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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!

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You can embed any field $k$ into the polynomial ring $k[x]$, but the latter is not a field.

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Take a field $K$ and product ring $K\times K$.

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