# The Stacks Project

## Tag 08XV

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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