As homework, I need to find all the prime ideals of $\mathbb Z[\frac 1n]$.
For starters, I needed to prove that given any two rational numbers $q_1,q_2$, there exists a natural number satisfying $\mathbb Z[q_1,q_2]=\mathbb Z[\frac 1n]$. I did this by writing $q_i=\frac{a_i}{b_i}$ and taking $n=b_1b_2$. Unfortunately, I don't really see how this helps me find all the primes ideals at all.
If $I\subset \mathbb{Z}[1/n]$ is an ideal, then $J=I\cap\mathbb{Z}$ is an ideal of $\mathbb{Z}$. Furthermore, if $x=a/n^k\in I$ (with $a\in\mathbb{Z}$), then $a\in J$ and $x=a\cdot (1/n^k)$. Thus $J$ generates $I$ as an ideal in $\mathbb{Z}[1/n]$. Since every ideal in $\mathbb{Z}$ is principal, this means every ideal in $\mathbb{Z}[1/n]$ is principal (and in fact generated by an integer). Now it shouldn't be too hard to figure out which of the principal ideals are prime.
