## Tag `08XU`

Chapter 106: Obsolete > Section 106.19: Duplicate and split out references

Lemma 106.19.5. Let $R$ be a ring. Let $E$ be an $R$-module. The following are equivalent

- $E$ is an injective $R$-module, and
- 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}
```

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