The Stacks project

Lemma 115.4.11. Let $(R, \mathfrak m)$ be a reduced Noetherian local ring of dimension $1$ and let $x \in \mathfrak m$ be a nonzerodivisor. Let $\mathfrak q_1, \ldots , \mathfrak q_ r$ be the minimal primes of $R$. Then

\[ \text{length}_ R(R/(x)) = \sum \nolimits _ i \text{ord}_{R/\mathfrak q_ i}(x) \]

Proof. Special (very easy) case of Chow Homology, Lemma 42.3.2. $\square$

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