Let be $A$ and $B$ two rings and let $f$ be a "rule" that associates elements of $A$ to elements of $B$, but not necessarily in a unique way, so that $f$ is a multifunction.
If I want to show that $f$ is a well defined homomorphism, is it enough to verify the following four statements?
- $f(0)=0$
- $f(a+b)=f(a)+f(b)$
- $f(ab)=f(a)f(b)$
- $f(1_A)=f(1_B)$
The last three statements ensure that the multifunction behaves well respect the ring properties and the second statement with the first ensures that $f$ is indeed a function.