Exercise 11, § 4

Let A be a principal ideal domain, let H be a submodule of $E = A^{n}$, and let $H^{0}$ be the submodule of the dual $E^{*}$ of E orthogonal to H (Ex. 7). Let M be a submodule of rank p of E and let $Aa_{i}$ ($1 \leq i \leq p$) be the invariant factors of M with respect to E, arranged in decreasing order. Let $(\nu_{H})$ be the gcd of the ideals $f(M)$ of A as f runs through $H^{0}$; show that, for $1 \leq k \leq p$, the element $a_{k}$ is a gcd of the $\nu_{H}$ for all submodules H of E of rank $k - 1$ which admit a complement in E (first consider the case $k = 1$; apply the result to the submodule $(M + H)/H$ of $E/H$, and use Ex. 8, d), applied to the quotient module $E/(M + H)$ of $E/M$.