Inverse limit and union of $\mathbb{Z} / p^{n}\mathbb{Z} $

Question

Let $p$ be a prime number and let the natural embeddings $\mathbb{Z}/p\mathbb{Z} \subset \mathbb{Z}/p^{2}\mathbb{Z} \subset \dots \subset \mathbb{Z}/p^n\mathbb{Z} \subset \dots $

Questions: Does the object $\bigcup_n \mathbb{Z}/p^n\mathbb{Z}$ make sense?
If so, what's the difference with the inverse limit $\varprojlim \mathbb{Z}/p^n\mathbb{Z} =: \mathbb{Z}_p$ (the $p$-adic integers)?

Answer 1

The object $\bigcup_n \mathbb{Z}/p^n\mathbb{Z}$ is the colimit of the sequence

$\mathbb{Z}/p\mathbb{Z} \hookrightarrow \mathbb{Z}/p^2\mathbb{Z} \hookrightarrow \cdots$

where the maps are given by multiplication by $p$. This is known as the Prüfer $p$-group (see nLab). On the other hand, the $p$-adic integers $\mathbb{Z}_p$ are the limit of the inverse system

$\mathbb{Z}/p\mathbb{Z} \leftarrow \mathbb{Z}/p^2\mathbb{Z} \leftarrow \cdots$

where the transition maps are given by taking residue classes.

Wikipedia claims that the ring of endomorphisms of the Prüfer $p$-group is given by $\mathbb{Z}_p$ (see here).