During my university studies, I didn't delve deeply into ring modules. However, in the context of Euclidean rings, particularly those of the form $\mathbb{Z}[i\sqrt{2}]$, we often create what is commonly referred to as a 'lattice' (I am uncertain if this is the correct term in English as we have two distinct words for it in my language).
In the case of $\mathbb{Z}[i\sqrt{2}]$, what I tend to do (and please correct me if I am mistaken) is to represent the complex plane with the basis $(1, i\sqrt{2})$, and mark a point every time there is an element of $\mathbb{Z}[i\sqrt{2}]$. In this way, it yields a tessellation of the complex plane, creating rectangular shapes.
I have encountered several references in textbooks to the process of creating a 'lattice' for $\mathbb{Z}[\sqrt{2}]$. However, since it is 'one-dimensional', I am at a loss to understand what I should label as the 'x-axis' and 'y-axis' in order to produce a meaningful illustration.
Any assistance or reference materials that might help me comprehend how to sketch lattices in such fundamental instances would be greatly appreciated.
If you seek a two-dimensional distribution of points, then indeed you won't get that with $\mathbb{Z}[\sqrt2]$ alone. Instead, you need $\mathbb{Z}[\sqrt2]×\mathbb{Z}[\sqrt2]$. I suspect that the textbooks rather sloppily use "lattice of $\mathbb{Z}[\sqrt2]$" as shorthand for "lattice of $\mathbb{Z}[\sqrt2]×\mathbb{Z}[\sqrt2]$".
The $\mathbb{Z}[\sqrt2]×\mathbb{Z}[\sqrt2]$ lattice is thus a dense collection of points in the two-dimensional plane. If you superpose this set on its own $45°$ rotation about the origin (or about any other lattice point), you get the octagonal quasilattice.