The Stacks Project


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

  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 WeiƟmann 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.

    Add a comment on tag 08XU

    Your email address will not be published. Required fields are marked.

    In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?