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The Stacks project

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

  1. M \subset N is an essential extension,

  2. 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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