Does there exist non free $R$-modules $F_0,F_1$ such that $F=F_0\oplus F_1$ be a free $R$-module?
1- If yes then for what kind of rings $R$ there exist such $R$ modules?
2- If yes then does it holds for both finitely and infinitely generated free $R$-module $F$?
Consider the ring $F_2\times F_2$.
$I=F_2\times \{0\}$ is not free, because it has a nonzero annihilator $J=\{0\}\times F_2$. The same can be said for $J$, having $I$ as its annihlator. But $I\oplus J\cong R$ is free.
To produce an infinitely generated free module example, just consider $I'=\oplus_{i=1}^\infty I$ which also has annihilator $J$, as a summand of $\oplus_{i=1}^\infty R$ with complement $J'=\oplus_{i=1}^\infty J$.