I'm currently reading "Algebra: Chapter 0" by Paolo Aluffi. Before I state my problem, I would like to give here the definition of split exact sequences from the book:
A short exact sequence
$0 \to M_1 \to N \to M_2 \to 0$
"splits" if there is a commutative diagram
in which the vertical maps are all isomorphisms.
Now, I'm having trouble with understandg a part of a given proof of the following proposition:
Let $\phi: M \to N$ be an injective $R$-module homomorphism. Then $\phi$ has a left-inverse(as a homomorphism, not a function) if and only if the sequence
$0 \to M \stackrel{\phi_1}{\rightarrow} N \to coker \ \phi \to 0$
splits.
Here is the proof from the book(I will state only the part I'm having trouble understanding with, that is, the "if" part of the statement):
If the sequence splits, then $\phi$ may be identified with the embedding of $M$ into a direct sum $M \oplus M'$, and the projection $M \oplus M' \to M$ gives a left-inverse of $\phi$.
Well, I'm a bit a lost here:
First of all, why $\phi$ can be identified with the embedding of $M \to M \oplus M'$? According to the definition above, it can be identidied with a surjective function $M'_1 \oplus M'_2$ where $M'_1 \cong M$ and $M'_2 \cong \frac{N}{im \ \phi} \cong coker \ \phi$