Does a euclidian size function distribute over addition?

Question

Let $R$ be a euclidian domain with size function $\sigma$. I'm curious as to if the following property is true: given $a,b \in R,\quad \sigma(a+b)=\sigma(a)+\sigma(b)$. From the definition of the euclidian size function such a fact is true for multiplication, but I'm unsure if this condition of distributing over addition is left out for a reason or is just trivially true.

Thanks

Answer 1

No, this is not true - in $\mathbb{Z}$, the absolute value is an Euclidean function, but $|{-1 + 1}| = 0 \neq 2 = |{-1}| + |1|$.