Lemma 10.15.1 (07K1)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.15: Miscellany
Lemma 10.15.1 (cite)

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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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