Let $A$ be a factorial ring and $S$ a multiplicative subset with $0 \notin S$. Show that $S^{-1}A$ is factorial, and that the prime elements of $S^{-1}A$ are of the form $up$ with primes $p$ of $A$ such that $(p)\cap S$ is empty, and units $u$ in $S^{-1}A$.
I'm confused as to why the prime elements are of the form given above. I'm thinking that the primes of $S^{-1}A$ should also include $q^{-1}$ where $q$ is a prime in $S$.
If $q \in S$, we have $qq^{-1} = 1$ in $S^{-1}A$. So $q^{-1}$ is a unit in $S^{-1}A$. Units are not prime.