On page 124 of the book Elements of representation theory of associative algebras, volume 1, Proposition 3.11, I have a question about the proof.
On Line 8 of the proof, it is said that if $u: R \to U$ is a monomorphism, then because $P$ is injective, then $u$ factor through $P$. But by the definition of an injective module, we only know that there exists a map $\gamma$ from $U$ to $P$ ($\gamma$ is not from $P$ to $U$) such that $\gamma u = f_{P}$, where $f: R \to R/S \oplus P$ is the map in the short exact sequence in Proposition 3.11 and $f_{P}$ is the restriction of $f$ on $P$. I am confused about this. By the definition of a left almost split map, we need to show that there is a map $\alpha$ from $R/S \oplus P$ to $U$ such that $\alpha f = u$.
How can we prove this? Thank you very much.
