Suppose that $M\subseteq \mathbb{Z}^n$ is a module such that $\mathbb{Z}^n/M$ is free and $S\subseteq \mathbb{R}^n$ is a bounded, symmetric convex set. Let $M'$ be the module generated by $S\cap M$.

Question: Is $\mathbb{Z}^n/M'$ free?

I think it is free if the following is true: for any $x\in M\setminus M'$ and $y\in M' (y\neq 0)$, it holds that $x$ and $y$ are linearly independent.

Is this true?

It seems true to me, but I haven't found a proof yet...