How can one express $\mathbb{Q} / \mathbb{Z}$ as a direct sum of $\mathbb{Q}$ and $\mathbb{Z} / p$ for $p$ prime?

Question

Consider the $\mathbb{Z}$ modules $\mathbb{Q}$ and $\mathbb{Z} / p$ for $p$ prime. I have a result that says that every injective $\mathbb{Z}$ module is a direct sum of these modules. I also know that $\mathbb{Q} / \mathbb{Z}$ is an injective $\mathbb{Z}$ module, being divisible. How can I reconcile these two notions and express $\mathbb{Q} / \mathbb{Z}$ as a direct sum of $\mathbb{Q}$ and $\mathbb{Z} / p$ for $p$ prime? This doesn't seem possible to me. Am I interpreting one of my stated results in the wrong way?

Answer 1

$\newcommand{\Z}{\mathbb{Z}}$It's not $\Z / p$, but rather the Prüfer group $\Z(p^{\infty})$.

Answer 2

Expanding on what Andreas has said.

$\mathbb{Z}/p$ is not injective (tensor it in $0\to\mathbb{Z}\to\mathbb{Z}\to\mathbb{Z}/p\to 0$). What you should have gotten instead is $\mathbb{Z}(p^\infty)=\injlim_{n\to\infty}\mathbb{Z}/p^n$ being divisible (i.e. injective). Then $$ \mathbb{Q}/\mathbb{Z}=\bigoplus_{p\text{ prime}} \mathbb{Z}(p^\infty) $$