Subadditive Euclidean norm

Definition
Let $$R$$ be a commutative unital ring and $$N$$ be a Euclidean norm on $$R$$. We say that $$N$$ is subadditive if for any $$a,b \in R$$ such that $$ab(a+b) \ne 0$$, we have:

$$N(a + b) \le N(a) + N(b)$$.

Stronger properties

 * Weaker than::Filtrative Euclidean norm