Exercise 8, § 4

Let G be a module of finite length over a principal ideal domain A. For any pair of free modules M and N such that $N \subset M$ and G is isomorphic to $M/N$, the invariant factors of N with respect to M which are distinct from A are identical to the invariant factors of G (VII, p. 20, Def. 2); their number is called the rank of G.

a) Show that the rank $r$ of G is the least number of cyclic submodules of G of which G is the direct sum, and is equal to the greatest of the ranks of the $\pi$-primary components of G.

b) Show that the rank of a submodule or a quotient module of G is at most equal to the rank of G (consider G as a quotient of two free modules of rank r).

c) For each $A \in A$, show that the rank of the submodule AG is equal to the number of invariant factors of G which do not divide A (notice that, if $E = A/(A\alpha)$, then the module $\lambda E$ is isomorphic to $(A\lambda)/((A\alpha) \cap (A\lambda))$). Deduce that the k-th invariant factor of G (in decreasing order) is a gcd of those $A \in A$ such that AG has rank $\leq r - k$.

d) Let $A\alpha_k$ ($1 \leq k \leq r$) be the invariant factors of G arranged in decreasing order. Let H be a submodule of G of rank $r - q$, and let $A\beta_k$ ($1 \leq k \leq r - q$) be its invariant factors, arranged in decreasing order. Deduce from b) and c) that $\beta_k$ divides $\alpha_{k-q}$ for $1 \leq k \leq r - q$. Similarly show that, if $G/H$ has rank $r - p$ and invariant factors $Ay, (1 \leq k \leq r - p)$ in decreasing order, then $\gamma_k$ divides $a_{r-p}$ for $1 \leq k \leq r - p$.

e) Conversely, let L be a torsion module of rank $r - q$, whose invariant factors $A\lambda_k$ ($1 \leq k \leq r - q$) are such that $\lambda_k$ divides $\alpha_{k+q}$ for $1 \leq k \leq r - q$. Show that there exist submodules M and N of G such that L is isomorphic to M and to G/N (decompose G into a direct sum of submodules isomorphic to $A/A\alpha_k$).