The Stacks project

Lemma 71.6.7. The sum of two effective Cartier divisors is an effective Cartier divisor.

Proof. Omitted. Étale locally this reduces to the following simple algebra fact: if $f_1, f_2 \in A$ are nonzerodivisors of a ring $A$, then $f_1f_2 \in A$ is a nonzerodivisor. $\square$

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