The Stacks project

Lemma 115.26.6. Let $R$ be a ring. Let $E$ be an $R$-module. The following are equivalent

  1. $E$ is an injective $R$-module, and

  2. given an ideal $I \subset R$ and a module map $\varphi : I \to E$ there exists an extension of $\varphi $ to an $R$-module map $R \to E$.

Proof. This is Baer's criterion, see Injectives, Lemma 19.2.6. $\square$

Comments (2)

Comment #2764 by Dario Weißmann on

This result is also stated in Lemma 19.2.6, is this intentional? In Lemma 19.2.6 (or rather directely above) this is called "criterion of Baer". I think it's quite nice that the result has a name, although "Baer's criterion" sounds better.

Comment #2877 by on

Sometimes it is useful for reasons of exposition to have duplicates, but in this case I agree that we don't need it. I also added a reference to Baer's original paper and I changed the name as you suggested. Thanks! See here for the changes.

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View Lemma 115.26.6 as pdf