Let $V=\mathbf{Z}^N$ be a free $\mathbf{Z}$-module of rank $N$. Let $S\subseteq V$ be a fixed finite subset.

Consider the submodule $M:=\langle S\rangle\leq V$ generated by $S$. We know form the general theory that $M$ is a free $\mathbf{Z}$-module of rank $\leq N$.

Q: Is there a theoritical criterion to determine when it is possible to find a subset $B\subseteq S$ such that $B$ is a $\mathbf{Z}$-basis of $M$?

If $\#S=r$, then taking the standard basis of $V$, one may associate to $S$ an $r\times n$ matrix. So a possible criterion (here I'm specalutating) could consist (partly) at looking at the gcd of determinants of sufficiently many minors of suitable sizes.

Let $V=\mathbf{Z}^N$ be a free $\mathbf{Z}$-module of rank $N$. Let $S\subseteq V$ be a fixed finite subset.

Consider the submodule $M:=\langle S\rangle\leq V$ generated by $S$. We know form the general theory that $M$ is a free $\mathbf{Z}$-module of rank $n\leq N$.

Q: Is there a theoritical criterion to determine when it is possible to find a subset $B\subseteq S$ such that $B$ is a $\mathbf{Z}$-basis of $M$ (in otherwords a subset $B\subseteq S$ of cardinality $n$ such that $\langle B\rangle=M$)?

In general, such a subset $B\subseteq S$ needs not to exist. For example take $V=\mathbf{Z}$ and $S=\{2,3\}$. Then $\langle 2,3\rangle=\mathbf{Z}$ has rank one, but $2\mathbf{Z}$ and $3\mathbf{Z}$ are not equal to $\mathbf{Z}$.

If $\#S=r$, then taking the standard basis of $V$, one may associate to $S$ an $r\times N$ matrix. So a possible criterion (here I'm specalutating) could consist (partly) at looking at the gcd of determinants of sufficiently many minors of suitable sizes.