Exercise 7, § 4

Let A be a principal ideal domain with field of fractions K, and let E be a finitely generated, torsion-free A-module of rank $n$. Let $E^*$ be the dual of E; then E and $E^*$ are each isomorphic to $A^n$.

a) Let M be a submodule of E; show that, if $M^0$ is the submodule of $E^*$ orthogonal to M, then the module $E^*/M^0$ is torsion-free, and consequently $M^0$ is a direct factor in $E^*$; the submodule $M^{00}$ of E orthogonal to $M^0$ is equal to $E \cap KM$ (considering E as being canonically embedded in an n-dimensional vector space over K). We can consider $E^*/M^0$ as being canonically embedded in the dual $M^*$ of M; show that the invariant factors of $E^*/M^0$ with respect to $M^*$ are equal to the invariant factors of M with respect to E (use Th. 1 of VII, p. 18).

b) Let F be a second finitely generated, torsion-free A-module, and let $u$ be a linear transformation from E into F. Show that the invariant factors of $'u(F^*)$ with respect to $E^*$ are equal to those of $u(E)$ with respect to F (proceed as in Prop. 5 of VII, p. 21).