Lattices over modules The standard construction of lattices in $\mathbb{R}^n$ can be generalised by taking any finite-dimensional vector space $V$ over a field $F$ and a subring $R$ of $F$, as per Wikipedia (https://en.m.wikipedia.org/wiki/Lattice_(group).
Is there any reason why only finite-dimensional vector spaces are considered here, and not free modules of finite rank? The lattice should still form an additive subgroup, and I think the property for bases to give isomorphic lattices should also be the same still. Is there any important property that is lost with this further generalisation that I'm missing?
 A: A important property of theory of lattices is the inner product that you get from $\mathbb{R}^n$, as well as related objects like the volume.
Many of the interesting  and useful theorems come from the interaction between the algebraic and geometric structures, e.g. Minkowski's theorem and it's application to the geometry of numbers. The geometry is also important for its applications in cryptography, since one needs a norm to talk about things like the shortest vector problem.
An equivalent definition for lattices is to define them as a free $\mathbb{Z}$ module equipped with a quadratic form. The following generalization appears on page 3 of Ebeling's "Lattices and Codes."
Definition: Let $R$ be a commutative ring with unity. A symmetric bilinear form module $(S,b)$ over $R$ is a pair consisting of a free $R$-module $S$ of rank $n$, and a symmetric bilinear form $b : S \times S \to R$.
Proposition 1.1 (Ebeling) The integral lattices in $\mathbb{R}^n$ (lattices where every dot product is an integer) are the symmetric bilinear form modules over the integers where $b$ is a positive definite symmetric bilinear form.
The text mentions that symmetric bilinear form modules over $\mathbb{Z}$ without the positive-definite property are studied in chapter 4, and examples over rings of integers are studied are studied in chapter 5. You may find them interesting to peruse.
