Euclideanness is localization-closed

This article gives the statement, and possibly proof, of a commutative unital ring property (i.e., Euclidean ring) satisfying a commutative unital ring metaproperty (i.e., localization-closed property of commutative unital rings)
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Statement

Suppose $R$ is a Euclidean ring and $S$ is a multiplicatively closed subset of $R$ not containing any zero divisors (without loss of generality, we may assume that $S$ is a saturated multiplicatively closed subset. Let $Q = S^{-1}R$ be the localization of $R$ at $S$. Then, $Q$ is also a Euclidean ring. Further, if $N$ is a Euclidean norm on $R$, we can define a new Euclidean norm $\tilde{N}$ on $Q$ as follows: $\tilde{N}(q) = \min \{ N(qx) \mid x \in Q, qx \in R \setminus \{ 0 \} \}$.

To see that this is well-defined, observe that any $q \in Q$ can be expressed as $s^{-1}r$ for some $s \in S$, $r \in R$, and if $q \ne 0$, $r \ne 0$. Thus, $qs \in R \setminus \{ 0 \}$. Hence, the set on the right side is nonempty. A minimum over a nonempty well-ordered set is well-defined, so the expression is well-defined.

Note that doing this operation for $S = \{ 1 \}$ does not necessarily give back the same Euclidean norm as we started with. It gives back the same norm only if the original norm was multiplicatively monotone.

Proof

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