Injective module characterisation Is there an easy way to prove one of the following two implications?
Every proof I saw uses some deeper embedding results or uses the functor approach ($Hom(R,.)$ exact functor) but I'm looking for an elementary proof of one of these statements.
(1) If every short exact sequence of left-$R$ modules $0 \to I \to M \to N \to 0$ splits, then $I$ is injective.
(2) Assume that for any left $R$-module $M$ containing $I$ as a submodule, $M$ has a submodule $K$ such that $M = K \oplus I$, then $I$ is an injective module.
 A: It's easy to prove that an injective module $I$ is such that every short exact sequence $0 \to I \to M \to N \to 0$ splits, because you just use the definition. On the contrary, the viceversa, i.e. point $(1)$, is based on a very deep result that says that:
Every module is injectable in an injective module.
If you know this fact then $(1)$ follows easily:
Let $I$ be such that every short exact sequence $0 \to I \to M \to N \to 0$ splits. Let $I \xrightarrow{i} Q$ be an injection of modules with $Q$ injective (here I used the result). Then of course $0 \to I \xrightarrow{i} Q \to Q/I \to 0$ is exact and then it splits, so there is a map of modules $Q \xrightarrow{r} I$ such that $ri=1_I$.
Now we are left to verify that $I$ verifies the definition of injective module: let $N_1 \xrightarrow{j} N_2$ be an injection of modules and let $N_1 \xrightarrow{f} I$ be a map of modules. Then $N_1 \xrightarrow{if} Q$ is also a map of modules and $Q$ is injective. Hence there is an arrow $N_2 \xrightarrow{g} Q$ such that $ gj=if$, that is: $$(pg)j=(pi)f=f $$ and we are done.
