# Constructing injection into injective group

$$\newcommand{\ZZ}{\mathbb{Z}}$$ $$\newcommand{\QQ}{\mathbb{Q}}$$ $$\newcommand{\Hom}{\mathrm{Hom}}$$ Some time ago I tried to construct for a given abelian group $$M$$ functorially a group $$I(M)$$ which should be an injective object (that is a divisible abelian group) and provide an injection $$0 \to M \hookrightarrow I(M)$$.

I formed $$I(M) = \Hom_{\ZZ}(F_\ZZ \Hom_\ZZ(M, \QQ/\ZZ), \QQ/\ZZ)$$ where $$F_\ZZ \Hom_\ZZ(M,\QQ/\ZZ)$$ is the free abelian group over the abelian group $$\Hom_\ZZ(M,\QQ/\ZZ)$$.

The group $$I(M)$$ is an injective (divisible) group because of the following equation $$\Hom_\ZZ(N^\bullet, \Hom_\ZZ (F_\ZZ\Hom_\ZZ(M,\QQ/\ZZ), \QQ/\ZZ)) = \Hom_\ZZ(N^\bullet \otimes_\ZZ F_\ZZ(\Hom_\ZZ(M, \QQ/\ZZ)), \QQ/\ZZ)$$ where $$N^\bullet$$ is the exact sequence $$0 \to N' \to N \to N'' \to 0$$. Now tensoring with a free group is exact and $$\QQ/\ZZ$$ is an injective group. So $$\Hom_\ZZ(N^\bullet, I(M))$$ preserves exactness, especially on $$0 \to N' \to N$$ and so $$I(M)$$ is an injective group.

Now the map $$M \to I(M)$$ is given as $$(*) \quad m \mapsto ((\sum_i n_i \phi_i) \mapsto \sum_i n_i \phi_i(m))$$ where $$\sum_i n_i \phi_i$$ is an element of $$F_\ZZ \Hom_\ZZ(M, \QQ/\ZZ)$$.

Now for every $$m \in M$$ with $$m \neq 0$$ one can construct a map $$\phi_m:M \to \QQ/\ZZ$$ with $$\phi_m(m) \neq 0$$ (because $$\QQ/\ZZ$$ is an injective group).

So the map $$(*)$$ is injective by choosing for given $$m \in M$$ the map $$\phi_m$$ as $$\sum_i n_i \phi_i$$ in $$(*)$$.

Two questions

1) Is this proof valid? It seems so simple that it is somewhat embarassing to ask such a question, but:

going through all my commutative algebra books I could not find this proof, although it seems obvious and simple and has the advantage of being functorial in $$M$$. So

2) Is there a place in the literature where a proof of enough injective objects in the abelian groups (or for modules over a commutative ring) is done with this idea (especially forming a free group with the intent of moving it to the left and tensor with it)?