Let $R$ be a commutative ring with $1 \neq 0$ and suppose $S$ is a multiplicatively closed subset of $R \backslash \{\, 0 \,\}$ containing no zero divisors. We have the relation $\sim$ defined on $R\times S$ via $$(a, b) \sim (c, d)\qquad\Leftrightarrow\qquad ad = bc.$$
If we define the $p$-integers as $R_S$ for $R=\mathbb{Z}$ and $S = R \backslash p\mathbb{Z}$, I want to verify that there are two prime ideals of the $p$-integers, one corresponding to the zero ideal and one corresponding to the prime $p$. I have been given the hint that it may help to regard $R_S \subset \mathbb{Q}$ via $$R_S=\{\, \tfrac ab:a,b \in \mathbb{Z}, p \nmid b \,\}.$$
I am not sure where to start, can anyone give any guidance? Thanks!
The construction of $R_S$ from $R$ is known as the localisation of $R$ at $S$, and is also denoted $S^{-1}R$.
A basic fact on localisations, which is worth proving yourself, is that there is a canonical morphism $$R\ \longrightarrow\ S^{-1}R,$$ and that it induces a bijection between the prime ideals of $S^{-1}R$ and the prime ideals of $R$ that do not intersect $S$. This bijection then immediately shows you what the ideals corresponding to the zero ideal and the ideal $p\Bbb{Z}$ are.