Filtrative Euclidean norm: Difference between revisions

Revision as of 17:10, 22 January 2009

This article defines a property that can be evaluated for a Euclidean norm on a commutative unital ring

Definition

A Euclidean norm on an integral domain is said to be filtrative if it satisfies the following condition:

The set of elements of norm at most $r$ , along with zero, forms an additive subgroup. Thus, the association to each $r$ of the corresponding subgroup forms a filtration of additive subgroups of the integral domain.

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 ==Definition==
-A [[Euclidean norm]] on an [[integral domain]] is said to be '''filtrative''' if it satisfies the following equivalent conditions:
+A [[Euclidean norm]] on an [[integral domain]] is said to be '''filtrative''' if it satisfies the following condition:
-* For any two elements of the domain, either their sum is zero or the norm of their sum is at most the maximum of the norms
+The set of elements of norm at most<math>r</math>, along with zero, forms an additive subgroup. Thus, the association to each <math>r</math> of the corresponding subgroup forms a filtration of additive subgroups of the integral domain.
-* The set of elements of norm at most<math>r</math>, along with zero, forms an additive subgroup. Thus, the association to each <math>r</math> of the corresponding subgroup forms a filtration of additive subgroups of the integral domain.
-Clearly, any filtrative Euclidean norm is also uniquely Euclidean.
-==Relation with other properties==
-===Weaker properties===
-* [[Uniquely Euclidean norm]]