I am reading Rings and Categories of Modules by Anderson and Fuller. In the book page 293:
An indecomposable injective left $R$-module must be the injective envelope of each of its non-zero submodules.It follows that the set$\quad${${E(R/I)|I\leq R}$}$\quad$contains an isomorphic copy of each indecomposable injective.
I want someone to explain why the set$\quad${${E(R/I)|I\leq R}$}contains an isomorphic copy of each indecomposable injective. Thanks very much.
I assume $E(-)$ denotes injective envelopes, and $I \leq R$ means $I$ is a left ideal of $R$.
Recall Lemma 25.2 on page 290 in Anderson-Fuller. It says $M$ cogenerates $R/I$ iff $I$ is the annihilator of $M$. We apply this to our situation.
Take any indecomposable injective $E$, and let $I \leq R$ be its annihilator. Then $E$ cogenerates $R/I$ by the lemma. That implies there is a monomorphism $\iota \colon R/I \to E^n$ for some $n \in \mathbb{N}$. Take a projection $\pi \colon E^n \to E$ such that $\pi \iota \neq 0$. Then the composition induces a monomorphism $(R/I) /\ker(\pi \iota) \to E$. Since $E$ is the injective envelope of all of its submodules, this shows $E$ is isomorphic to the injective envelope of $(R/I) /\ker(\pi \iota)$, which is a factor module of $R$.