Lemma 47.2.4 (08XM)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 47: Dualizing Complexes
Section 47.2: Essential surjections and injections
Lemma 47.2.4 (cite)

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Lemma 47.2.4. Let $R$ be a ring. Let $M \subset N$ be $R$ -modules. The following are equivalent

$M \subset N$ is an essential extension,
for all $x \in N$ nonzero there exists an $f \in R$ such that $fx \in M$ and $fx \not= 0$ .

Proof. Assume (1) and let $x \in N$ be a nonzero element. By (1) we have $Rx \cap M \not= 0$ . This implies (2).

Assume (2). Let $N' \subset N$ be a nonzero submodule. Pick $x \in N'$ nonzero. By (2) we can find $f \in R$ with $fx \in M$ and $fx \not= 0$ . Thus $N' \cap M \not= 0$ . $\square$

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