Show that $\operatorname{End}_{\mathbb{Z}}(\mathbb{Q})$ is isomorphic to the field $\mathbb{Q}$

Question

I have problem in showing that $\operatorname{End}_{\mathbb{Z}}(\mathbb{Q})$ is isomorphic as a ring to the field $\mathbb{Q}$.

Any idea?

Thanks

Answer 1

Hint: An endomorphism of $\mathbb Q$ is determined by the image of $1$.

Here are some details:

Let $\sigma$ be an endomorphism of $\mathbb Q$ and consider $x=m/n \in \mathbb Q$. Then $m\sigma(1)=\sigma(m)=\sigma(nx)=n\sigma(x)$, and so $\sigma(m/n)=\sigma(1)(m/n)$. This gives us an explicit ring isomorphism $f: \mathbb Q \to End(\mathbb Q)$ defined by $f(q)(x)=qx$. The hint proves that $f$ is surjective. $f$ is injective because $\mathbb Q$ is a field. You need to verify that $f$ is indeed a ring isomorphism.