Exercise 3, § 4

Let A be a principal ideal domain with field of fractions K, let E be a vector space over K, let M be a finitely generated A-module of rank n contained in E, and let N be a finitely generated A-module of rank $p \leq n$ contained in KM. Show that there exists a basis $(e_i), \ldots,$ of M and a basis of N consisting of vectors $\alpha_i e_i$ ($1 \leq i \leq p$), where $a_i \in K$ and $a_i$ divides $a_{i-1}$ (with respect to the ring A) for $1 \leq i \leq p - 1$; show also that the fractional ideals $\Lambda \alpha_i$ are uniquely determined (invariant factors of N with respect to M).