Def: [$p$ is prime] $\iff$ [$p | ab$ $\Rightarrow$ $(p | a)$ or $(p | b)$]
Assume $$S_1(n) := \{n^k | k \geq 0\}$$ $$S_2(n) := \{m \in \mathbb{Z} | (m,n) = 1\}$$
I want to describe all prime elements in localisations $S_1^{-1}\mathbb{Z}$ and $S_2^{-1}\mathbb{Z}$. Which technique can I use here ?
In the localization $S_1^{-1} \Bbb Z$, the number $n \in \mathbb Z$ becomes a unit. Now all the factors of $n$ in $\mathbb Z$ become units: Write $n = k \ell$. Then $$\ell^{-1} = n^{-1} k \in S_1^{-1}\Bbb Z.$$ So all factors of $n$ become units, i.e., the primes in $S_1^{-1}\Bbb Z$ are all primes in $\mathbb Z$ except for the ones that are factors of $n$.
On the other hand, the units in $S_2^{-1} \Bbb Z$ are all numbers that are coprime to $n$. So the primes in $S_2^{-1}$ are the primes in $\mathbb Z$ which are a factor of $n$.