Consider a lattice $\Lambda \cong \mathbb{Z}^n$. To fix such an isomorphism one should fix elements $v_1, \dots, v_n \in \Lambda$ such that the whole of $\Lambda$ is generated by these vectors.
Question How should a call such a set of generators of a lattice? Is the word "basis" good?
Remark. Sorry if I am too pedagogical. But let me point out, that if you embed $\Lambda \subset \mathbb{R}^n$, then set $v_1, \dots, v_n$ is a basis of $\mathbb{R}^n$. But not any basis $w_1, \dots, w_n$ of $\mathbb{R}^n$ (even if you require $w_i \in \Lambda$) is a basis of $\Lambda$. For instance $(2,0)$, $(0,1)$ is a basis of $\mathbb{R}^2$ but not a basis of $\mathbb{Z}^2$.
Best Answer
Yes, it's the right terminology given that a lattice is a free $\Bbb Z$-module.