Why this s.e.s. splits?

Question

Suppose that $$0\to(M/N)^+\to M^+\to N^+\to 0$$

is a short exact sequence of modules with $(M/N)^+$ and $N^+$ injective as defined here.

Why $(M/N)^+\to M^+$ splits and what does it mean that it splits?

Answer 1

One might say the most "trivial" short exact sequence one could ask for is one of the form $0\to L\to L\oplus N\to N\to 0$, where the first map is the inclusion $x\mapsto(x,0)$ and the second map is the projection $(x,y)\mapsto y$. To say an arbitrary short exact sequence $0\to L\to M\to N\to 0$ splits is a way of saying that it "looks like" one of these trivial exact sequences; more precisely, the sequence splits if there is an isomorphism $\theta:M\to L\oplus N$ making the following diagram commute

$\require{AMScd}$ \begin{CD} 0 @>>> L @>>> M@>>>N@>>>0 \\ & @V{\mathrm{id}_L} V V @V{\theta} V V @V{\mathrm{id}_N} V V \\ 0 @>>> L @>>>L\oplus N@>>>N@>>>0 \end{CD}

Now it is a theorem, which you can find in just about any textbook discussing split exact sequences, that the sequence $0\to L\overset{\varphi}{\to} M\to N\to 0$ splits if and only if there is a homomorphism $r:M\to L$ such that $r\circ\varphi=\mathrm{id}_L$. Recreating the proof here would be pointless in my opinion, so either look it up or accept it as a black box for the time being; one place to look is here.

From here, reexamine the definition of $L$ being an injective module: let's take for instance the third definition on the Wikipedia page you've linked. Then because $L$ is injective and $\varphi:L\to M$ is an injective module homomorphism and $\mathrm{id}_L:L\to L$ is a module homomorphism, the definition says there is a homomorphism $r:M\to L$ such that $r\circ\varphi=\mathrm{id}_L$, and this is exactly the criteria for the sequence to split.