${\displaystyle R\setminus \{0\}\longrightarrow \mathbb {N} ,\,f\longmapsto \operatorname {ord} (f),}$

folgende Eigenschaften.

1. ${\displaystyle {}\operatorname {ord} \,(fg)=\operatorname {ord} \,(f)+\operatorname {ord} \,(g)}$.
2. ${\displaystyle {}\operatorname {ord} \,(f+g)\geq \min\{\operatorname {ord} \,(f),\operatorname {ord} \,(g)\}}$.
3. Es ist ${\displaystyle {}f\in {\mathfrak {m}}}$ genau dann, wenn ${\displaystyle {}\operatorname {ord} \,(f)\geq 1}$ ist.
4. Es ist ${\displaystyle {}f\in R^{\times }}$ genau dann, wenn ${\displaystyle {}\operatorname {ord} \,(f)=0}$ ist.