All homomorphisms from $\mathbb{Q}$ to $\mathbb{Q}$ are isomorphisms

Question

I came across a problem which asked to prove that the ring of endomorphisms of $\mathbb{Q}$ (viewed as an abelian group) is isomorphic to $\mathbb{Q}$, i.e $\operatorname{End}\mathbb{Q} \cong \mathbb{Q}$. Supposing this was true, since $\mathbb{Q}$ is a field this will imply that $\operatorname{End} \mathbb{Q}$ is a field. This means every element has an inverse or more specifically every element(non zero) in $\operatorname{End} \mathbb{Q}$ is an isomorphism. So given any group homorphism from $\phi:\mathbb{Q} \to \mathbb{Q}$, it turns out to be an isomorphism. Can anyone explain what is happening here?

Answer 1

The thing is, an abelian group morphism $\mathbb{Q}\to \mathbb{Q}$ is automatically $\mathbb{Q}$-linear, and we know what $K$-linear maps $K\to K$ are for a field $K$.

A group morphism $\mathbb{Q}\to\mathbb{Q}$ is necessarily of the form $x\mapsto rx$, so yes, unless it's $0$, it's an isomorphism.