Filtrative Euclidean norm
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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