Hom of the direct product of $\mathbb{Z}_{n}$ to the rationals is nonzero.

Question

Why is $\mathrm{Hom}_{\mathbb{Z}}\left(\prod_{n \geq 2}\mathbb{Z}_{n},\mathbb{Q}\right)$ nonzero?

Context: This is problem $2.25 (iii)$ of page $69$ Rotman's Introduction to Homological Algebra:

Prove that $$\mathrm{Hom}_{\mathbb{Z}}\left(\prod_{n \geq 2}\mathbb{Z}_{n},\mathbb{Q}\right) \ncong \prod_{n \geq 2}\mathrm{Hom}_{\mathbb{Z}}(\mathbb{Z}_{n},\mathbb{Q}).$$

The right hand side is $0$ because $\mathbb{Z}_{n}$ is torsion and $\mathbb{Q}$ is not.

Answer 1

Let $G=\prod_{n\geq2}\mathbb Z_n$ and let $t(G)$ be the torsion subgroup, which is properly contained in $G$ (the element $(1,1,1,\dots)$ is not in $t(G)$, for example) Then $G/t(G)$ is a torsion-free abelian group, which therefore embeds into its localization $(G/t(G))\otimes_{\mathbb Z}\mathbb Q$, which is a non-zero rational vector space, and in fact generates it as a vector space. There is a non-zero $\mathbb Q$-linear map $(G/t(G))\otimes_{\mathbb Z}\mathbb Q\to\mathbb Q$ (here the Choice Police will observe that we are using the axiom of choice, of course...). Composing, we get a non-zero morphism $$G\to G/t(G)\to (G/t(G))\otimes_{\mathbb Z}\mathbb Q\to\mathbb Q.$$

Remark. If $H$ is a torsion-free abelian group, its finitely generated subgroups are free, so flat. Since $H$ is the colimit of its finitely generated subgroups, it is itself flat, and tensoring the exact sequence $0\to\mathbb Z\to\mathbb Q$ with $H$ gives an exact sequence $0\to H\otimes_{\mathbb Z}\mathbb Z=H\to H\otimes_{\mathbb Z}\mathbb Q$. Doing this for $H=G/t(G)$ shows $G/t(G)$ embeds in $(G/t(G))\otimes_{\mathbb Z}\mathbb Q$, as claimed above.

Answer 2

$\mathbb{Q}$ is an injective $\mathbb{Z}$-module. The exact sequence $$0\rightarrow \mathbb{Z} \rightarrow \prod \mathbb{Z}_n\rightarrow C\rightarrow 0$$ yields the exact sequence $$0\rightarrow \mathrm{Hom}_{\mathbb{Z}}(C,\mathbb{Q}) \rightarrow \mathrm{Hom}_{\mathbb{Z}}(\prod \mathbb{Z}_n, \mathbb{Q})\rightarrow \mathrm{Hom}_{\mathbb{Z}}(\mathbb{Z},\mathbb{Q})\rightarrow 0.$$ The last term in the latter exact sequence is just $\mathbb{Q}$, hence $\mathrm{Hom}_{\mathbb{Z}}(\prod \mathbb{Z}_n,\mathbb{Q})\neq 0.$