The Stacks project

Proof. The second statement is Lemma 47.5.8. For the first statement, let $I$ be an injective $R$-module. We will use transfinite recursion to construct $I_\alpha \subset I$ for ordinals $\alpha $ which are direct sums of indecomposable injective $R$-modules $E_{\beta + 1}$ for $\beta < \alpha $. For $\alpha = 0$ we let $I_0 = 0$. Suppose given an ordinal $\alpha $ such that $I_\alpha $ has been constructed. Then $I_\alpha $ is an injective $R$-module by Lemma 47.3.7. Hence $I \cong I_\alpha \oplus I'$. If $I' = 0$ we are done. If not, then $I'$ has an associated prime by Algebra, Lemma 10.63.7. Thus $I'$ contains a copy of $R/\mathfrak p$ for some prime $\mathfrak p$. Hence $I'$ contains an indecomposable submodule $E$ by Lemmas 47.5.3 and 47.5.7. Set $I_{\alpha + 1} = I_\alpha \oplus E_\alpha $. If $\alpha $ is a limit ordinal and $I_\beta $ has been constructed for $\beta < \alpha $, then we set $I_\alpha = \bigcup _{\beta < \alpha } I_\beta $. Observe that $I_\alpha = \bigoplus _{\beta < \alpha } E_{\beta + 1}$. This concludes the proof. $\square$