To prove that all sub rings of $Q$ Euclidean Domain , what will be the Euclidean Valuation? Can anyone please give me a hint.
My attempt : Can I take $v(a/b) = |a|$?
If I take this how would the following properties would be satisfied? If $x, y \in S$ then $v(x) \le v(xy)$ .
By this answer of mine, any localization of $\Bbb Z$ is a Euclidean domain. Now if $R\subseteq\Bbb Q$ is any subring, it is some localization of $\Bbb Z$, simply take $S$ to be the set of denominators of all completely reduced elements or $R$, then $R=S^{-1}\Bbb Z$ is a Euclidean domain. That old answer also details the construction of the Euclidean degree function.