Every projective f. g. module is f. p.

Question

I want to show that if $P$ is a finitely generated (f.g.) projective module then $P$ is finitely presented (f.p.).

Answer 1

As $P$ is f. g. we have an exact sequence $0\rightarrow Q\rightarrow A^n\rightarrow P\rightarrow 0$, $Q$ denoting the kernel of the map $A^n\rightarrow P$. As $P$ is projective, this exact sequence splits, $A^n\cong Q\oplus P$. The exact sequence $A^n\cong Q\oplus P\rightarrow A^n\rightarrow P\rightarrow 0$ shows $P$ to be f. p. (where $Q\oplus P\rightarrow A^n$ is the projection onto $Q$).