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Math Help - projective, module homomorphism

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by zelda2139 View Post

    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.
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