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Tag 08XV

Chapter 45: Dualizing Complexes > Section 45.3: Injective modules

Lemma 45.3.8. Let $R$ be a Noetherian ring. A direct sum of injective modules is injective.

Proof. Let $E_i$ be a family of injective modules parametrized by a set $I$. Set $E = \bigcup E_i$. To show that $E$ is injective we use Lemma 45.3.7. Thus let $\varphi : I \to E$ be a module map from an ideal of $R$ into $E$. As $I$ is a finite $R$-module (because $R$ is Noetherian) we can find finitely many elements $i_1, \ldots, i_r \in I$ such that $\varphi$ maps into $\bigcup_{j = 1, \ldots, r} E_{i_j}$. Then we can extend $\varphi$ into $\bigcup_{j = 1, \ldots, r} E_{i_j}$ using the injectivity of the modules $E_{i_j}$. $\square$

    The code snippet corresponding to this tag is a part of the file dualizing.tex and is located in lines 278–282 (see updates for more information).

    \begin{lemma}
    \label{lemma-sum-injective-modules}
    Let $R$ be a Noetherian ring. A direct sum of injective modules
    is injective.
    \end{lemma}
    
    \begin{proof}
    Let $E_i$ be a family of injective modules parametrized by a set $I$.
    Set $E = \bigcup E_i$. To show that $E$ is injective we use
    Lemma \ref{lemma-characterize-injective}.
    Thus let $\varphi : I \to E$ be a module map from an ideal of $R$
    into $E$. As $I$ is a finite $R$-module (because $R$ is Noetherian)
    we can find finitely many elements $i_1, \ldots, i_r \in I$
    such that $\varphi$ maps into $\bigcup_{j = 1, \ldots, r} E_{i_j}$.
    Then we can extend $\varphi$ into $\bigcup_{j = 1, \ldots, r} E_{i_j}$
    using the injectivity of the modules $E_{i_j}$.
    \end{proof}

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