Eisenstein Plane

Question

Suppose I plotted points of the Eisenstein integers $\mathbb{Z}\left[\frac{-1+\sqrt{-3}}{2}\right]$, where the point $a+b\omega$ was the point $(a,b)$. How could I describe this plane in the language of vector spaces and modules? In terms of vector spaces, we only care about integer multiples of points, but $\mathbb{Z}$ isn't a field, so we couldn't really call $\mathbb{Z}[\omega]$ as a "$\mathbb{Z}$-vector space". In terms of modules, it seems like we could call $\mathbb{Z}[\omega]$ as a $\mathbb{Z}$-module, but $\mathbb{Z}[\omega]$ isn't an Abelian group... Could someone help me out here? Thanks in advance!