Injective hull of quotient of modules Let $R$ be a Noetherian commutative ring. If $N\leq M$ are $R$-modules, then is $E(M)$ isomorphic to a submodule of $E(N)\oplus E(M/N)$?
Here $E(M)$ denotes the injective hull of $M$.
 A: Yes, this is true for any ring (no commutativity or noetherianess required). 
Proof: Suppose we have an embedding $\phi: M \hookrightarrow E(N) \oplus E(M/N)$. Then, since  $E(N) \oplus E(M/N)$ is injective, by the universal property of the injective hull, there is an embedding $E(M) \hookrightarrow E(N) \oplus E(M/N)$ as stated. 
Hence it suffices to construct $\phi$: Label the maps as 
$$ 0 \to N \xrightarrow{i}M \xrightarrow{p} M/N \to 0$$
and let $$f: N \hookrightarrow E(N),\qquad g: M/N \hookrightarrow E(M/N)$$ 
be embeddings. By the universal property of injectives (draw a triangle), there is a homomorphismus $h: M \to E(N)$ such that $h \circ i = f$. Define 
$$\phi: M \to E(N) \oplus E(M/N), m \mapsto (h(m), (g\circ p)(m))$$
Now a simple computation shows that $\phi$ is injective. q.e.d. 
Remark: In disguise this is just an application of the horse lemma for injectives (the construction of $\phi$ is the crucial step in the proof of the horse lemma): See the first version of my answer. 
