It's possible to produce an example of an integral domain $R$ and a free $R$-module $M$ with free submodules $L, L'$ such that $L+L'$ is not free. We can take $R=M=K[x,y]$ , $L=\left<x\right>$ , $L'=\left<y\right>$.
If $R$ is a PID and $M$ is free $R$-module, then for every pair of submodules $L, L'$ of $M$ we know that $L+L'$ is free.
My question is the following.
If $R$ is a PID and $M$ is an $R$-module, is it true that $L+L'$ is free whenever $L$ and $L'$ are free submodules of $M$?
Thanks!