Definition 10.121.2. Suppose that $K$ is a field, and $R \subset K$ is a local1 Noetherian subring of dimension $1$ with fraction field $K$. In this case we define the order of vanishing along $R$

$\text{ord}_ R : K^* \longrightarrow \mathbf{Z}$

by the rule

$\text{ord}_ R(x) = \text{length}_ R(R/(x))$

if $x \in R$ and we set $\text{ord}_ R(x/y) = \text{ord}_ R(x) - \text{ord}_ R(y)$ for $x, y \in R$ both nonzero.

[1] We could also define this when $R$ is only semi-local but this is probably never really what you want!

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