Let $P$ be a projective module over a commutative ring $R$ with unity. Then for some free $R$-module $M$ , $M=P\oplus Q$, and we can think of $P,Q$ as submodules of $M$. Now let $N$ be a finitely generated free submodule of $M$ such that $P \subseteq N$; then is it true that $P$ is finitely generated ? I know that if I can show that there exists a surjective module homomorphism of $N$ onto $P$, then we are done, but I am unable to show that. Please help. Thanks in advance
Let $\pi:M\to P$ be the projection given by the splitting $M=P\oplus Q$, so in particular $\pi(p)=p$ for all $p \in P$. We can then restrict $\pi$ to $N$, and it will still be surjective, since $P\subseteq N$ and $\pi$ is the identity on $P$. Thus since $N$ is finitely generated, $P$ is finitely generated.
A different way to see the same thing in Eric Wofsey's answer is that if you have $A\oplus P=M$, then $(A\cap N)\oplus P=N$, so that $P$ is a summand of $N$, and is hence finitely generated.