class page exercises

If $P$ and $P'$ are projective, then $P \oplus P'$ is projective.

Proof (thanks to Robin Sulzgruber) Let $M$ be an $R$-module such that $\phi: M \rightarrow P \oplus P'$ is a surjection. Then $\pi_1 \circ \phi$ is a surjection onto $P$ so $M \simeq P \oplus ker(\pi_1 \circ \phi)$. Then let $\psi = \pi_2 \circ \phi |_{ker(\pi_1 \circ \phi)}$. Since $\phi$ is a surjection onto $P \oplus P'$ then $\psi$ is also a surjection from $ker(\pi_1 \circ \phi)$ onto $P'$ and since $P'$ is projective, then $ker(\pi_1 \circ \phi) \simeq P' \oplus ker~\psi$. But $ker~\psi = ker(\pi_1 \circ \phi) \cap ker(\pi_2 \circ \phi)$. Therefore $M \simeq P \oplus P' \oplus (ker(\pi_1 \circ \phi) \cap ker(\pi_2 \circ \phi))$.