Proposition 3

(i) *If $M_1$ and $M_2$ are lattices of V, so are $M_1 \cap M,$ and $M_1 + M_2$.

(ii) If W is a vector subspace of V and M is a lattice of V, $M \cap W$ is a lattice of W.

(iii) Let $V, V_1, \ldots, V_k$ be vector spaces of finite rank over $K$ and let

be a multilinear mapping whose image generates $V$. If $M_i$ is a lattice of $V_i$ for $1 \leq i \leq k$, the sub-A-module of $V$ generated by $f$ ($(M, x \ldots x M_k)$) is a lattice of $V$.

(iv) Let $V$ and $W$ be two vector spaces of finite rank over $K$, $M$ a lattice of $V$ and $N$ a lattice of $W$. The sub-A-module $N$: $M$ of $\mathrm{Hom}_K(V, W)$, consisting of the K-linear mappings $f$ such that $f(M) \subset N$, is a lattice of $\mathrm{Hom}_K(V, W)$.

(i) By virtue of Proposition 2, there exist non-zero $a$ and $b$ in $A$ such that $aM_1 \subset M_2 \subset b^{-1}M_1$; we conclude that $M_1 \cap M_2$ and $M_1 + M_2$ lie between $aM_1$ and $b^{-1}M_1$ and are therefore lattices by virtue of Proposition 2.

(ii) Let $S$ be a complement of $W$ in $V$, $L_w$ a free lattice of $W$ and $L_s$ a free lattice of $S$, so that $L = L_w \oplus L_s$ is a free lattice of $V$. Then there exist $x, y$ in $K^*$ such that $xL \subset M \subset yL$. We deduce that $xL_w \subset M \cap W \subset yL_w$, which shows that $M \cap W$ is a lattice of $W$ (Proposition 2).

(iii) As $KM_i = V$, clearly by linearity $f$ ($M_1 x \ldots x M_n$) generates the vector K-space $V$; on the other hand, for all $i$, there exists a finitely generated sub-A-module $N_i$ of $V$, such that $M_i \subset N_i$; the sub-A-module $N$ of $V$ generated by $f(N_1 x \ldots x N_n)$ is finitely generated and contains $M$ and hence $M$ is a lattice of $V$ (Proposition 1).

(iv) Let $P$ (resp. $Q$) be a free lattice of $V$ (resp. $W$) containing $M$ (resp. contained in $N$); obviously $N : M \supset Q : P$. Now it is immediate that $Q : P$ is isomorphic to $\mathrm{Hom}_A(P, Q)$, hence is a free A-module of rank $(\mathrm{rg}_A P)(\mathrm{rg}_A Q)$ (Algebra, Chapter II, § 1, no. 6, Corollary 1 to Proposition 6) and therefore a lattice of $\mathrm{Hom}_K(V, W)$. Similarly, if $P'$ (resp. $Q'$) is a free lattice of $V$ (resp. $W$) contained in $M$ (resp. containing $N$), then $Q' : P' \supset N : M$ and $Q' : P'$ is a lattice of $\mathrm{Hom}_K(V, W)$; whence the conclusion.

Remark (2) Proposition 3 (i) shows that the set $R(V)$ of lattices of $V$ is lattice-ordered with respect to inclusion; moreover, if $M$ is a fixed lattice of $V$, the $xM$, where $x$ runs through $K^*$, form a subset of $R(V)$ which is both coinitial and cofinal (Set Theory, Chapter III, § 1, no. 7).

(3) In the notation of Proposition 3 (iv), the canonical mapping

which maps every K-linear mapping $f \in N : M$ to the A-linear mapping from $M$ to $N$ which has the same graph as $f|_M$, is bijective, for every A-linear mapping $g : M \to N$ can be imbedded in a K-linear mapping

and it has been seen that $M \otimes_A K$ and $N \otimes_A K$ are respectively identified with $V$ and $W$.

In particular, if we take $W = K$, $N = A$, $\operatorname{Hom}_K(V, W)$ is just the dual vector K-space $V^*$ of $V$ and $A$: $M$ is identified with the dual A-module $M^*$ of $M$; we shall henceforth make this identification and we shall say that $M^*$ is the dual lattice of $M$: it is therefore the set of $x^* \in V^*$ such that $\langle x, x^* \rangle \in A$ for all $x \in M$.