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. 



General criterion to find a Z-basis in a fixed generating subset
The condition is the following. Let $A$ be the matrix whose rows are
the elements of $S$. Then there is some ordering $v\_1,\dots,v\_n$ of
some $n$ elements of $S$ such that the for all $1\leq k\leq n$, the
gcd of the $k\times k$ minors of the matrix whose rows are
$v\_1,\dots,v\_k$ equals the gcd of the $k\times k$ minors of $A$.


