Let $R$ be a ring, $F$ a field, and $\phi:R\rightarrow F$ a ring homomorphism. Suppose that exists a bijection $f:R\rightarrow F$ such that $$f(rx+sy) = \phi(r)f(x)+\phi(s)f(y)$$ for all $r,s,x,y\in R$. I think that this does not imply that $R$ is a field, but I am failing to find a (counter) example. Any ideas?
To simplify things, think of $F$ as field containing $R$, or $R$ being a subring of $F$.