$\mathbb{Q}_{\mathbb{Z}}$ is an injective hull of $\mathbb{Z}$

Question

I want to prove that $\mathbb{Q}_{\mathbb{Z}}$ is an injective hull of $\mathbb{Z}$. Suppose that $H$ is an injective hull of $Z$. If I can show that $\mathbb{Q}$ contains an isomorphic copy of $H$, then I complete the proof by using divisibility of $H$.

Does the quotient field of a ring always contain its injective hull?

Answer 1

Indeed, for any integral domain $R$ (commutative ring containing 1), the fraction field $Q$ of $R$ is an injective $R$-module.

This can be seen easily by Baer’s Criterion:

Baer’s Criterion: Let $R$ be a ring. An $R$-module $Q$ is injective if and only if, given any ideal $\mathfrak a\subset R,$ and $R$-module map $\phi: \mathfrak a\to Q,$ there is an extension of $\phi$ to all of $R.$

Let $I \subset R$ be an $R$-ideal, and $f: I \to Q$ is an $R$-linear map. The extension $f^\sim :R \to Q$ is determined by $f^\sim (1)$. Choose $a \in I$ such that $f(a) \neq 0$. Then one can see that $f^\sim (1) = f(a)/a$ does the job.