Ring homomorphisms between $\mathbb{Z}[i]$ and $\mathbb{Z}[\sqrt{-3}]$.

Question

This is a question from my past Qual exams.

"Let $R=\mathbb{Z}[i]$ and $S=\mathbb{Z}[\sqrt{-3}]$. Does there exist a ring homomorphism $R\to S$. Does there exist a ring homomorphism $S\to R$?"

I think I solved it but my way is insanely easy.

Suppose there is a map $f:R\to S$. Then $f(i)=a+b\sqrt{-3}$. We have $-1=(f(i))^2=a^2-3b^2+2ab\sqrt{-3}$ but there are no solution here, so a contradiction. The other way uses the same method.

Am I wrong somewhere?

Answer 1

Your solution is right. In fact, those rings are isomorphic as additive groups. They are different exactly when it comes to the monoid structure of multiplication, and your proof is correct.