is false, as I can imagine a scenario where some element $s\in S$ is not $f(r)$ for some $r\in R$, and this element $s$ is not a unit. However, I'm having trouble coming up with an explicit counterexample.
Any help appreciated!