I want to show that the integers localized at some prime natural number $p$:
$$R=\Biggl\{\frac mn \in \Bbb Q ~\Bigg\vert~ m,n \in\Bbb Z,\ n\notin p\Bbb Z\Biggr\}$$
is a Euclidean Domain, but I can't figure out how to show that my norm function works.
I have $f:R\to \Bbb N$ defined as $f(\frac ab)=|a|$. Could someone show me how $f$ is a Euclidean norm function?