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Homework Help: Question regarding finitely generated modules

  1. Oct 13, 2006 #1
    I am supposed to show that the following are equivalent for a finitely generated module P:

    1. P is Projective
    2. P is isomorphic to direct summand of a free module
    (There are 2 others but they refer to a diagram)

    I am stuck on showing 1 => 2.

    I know that since P is projective there is α: M -> P so that
    M is isomorphic to ker (α) (direct sum) K,
    where K is a subset of P.
    Also since P is finitely generated P = Rx1 (direct sum) … (direct sum)Rxn.

    I also know that K is isomophic to M/ker(α)

    I believe I need to show that P = K, because then P would be isomorphic to a direct summand, but I don’t know how to show this.
  2. jcsd
  3. Oct 14, 2006 #2
    I have recently noticed that my definition of Projective is incorect.
    A module P is projective provided:
    If f:M -> P is a homomorphism and onto then M = ker(f) (direct sum) K,
    K contained in P.

    Hopefully that makes my question easier.
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