I am reading the chapter $12$ of Artin's Algebra,in this chapter following is written:
'Lattices in $\mathbb R^2$ are free $\mathbb Z-$ modules' ?
Could someone please explain me the meaning of this? I understand the definition of free modules.I don't understand the lattice meaning in this context
A lattice in $\mathbb{R}^n$ is the subgroup of $(\mathbb{R}^n,+)$ generated by a basis of $\mathbb{R}^n$ as a vector space. For instance, if we have the basis $\{(1, 0), (0, 1)\}$ for $\mathbb{R}^2$, this generates the subgroup $$\{(a, b) \in \mathbb{R}^n \; | \; a, b \in \mathbb{Z}\}.$$ A lattice is a free $\mathbb{Z}$-module because if it weren't then the relation would imply a linear dependence between elements of the basis.