Exercise 10, § 4

Let A be a principal ideal domain and let M be a submodule of rank $n$ in $E = A^{n}$; let $A\alpha_{i}$ ($1 \leq i \leq n$) be the invariant factors of M with respect to E, arranged in decreasing order. Let N be a submodule of M admitting a complement in M (that is, such that $N = M \cap KN$, where K is the field of fractions of A). a) Let P be a complement of N in M. Show that the submodule $(P + (E \cap KN))/M$ of $E/M$ is isomorphic to $(E \cap KN)/N$. Deduce that, if N has rank p and $A\beta_{k}$ ($1 \leq k \leq p$) are the invariant factors of N with respect to E, then $\beta_{k}$ divides $\alpha_{k+n-p}$ for $1 \leq k \leq p$ (use Ex. 8, d)). b) Give an example to show that the module $E/M$ is not necessarily isomorphic to the product of the modules $(E \cap KN)/N$ and $E/(P + (E \cap KN))$ (take $E/M$ to be indecomposable).