Euclideanness is localization-closed

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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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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.


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