Let $F$ be a field and $\phi$ be a trivial homomorphism such that $\phi(x) = 0$ for every $x\in F$. The homomorphic image of $F$ in such case is therefore a ring, not a field.
Is this correct?
(I ask for there is a question, F6 of Chapter 18, in Pinter's Abstract Algebra to prove that any homomorphic image of a field is a field, which doesn't seem to hold in all cases.)
Yep, that's correct; the zero ring isn't a field. (This is an example of the "too simple to be simple" phenomenon).
The exercise is close though, the homomorphic image of a field is either 0 or another field.