Direct sum of projective module Let $\left\{ P_{i}\right\} _{i\in I}$ be a family of $R$-module. I know that if each $P_{i}$ is projective then $\oplus_{i\in I}P_{i}$ is projective. Is the converse true, i.e if $\oplus_{i\in I}P_{i}$ is a projective module, is $P_{i}$ projective module for all $i\in I$?
Yes its true. $P$ is projective iff there is free module $F$ such that $F=P\oplus Q$. Now if $\bigoplus P_i$ is projective then there is free module $F$ such that $F=\bigoplus P_i\oplus Q=P_j\oplus(\bigoplus_{j\neq i}P_i\oplus Q)$. This means that $P_i$ is projective.
This is true. For injective modules,the corresponding results hold.