Filtrative Euclidean norm

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$$r$$, 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.

Facts
A filtrative Euclidean norm on an integral domain that is also multiplicatively monotone is a uniquely Euclidean norm.