Lemma 47.2.3. Let $R$ be a ring. Let $M$ be an $R$-module. Let $E = \mathop{\mathrm{colim}}\nolimits E_ i$ be a filtered colimit of $R$-modules. Suppose given a compatible system of essential injections $M \to E_ i$ of $R$-modules. Then $M \to E$ is an essential injection.

Proof. Immediate from the definitions and the fact that filtered colimits are exact (Algebra, Lemma 10.8.8). $\square$

Comment #1365 by jojo on

I feel like the conclusion should be either

$M \to E$ is an essential injection.

or

$E$ is an essential extension of $M$.

There are also:

• 4 comment(s) on Section 47.2: Essential surjections and injections

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