Filtrative Euclidean norm: Difference between revisions
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{{Euclidean norm property}} | {{Euclidean norm property}} | ||
==Definition== | ==Definition== | ||
A [[Euclidean norm]] on | A [[Euclidean norm]] on a [[commutative unital ring]] is said to be '''filtrative''' if it satisfies the following condition: | ||
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. | ||
==Facts== | |||
A filtrative Euclidean norm on an [[integral domain]] that is also [[multiplicatively monotone Euclidean norm|multiplicatively monotone]] is a [[uniquely Euclidean norm]]. {{proofat|[[Filtrative and multiplicatively monotone implies uniquely Euclidean]]}} | |||
Latest revision as of 17:21, 22 January 2009
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This article defines a property that can be evaluated for a Euclidean norm on a commutative unital ring
Definition
A Euclidean norm on a commutative unital ring is said to be filtrative if it satisfies the following condition:
The set of elements of norm at most, along with zero, forms an additive subgroup. Thus, the association to each of the corresponding subgroup forms a filtration of additive subgroups of the integral domain.
Facts
A filtrative Euclidean norm on an integral domain that is also multiplicatively monotone is a uniquely Euclidean norm. For full proof, refer: Filtrative and multiplicatively monotone implies uniquely Euclidean