Definition 1

Let $V$ be a finite-dimensional vector space over the field $K$. A lattice of $V$ with respect to $\mathbf{A}$ (or simply a lattice of $V$) is defined to be any sub-$\mathbf{A}$-module $M$ of $V$ satisfying the following condition:

There exist two free sub-$\mathbf{A}$-modules $L_1,\ L_2$ of $V$ such that $L_1 \subset M \subset L_2$ and $\mathrm{rg}_{\mathbf{A}}(L_1) = \mathrm{rg}_K(V)$.

Examples (1) If we take $V = K$, the lattices of $K$ are just the fractional ideals $\neq (0)$ of $K$ ($\S$ 1, no. 1, Definition 1).

(2) If $\mathrm{rg}_K(V) = n$, every free sub-A-module L of V has a basis containing at most n elements, every subset of V which is free over A being free over K; for L to be a lattice of V, it is necessary and sufficient that L have a basis of n elements (in other words, that $\mathrm{rg}_A(L) = n$).

(3) If A is a principal ideal domain, every lattice M of V is a finitely generated A-module (since A is Noetherian) which is torsion-free and hence a free A-module (Algebra, Chapter VII, § 4, no. 3, Corollary 2 to Theorem 2).