Proposition 2

Let M be a lattice of V and M, a sub-A-module of V. If there exist two elements x, y of $K^*$ such that $xM \subset M_1 \subset yM$, $M_1$ is a lattice of V; conversely, if $M_1$ is a lattice of V, there exist two non-zero elements a, b of A such that $aM \subset M, \subset b^{-1}M$.

If $L_1, L_2$ are two free lattices of V such that $L, \subset M \subset L,$, the relations $xM \subset M_1 \subset yM$ imply $xL_1 \subset M, \subset yL_2$ and $xL_1$ and $yL_2$ are free lattices; conversely, if M, is a lattice and $(e_i)_{1 \leq i \leq n}$ a basis of L, over A, the relation KM, = V implies the existence of $x = a/s \in K^*$ (where a and s are non-zero elements of A) such that $xe_i \in M,$ for all $i$, whence $xM \subset xL_2 \subset M_1$ and a fortiori $aM \subset M,$; exchanging the roles of M and M, it can be similarly shown that there exists $b \neq 0$ in A such that $bM_1 \subset M$.