Question regarding finitely generated modules

Amked

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.

Amked

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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