Filtrative norm

Definition
A norm on a commutative unital ring is termed filtrative if it is a function from the nonzero elements of the ring to nonnegative integers, and further, for any natural number $$n$$, the set of elements with norm less than $$n$$, along with zero, form an additive subgroup of the ring.

Stronger properties

 * Weaker than::Filtrative Euclidean norm