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.