Let $R$ be a PID and $F$ be a free module over $R$. Suppose we have $F=A\oplus B$ for some $R$-modules $A$ and $B$. Then are $A$ and $B$ necessarily free?
If $F$ is finitely generated, then I know that $A$ too is finitely generated. Thus we can write $A$ as a direct sum of a torsion module and a free module. Since there is no torsion in $F$, the torsion part of $A$ must be zero. Thus $A$ must be free and same for $B$.
Is the statement true even when $F$ is not finitely generated?