Is it necessary for an Euclidean domain to satisfies Triangle inequality

Question

I am wondering whether there always exist a Euclidean function f that satisfies $$f(a+b) \le f(a)+f(b).$$ In all the ED I know, it seems to be true. For example, integers, Gaussian integers, all the Fields I know. Intuitively, I guess that it should be true, though I haven't try hard in proof; so I want to know whether there is any existed theory suggesting this to be true or there is any counter example.

Answer 1

Here is a counterexample for $R=\Bbb Z[\sqrt{2}]$. This is a Euclidean ring with Eulcidean norm function $$ f(x+y\sqrt{2})=x^2-2y^2. $$ For $a=1+\sqrt{2}$ and $b=1-\sqrt{2}$ we have $a+b=2$, so that $f(a+b)=f(2)=4$, but $f(a)=f(b)=1^1-2\cdot 1^2=-1$. So we have $$ f(a+b)=4>-2=f(a)+f(b). $$