Lemma 31.13.9. Let $X$ be a scheme. Let $Z, Y$ be two closed subschemes of $X$ with ideal sheaves $\mathcal{I}$ and $\mathcal{J}$. If $\mathcal{I}\mathcal{J}$ defines an effective Cartier divisor $D \subset X$, then $Z$ and $Y$ are effective Cartier divisors and $D = Z + Y$.

Proof. Applying Lemma 31.13.2 we obtain the following algebra situation: $A$ is a ring, $I, J \subset A$ ideals and $f \in A$ a nonzerodivisor such that $IJ = (f)$. Thus the result follows from Algebra, Lemma 10.120.16. $\square$

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