Subrings and ideals of subring $\mathbb{Q}_p$ of $\mathbb{Q}$

Question

Let $p$ be a prime number and $\mathbb{Q}_{p} = \{ \frac{a}{b} \in \mathbb{Q} : (a,b)=1 \Rightarrow p \nmid b \}$. How would I find all the subrings and ideals of $\mathbb{Q}_p$? I already know that $p\mathbb{Q}_p$ is the only maximal ideal. Now, if $I$ is a proper ideal of $\mathbb{Q}_p$ and $\frac{a}{b} \in I$ then $p\mid a$. With this, my guess is that $\mathbb{Q}_p$ is a PID but I don't know how to proceed and definitely don't know what to do about the subrings. I haven't found anything about this ring anywhere and I'm desperate. Any help would be very much appreciated.

Answer 1

Recognize that this is just the localization of $\Bbb{Z}$ at the prime ideal $(p)$ and use the standard theorems on the correspondence between ideals in a localization and in the original ring.

Answer 2

Your ring $\Bbb Z_{(p)}$ allows for denominators all numbers prime to $p$. Among its subrings there are the following:
Let $S$ be any set of primes not including $p$. Now consider numbers $m/n$ where the only primes dividing $n$ are from $S$. For instance, if $S=\{2\}$, then I’m talking about the ring frequently written $\Bbb Z[\frac12]$. But since there are uncountably many subsets $S$, there are uncountably many subrings of $\Bbb Z_{(p)}$. I’ll leave it to you to decide whether the rings I’ve named are the only subrings of $\Bbb Z_{(p)}$.

In most cases, if $R$ is a ring, its subrings are not of nearly as great interest as its quotients $R/I$, its localizations, and its extension rings. (I am open to be contradicted in this claim.)