It's $\mathbb{Q}/\mathbb{Z}$ an $Z_n$-injective module?

Question

It's $\mathbb{Q}/ \mathbb{Z}$ an $Z_n$-injective module, where $n\in \mathbb{N}$?

Answer 1

You can't endow $\mathbb{Q}/\mathbb{Z}$ with the structure of a module over $\mathbb{Z}_n$ exactly because it is divisible.

Since $n1_{\mathbb{Z}_n}=0$, we must have $nM=0$, for every module over $\mathbb{Z}_n$. Of course, for every divisible abelian group $G$ we have $nG=G$, by definition.

The fact that $\mathbb{Q}/\mathbb{Z}$ is divisible is obvious, because $\mathbb{Q}$ is divisible.

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The ring $\mathbb{Z}_n$ is self-injective, which you can prove with the help of Baer’s criterion. The ideals of $\mathbb{Z}_n$ have the form $m\mathbb{Z}_n$ for $m\mid n$. Suppose you have a homomorphism $f\colon m\mathbb{Z}_n\to\mathbb{Z}_n$; set $x=f(m)$. Then, for $q=n/m$, $$ qx=qf(m)=f(qm)=f(0)=0 $$ so actually the image of $f$ is contained in $m\mathbb{Z}_n$. Now you can end.