Exercise 24, § 4

Let $A$ be an integrally closed Noetherian domain. For a lattice $M$ (with respect to $A$) to be reflexive, it is necessary and sufficient that it satisfy the following condition: for every ordered pair $(a, b)$ of elements of $A$ such that $a \neq 0$ and the homothety of ratio $b$ on $A/aA$ is injective, the homothety of ratio $b$ on $M/aM$ is then injective. (To see that the condition is necessary, consider two elements $x, y$ of $M$ such that $ax + by = 0$ and show that, for all $p \in P(A)$, $y \in aM_p$. To show that the condition is sufficient, use criterion (c) of Theorem 2 and observe (using Proposition 8 of § 1, no. 4) that the hypothesis may be written as

for all $a \neq 0$ in $A$ and that for every module $E$ such that $M \subset E \subset V$ there exists $a \neq 0$ in $A$ such that $aM \subset aE \subset M.$)