Corollary

Let U, V, W be three vector spaces of finite rank over K and f: U × V → W a left non-degenerate K-bilinear mapping (Algebra, Chapter IX, § 1, no. 1, Definition 3). If M is a lattice of V and N a lattice of W, the set N: ,M of x ∈ U such thatf (x, y) ∈ N for ally ∈ M is a lattice of U.

Let $s_f: U \to \operatorname{Hom}_K(V, W)$ be the K-linear mapping left associated with $f$ (Algebra, Chapter IX, loc. cit.) such that $s_f(x)$ is the linear mapping $y \mapsto f(x, y)$; recall that to say that $f$ is left non-degenerate means that $s_f$ is injective. By Proposition 3 (iv), $N: M$ is a lattice of $\operatorname{Hom}_K(V, W)$; as $N: fM = s_f^{-1}(N: M)$ and $s_f$ is injective, the corollary follows from Proposition 3 (ii).

Examples

(4) Let $S$ be a (not necessarily associative) K-algebra of finite rank with a unit element; then the bilinear mapping $(x, y) \mapsto xy$ of $S \times S$ to $S$ is (left and right) non-degenerate. If $M$ and $N$ are lattices of $S$ with respect to $A$, so are $M.N$ (Proposition 3 (iii)) and the set of $x \in S$ such that $xM \subset N$ (Corollary to Proposition 3). Note that there exists a sub-A-algebra of $S$ containing the unit element of $S$ which is a lattice of $S$; for consider a basis $(e_i)_{1 \leq i \leq n}$ of $S$ such that $e_1$ is the unit element of $S$ and let $e_i e_j = \sum_k c_{ijk} e_k$ be the multiplication table of $S$ ($1 \leq i \leq n, 1 \leq j \leq n$), so that $c_{1jk} = \delta_{jk}, c_{i1k} = \delta_{ik}$ (Kronecker symbols). Let $s \in A$ be a non-zero and such that $c'_{ijk} = s \cdot c_{ijk} \in A$ for all triplets of indices $(i, j, k)$; if we write $e'_i = s^{-1} e_i$ for $i \geq 2$, then

for $i \geq 2$ and $j \geq 2$; the lattice of $S$ with basis $e_1$ and the $e'_i$ ($2 \leq i \leq n$) is a sub-A-algebra of $S$ with unit element $e_1$.

(5) Let $V$ be a finite-dimensional vector space over $K$ and $f$ a non-degenerate bilinear form on $V$. If $M$ is a lattice of $V$, it follows from the Corollary to Proposition 3 that the set $M^*_f$ of $x \in V$ such that $f(x, y) \in A$ for ally $\in M$ is also a lattice of $V$; if $s_f: V \to V^*$ is the linear mapping left associated with $f$ (which is bijective), $s_f(M^*_f)$ is just the dual lattice $M^*$ of $M$.