# The Stacks Project

## Tag 08XU

Lemma 106.19.5. 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. $\square$

The code snippet corresponding to this tag is a part of the file obsolete.tex and is located in lines 2295–2303 (see updates for more information).

\begin{lemma}
\label{lemma-characterize-injective}
Let $R$ be a ring. Let $E$ be an $R$-module. The following are equivalent
\begin{enumerate}
\item $E$ is an injective $R$-module, and
\item 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 $R \to E$.
\end{enumerate}
\end{lemma}

\begin{proof}
This is Baer's criterion, see
Injectives, Lemma \ref{injectives-lemma-criterion-baer}.
\end{proof}

## Comments (2)

Comment #2764 by Dario Weißmann on August 10, 2017 a 8:35 pm UTC

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 Johan (site) on October 6, 2017 a 1:07 pm UTC

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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