Proposition 5

Let $V, W$ vector spaces of finite rank over $K$, $M$ a lattice of $V$ and $N$ a lattice of $W$. If $M$ is finitely generated, then (in the notation of Proposition 3):

(1)

in $\operatorname{Hom}_K(V, W)$.

Clearly the left hand side of (1) is contained in the right hand side. Conversely, let $f \in S^{-1}N : S^{-1}M$ and let $(x_i)_{1 \leq i \leq n}$ be a system of generators of $M$. There exists $s \in S$ such that $f(x_i) \in s^{-1}N$ for all $i$ and hence $sf \in N : M$, which proves the proposition.