Describe units and maximal ideals in these two PIDs

Question

If $p$ is a fixed prime integer, let $R$ be the set of all rational numbers that can be written in the form

$(a)$ $\frac{a}{b}$ with $b$ not divisible by $p$.

$(b)$ $\frac{a}{b}$ with $b=p^k$ for a non-negative integer $k$.

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For (a), I can tell that all units have the form of $u=\frac{a}{b} \in R $ s.t. $ p \nmid a$, and that the only maximal ideal is $\mathfrak m= \{ \frac{a}{b} \in R:p \mid a , \ p \nmid b \}$.

I'm having a hard time visualizing part (b). Any help will be greatly appreciated, thanks in advance!

Answer 1

(b) $R=S^{-1}\mathbb Z$, where $S=\{1,p, \dots,p^n,\dots\}$. A prime ideal $Q$ of $R$ is of the form $S^{-1}q\mathbb Z$, where $q\in\mathbb Z$ is a prime such that $q\mathbb Z\cap S=\emptyset$. This shows that the prime ideals of $R$ are $(0)$ and $\{qa/p^k:a\in\mathbb Z,k\ge0\}$ with $q\ne p$.

The invertible elements are those from $R-\cup_{q\ne p, q \text{ prime }} S^{-1}q\mathbb Z$. You can figure out what form they have.