I denote by $\mathbb{Z}_{(p)}$ the localization at a prime p, and by $\mathbb{Z}_p$ the p-adic integers.
Question: what is the structure of $\mathbb{Z}_p$ as $\mathbb{Z}_{(p)}$- module? For example is it free?
A related question:what are the finite index subgroups of $\mathbb{Z}_p$? Of course being a $\mathbb{Z}_{(p)}$- module they must be killed by a power of p, but are they the only closed ones(in that cas the module is clearly not free)?Conversely if it is free there are many wich are not closed.
Many thanks!
Edit:ok i can partially answer my self; of course the finite index subgroup are closed. Indeed the quotient, as remarked, has to be killed by a power of p, say $p^m$. So you see that all the expansions starting from at least $p^m$ are in the group, thus it contains the closed subgroup generated by $p^m$ and from here is trivial to conclude that the group is closed. So this answer also the other question: no the module cannot be free.