I am trying to determine if the following statements are true or false
(i) There are free modules with non zero elements $x$ such that $\{x\}$ is linearly dependent.
(ii) There are non free modules such that for every non zero element $x$, $\{x\}$ is linearly independent.
(iii) If $M$ is a free $R$-module and $N$ is a submodule which is free as $R$-module, then $N$ is a direct summand of $M$.
For (i), I couldn't think of counterexamples. For example, I know that this is false for the case $\mathbb Z_6$ seen as $\mathbb Z$ module because $2.3=0 ( 6)$ so $\{3\}$ is linearly dependent. But $Z_6$ is not a free module, I can't think of counterexamples with free modules.
The same goes for (ii) and (iii), I would appreciate suggestions to construct counterexamples or help to prove the statement if any of them is true.
Hints:
For (i): consider any ring $R$ with nonzero zero divisors. (In other words, the example you gave was fine. It sounds like you are forgetting to specify which ring the module is over, and that will be critical in deciding if it is free or not.)
For (ii): consider any ring $R$ without zero divisors and with a nonprojective ideal $I$. $I$ is your candidate.
For (iii): For a principal ideal domain $R$, submodules of free $R$ modules are all free. Can nontrivial submodules of $R$ be summands?