Decomposition of the injective hulls I think this question is easy but I just cannot see how to solve it.
Let $R$ be a ring and $M$ an $R$-module. Suppose $0=\bigcap_{i=1}^n N_i$ is a decomposition of $0$ with irreducible submodules of $M$. Then $E(M)\cong \bigoplus_{i=1}^n E(M/N_i)$.  ($E(\_)$ denotes the injective hull).
Of course each $E(M/N_i)$ is indecomposable by one of my previous question: When the injective hull is indecomposable .
But I cannot see how to prove this generalization, any hints?
In my previous question they called irreducible meet-irreducible and indecomposable directly-indecomposable, I'm not familiar with this terminology but this is what I mean:
$N$ is irreducible in $M$ if $N=N_1\cap N_2$ with $N_1,N_2\subset M$ then one of them is $N$.
$N$ is indecomposable if $N=N_1\oplus N_2$ implies one of them is 0.
POSSIBLE BEGINNING OF A PROOF:
Let's consider the following maps:

$\varphi_i:M\rightarrow M/N_i$
$j_i:M/N_i\rightarrow E(M/N_i)$
$\psi_i=j_i\varphi_i$
$\psi:M\rightarrow\bigoplus E(M/N_i)\;\;\;\;\;$ $\psi=(\psi_i)_i$

Let's prove that $\psi$ is ingective: $\mathrm{ker}\;\psi=\bigcap\mathrm{ker}\;\psi_i=\bigcap\mathrm{ker}\;\varphi_i=\bigcap N_i=0$.
So the only thing I have to prove is that $\psi$ is an essential extension, i.e. if $N\subset\bigoplus E(M/N_i)$ with $N\cap M=0$ then $N=0$. I made a couple of observations that may be useful, first since finite direct sum of injectives is injective we have a $N^\prime$ such that $\bigoplus E(M/N_i)=N\oplus N^\prime$, and also an $N^{\prime\prime}$ such that $\bigoplus E(M/N_i)=E(N)\oplus N^{\prime\prime}$. And by my previous question (the one linked above) $E(M/N_i)$ is indecomposable for every $i$. But now I don't know how to continue, any hints?

Hint: Consider the $M\to M/N_i \to E(M/N_i)$ homomorphisms, these induce an $M\to\bigoplus_i E(M/N_i)$, and try to prove its universal property.
