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.