A morphism $f: X\to Y$ in mod A is called irreducible if
f is not a section,
f is not a retraction,
and whenever $f = gh $ for some morphisms $h: X \to Z$ and $g: Z \to Y$, then either $h$ is a section or $g$ is a retraction.
Now I am reading Schiffler's book "Quiver Representation",
Proposition 7.4 (2) states the fact that if $f: X\to Y$ is an irreducible morphsim, then $f$ admits no nontrivial factorization.
I have two questions:
- What is trivial factorization or nontrivial factorization of $f$?
- In the proof of Proposition 7.4 (2), $f$ is injective, then $f$ is not surjective, (Proposition 7.4 (1)) and thus $h$ cannot be a retraction. I don't know why $h$ is not a retraction.
$f=hg$. If $f$ is both injective and surjective, then it is an isomorphism i.e there exists $f^{-1}:Y \to X$ such that $f.f^{-1}=1_Y$. Thus $1_Y=f.f^{-1}=hgf^{-1}$, so $gf^{-1}$ is the right inverse of $h$ i.e. $h$ is a retraction.
A retraction necessarily is surjective. $f$ is not surjective, it seems that this will imply $h$ is not surjective.