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.

Examples

• Using the fact that the polynomial ring over a field is Euclidean, we can prove that the Laurent polynomial ring over a field is also Euclidean. The Euclidean norm of a Laurent polynomial is defined as the difference between the highest and lowest degrees among the constituent monomials with nonzero coefficients. Further information: Polynomial ring over a field is Euclidean with norm equal to degree, Laurent polynomial ring over a field is Euclidean with norm equal to degree gap

Related facts

• Euclideanness is quotient-closed
• Minimum over principal ideal of Euclidean norm is a smaller multiplicatively monotone Euclidean norm: In fact, this is precisely the new norm we get when we do the process outlined above, setting $S=\{1\}$.

Proof

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