Vipul: New page: {{curing metaproperty satisfaction| property = Euclidean ring| metaproperty = localization-closed property of commutative unital rings}} ==Statement== Suppose $R$ is a Euclide...

2009-02-05T16:57:16Z

New page: {{curing metaproperty satisfaction| property = Euclidean ring| metaproperty = localization-closed property of commutative unital rings}} ==Statement== Suppose <math>R</math> is a Euclide...

New page

{{curing metaproperty satisfaction|
property = Euclidean ring|
metaproperty = localization-closed property of commutative unital rings}}

==Statement==

Suppose <math>R</math> is a Euclidean ring and <math>S</math> is a [[fact about::multiplicatively closed subset]] of <math>R</math> not containing any zero divisors (without loss of generality, we may assume that <math>S</math> is a [[fact about::saturated multiplicatively closed subset]]. Let <math>Q = S^{-1}R</math> be the [[fact about::localization at a multiplicatively closed subset|localization]] of <math>R</math> at <math>S</math>. Then, <math>Q</math> is also a Euclidean ring. Further, if <math>N</math> is a Euclidean norm on <math>R</math>, we can define a new Euclidean norm <math>\tilde{N}</math> on <math>Q</math> as follows:

<math>\tilde{N}(q) = \min \{ N(qx) \mid x \in Q, qx \in R \setminus \{ 0 \} \}</math>.

To see that this is well-defined, observe that any <math>q \in Q</math> can be expressed as <math>s^{-1}r</math> for some <math>s \in S</math>, <math>r \in R</math>, and if <math>q \ne 0</math>, <math>r \ne 0</math>. Thus, <math>qs \in R \setminus \{ 0 \}</math>. 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 <math>S = \{ 1 \}</math> 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 norm|multiplicatively monotone]].

==Examples==

* Using the fact that the [[particular example::polynomial ring over a field]] is Euclidean, we can prove that the [[particular example::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|[[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 <math>S = \{ 1 \}</math>.

==Proof==

{{fillin}}

==

Euclideanness is localization-closed - Revision history

Vipul: New page: {{curing metaproperty satisfaction| property = Euclidean ring| metaproperty = localization-closed property of commutative unital rings}} ==Statement== Suppose R is a Euclide...

Vipul: New page: {{curing metaproperty satisfaction| property = Euclidean ring| metaproperty = localization-closed property of commutative unital rings}} ==Statement== Suppose $R$ is a Euclide...