# 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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## Contents

## Statement

Suppose is a Euclidean ring and is a multiplicatively closed subset of not containing any zero divisors (without loss of generality, we may assume that is a saturated multiplicatively closed subset. Let be the localization of at . Then, is also a Euclidean ring. Further, if is a Euclidean norm on , we can define a new Euclidean norm on as follows:

.

To see that this is well-defined, observe that any can be expressed as for some , , and if , . Thus, . 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 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 .

## Proof

*Fill this in later*

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