Show that every ring with underlying abelian group isomorphic to $\mathbb{Q}$ is isomorphic to $\mathbb{Q}$ as a ring.
I'm not sure how to get started here. I've tried using the same map for isomorphism of groups and show that it's also a ring isomorphism but I don't think it leads anywhere. Like $\varphi:R\to\mathbb{Q}$ is an isomorphism of groups and consider $\varphi(a) = 1$. Then $\varphi(a+a) = 2$ and so on. But I don't think this goes anywhere.
I think you can do the following (an outline). Fix a group isomorphism $\alpha:\mathbb Q\to R$.
(1) Consider the unique ring map $\mathbb Z\to R$. Note that this map is injective since $\alpha$ is injective ($\alpha$ being injective implies $R$ has characteristic $0$). Show that every nonzero element of $\mathbb Z$ maps to a unit in $R$. Hint: use the fact that $\alpha(q)=1$ for some $q\in\mathbb Q$.
(2) Then use the universal property of localization to get a unique ring map $\psi:\mathbb Q\to R$ extending $\mathbb Z\to R$.
(3) Show that $\psi$ is injective. Hint: the domain is a field.
(4) Show that $\psi$ is surjective. Hint: use surjectivity of $\alpha$.
