How many homomorphisms are there from a finite field to a ring?
I have some basic knowledge of this but I am not able to put it into use. I know
$1)$ If $\phi$$(1)=a$, then $|a|$ should divide both, order of the field as well as the order of the ring. But order of the ring isn't specified here.
$2)$ Also, since $\phi(1.1)=\phi(1).\phi(1)$ so $a^2=a$ i.e. a homomorphism maps 1 to an idempotent.
Please help me on how to proceed further.
There can be just the trivial homomorphism, as is the case with $F_3\to \Bbb Z$, or there could be infinitely many rng homomorphisms, as is the case with $F_2\to \prod_{i=1}^\infty F_2$.
There isn't a uniform answer for all pairs of fields and rings: it depends on your rings, or else what you want to require of the homomorphism.
Supposing that the homomorphism maps the identity of $F$ to the nonzero identity of $R$, you have in injection of $F$ into $R$, so $R$ must have the same characteristic as $F$. This is necessary but not sufficient because, for example, $F_4$ can't be injected into $F_2$ even though they have the same characteristic. If the field has a prime number of elements, then it is sufficient as well. The field is additively cyclic, and the image of $1$ will determine the images of everything else.