Lemma 10.121.1. Let $R$ be a semi-local Noetherian ring of dimension $1$. If $a, b \in R$ are nonzerodivisors then

$\text{length}_ R(R/(ab)) = \text{length}_ R(R/(a)) + \text{length}_ R(R/(b))$

and these lengths are finite.

Proof. We saw the finiteness in Lemma 10.119.11. Additivity holds since there is a short exact sequence $0 \to R/(a) \to R/(ab) \to R/(b) \to 0$ where the first map is given by multiplication by $b$. (Use length is additive, see Lemma 10.52.3.) $\square$

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