Questions about projective modules. Let $P$ be a projective module and $M$ a submodule of $P$. We know that $M$ is also a projective module. Can we conclude that $P=M\oplus N$ for some module $N$? Thank you very much.

As pointed out in the comments that submodules of projective modules are projective does not always hold. If this holds the algebra is called hereditary. An equivalent description of that is that $\operatorname{gldim} A \leq 1$.
But even if it holds, this does not guarantee that submodules of projectives are direct summands. For example take the hereditary path algebra of the quiver $1\to 2$. The projective module to the vertex $1$ has the composition series $0\subset S_2\subset P_1$. The simple module $S_2$ is also projective but the module $P_1$ is indecomposable, hence there is no complement. 
