Localization of euclidean ring is euclidean?

Question

I am trying to prove that a localization of a euclidean ring is euclidean, and the converse statement.

I feel the basic definition of the norm is enough but I do not know how.

Please note I am very much a beginner in abstract algebra.

Answer 1

For one direction, see this answer. The converse is not true: Given any integral domain $R$, you can localize at $S:=R\setminus\{0\}$ and obtain a field $S^{-1}R=\operatorname{Frac}(R)$, which is a Euclidean domain. However, there are integral domains which are not Euclidean, for example $R=\mathbb Q[X,Y]$. Note that $R$ is not Euclidean because it is not a principal ideal domain.