Let $R$ be a commutative ring. Show that there is a bijection between the set $L$ of ringhomomorphisms $\mathbb Z[i]\to R$ and the set $X$ of $x\in R$ with $x^2+1=0$.
I can show that $\phi\colon L\to X\colon f\mapsto f(i)$ is a well-defined ring homomorphism. Now I would either like to show directly that it is bijective, or I should show that its inverse exists; $$\phi^{-1}\colon X\to L\colon x\mapsto f:f(i)=x. $$ Now here I am a bit stuck, because I can't really make any progress either way. I also haven't used that $R$ is commutative, it seems. Could anyone give a hint on how to continue from this point?
Btw: My course has just started dealing with rings and fields, so I barely know any theorem, except the isomorphism theorems. So I am looking for an elementary proof, if possible, at this point in my course.