Filtrative Euclidean norm

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 conditions:

• 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 $r$, 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.

Clearly, any filtrative Euclidean norm is also uniquely Euclidean.