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.