1. projective, module homomorphism

A module P is projective if for every surjective module homomorphism $f : N \twoheadrightarrow M$ and every module homomorphism $g : P \rightarrow M$, there exists a homomorphism $h : P \rightarrow N$ such that $f \circ h = g$.

Prove that if $P$ is projective and $h: P \rightarrow M$ is a surjective $R-$module homomorphism, then there exists some $R-$module homomorphism $s: M \rightarrow P$ such that $h \circ s=1_M$.

This is a section map. I don't know how to prove this. I was thinking to let $N=M$, but it doesn't seem that easy.

2. Originally Posted by zelda2139

Prove that if $P$ is projective and $h: P \rightarrow M$ is a surjective $R-$module homomorphism, then there exists some $R-$module homomorphism $s: M \rightarrow P$ such that $h \circ s=1_M$.
the question, as you wrote it, is wrong! (check what you wrote carefully!) the correct one is this:

Prove that if $P$ is projective and $h: M \rightarrow P$ is a surjective $R-$module homomorphism, then there exists some $R-$module homomorphism $s: P \rightarrow M$ such that $h \circ s=1_P$.

the proof is trivial: we have the identity map $1_p: P \longrightarrow P$ and a surjection $h: P \longrightarrow M.$ thus by the definition of projectivity there exists a map $s: P \longrightarrow M$ such that $hs=1_P.$