Exercise 17, § 4

Let A be a principal ideal domain, let a be a nonzero element of A , and let E be the module $A/A\alpha$. Then the product $E^n$ is isomorphic to $A^n/(\alpha A^n)$. a) Let u be a linear map from $E^n$ into $E^m$; show that u can be obtained from a linear map v from $A^n$ into $A^m$ on passing to quotients. b) Let $A\beta_i$ ($1 \leq i \leq p \leq \inf(m, n)$) be the invariant factors of $v(A^n)$ with respect to $A^m$, arranged in decreasing order ; let q $\leq p$ be the greatest of the indices i such that or does not divide $\beta_i$, and let $\delta_i$ be a gcd of a and $\beta_i$ for $1 \leq i \leq q$. Show that the kernel $u^{-1}(0)$ of u is the direct sum of $n-q$ modules isomorphic to E and q cyclic modules isomorphic to $A/A\delta_i$ respectively ($1 \leq i \leq q$). If $a = \gamma_i \delta_i$ ($1 \leq i \leq q$), show that the module $u(E^n)$ has invariant factors $A\gamma_i$.