An integer lattice is a subgroup of $\mathbb{Z}^n$. Since $\mathbb{Z}$ is PID, each lattice has a well-defined rank and a generating set of rank many elements is a basis.
I wonder if there is a way to define lattice basis by inclusion minimality, like for vector spaces. Basically I would like to say: If $B \subset L$ spans and no proper subset of $B$ spans, then $B$ is a basis of $L$. This is not true since the sublattice of $\mathbb{Z}$ generated by $2,3$ is equal to $\mathbb{Z}$, but neither $2$ nor $3$ alone spans. One remedy seems to be
Def. $B\subset L$ is a lattice basis if it spans $L$, and is a basis of $L \otimes_\mathbb{Z} \mathbb{Q}$.
This seems like a lot of machinery. Are there any alternatives that avoid embedding $\mathbb{Z}$ into a vector space?