Prove that there is a unique homomorphism from $\mathbb{Z} [i]$ to $\mathbb{Z}/2\mathbb{Z}$.
I'm struggling to show uniqueness here. In the past I have shown that $\mathbb{Z}[i]/(1+i)$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$, and from this past work I am convinced that the homomorphism in question is in fact
$$ x+yi \mapsto x-y \mod 2. $$
How do I even begin to show uniqueness here? Thanks in advance.
Edit: To clarify, we are working with only commutative rings with unity in our course, and our definition of a ring homomorphism includes the clause that $\phi(1) = 1$.
This is only true if $1$ is sent to $1$. If that is the case, let $f$ be the homomorphism. Then $f(i)^2=f(-1)=-f(1)=1$, hence $f(i) =f(-i) =1$. Since the ring is generated by $1$ and $i$, the homomorphism is completely determined.