Lemma 10.15.1. Let $R$ be a ring, $I$ and $J$ two ideals and $\mathfrak p$ a prime ideal containing the product $IJ$. Then $\mathfrak {p}$ contains $I$ or $J$.

**Proof.**
Assume the contrary and take $x \in I \setminus \mathfrak p$ and $y \in J \setminus \mathfrak p$. Their product is an element of $IJ \subset \mathfrak p$, which contradicts the assumption that $\mathfrak p$ was prime.
$\square$

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