Proposition 1

For a sub-A-module M of V to be a lattice of V, it is necessary and sufficient that KM = V and that M be contained in a finitely generated sub-A-module of V.

The conditions are obviously necessary, for a free sub-A-module of V with the same rank as V generates V. Conversely, if KM = V, M contains a basis $(a_i)_{1 \leq i \leq n}$ of V over K and hence it contains the free sub-A-module L, generated by the $a_i$; on the other hand, if $M \subset M_1$, where M, is a sub-A-module of V generated by a finite number of elements b, and $(e_i)_{1 \leq i \leq n}$ is a basis of V over K, there exists an element s $\neq 0$ of A such that each of the b, is a linear combination of the $s^{-1}e_i$ with coefficients in A; if L, is the free sub-A-modules of V generated by the $s^{-1}e_i$, then $M \subset L_2$.