Let $R$ be a ring with $1$. We say that a right $R$-module $M$ is indecomposable if $M$ can not be decomposed as the internal direct sum $M=L \oplus K$ with $L$ and $K$ non-zero. Suppose that $R$ is a Noetherian ring and $M$ is a right $R$-module. How can I prove the following:
$M$ is injective if and only if $M \cong E(R/P)$ for some prime ideal $P\subset R$, where $E(R/P)$ is the injective envelope (or injective hull).
Or where can I find the proof of such a statement?.
Thanks in advance.