Inverse limit of $\left(\mathbb{Z}/p^n\mathbb{Z}\right)_{n \in \mathbb{N}}$ In the wikipedia article about the inverse limit it is stated that for a prime number $p$
$$\varprojlim_{n \in \mathbb{N}} \mathbb{Z}/p^n\mathbb{Z} = \mathbb{R}/\mathbb{Z},$$
where the arrow between the groups $\mathbb{Z}/p^n\mathbb{Z} \longrightarrow \mathbb{Z}/p^{n+1}\mathbb{Z}$ is given by multiplication by $p$.
I don't quite understand that: I interpret this "multiplication by $p$" as $[x]_{p^n} \mapsto [px]_{p^{n+1}}.$ But shouldn't the arrows point in the other direction from $\mathbb{Z}/p^{n+1}\mathbb{Z}$ to $\mathbb{Z}/p^n\mathbb{Z}$ for the inverse limit to make sense? On the other hand if one interprets it as a direct limit, then the arrows become inclusions and the limit is simply the group of all roots of unity of order $p^n$ for every $n \in \mathbb{N}$. I also find it a bit odd that one can get a "continous" object from "discrete" objects without invoking any topological notions at all.
How can this be reconciled, is this direct limit correct?
 A: This is false (I'll correct it on Wikipedia). The correct statement is that the direct limit / filtered colimit of $\mathbb{Z}/p^n \mathbb{Z}$ with these maps is the Prufer $p$-group $\mathbb{Z} \left[ \frac{1}{p} \right]/\mathbb{Z}$, which is the subgroup $\mu_{p^{\infty}}$ of $\mathbb{R}/\mathbb{Z}$ consisting of $p$-power roots of unity. The inverse limit / cofiltered limit (with a different set of maps going the other way, namely the quotient maps $\mathbb{Z}/p^{n+1}\mathbb{Z} \to \mathbb{Z}/p^n\mathbb{Z}$) is the $p$-adic integers $\mathbb{Z}_p$.
There's a reason people don't take inverse limits over diagrams that are "increasing to the right" like this diagram is:
$$\mathbb{Z}/p^0\mathbb{Z} \xrightarrow{p} \mathbb{Z}/p\mathbb{Z} \xrightarrow{p} \mathbb{Z}/p^2\mathbb{Z} \xrightarrow{p} \dots$$
and it's that the inverse limit over any such diagram is its first term.
The solenoid example is also wrong (and I will also correct it). $\mathbb{R}/p^n \mathbb{Z}$ is not a ring because $p^n \mathbb{Z}$ is not an ideal. You can in fact prove that every compact Hausdorff ring must be totally disconnected, or equivalently must be profinite, so rings like the $p$-adic integers are all you get.
