Exercise 14, § 4

Let M be a module of finite length over a principal ideal domain A with field of fractions K. a) Show that if N is a submodule of M then the rank of M is at most equal to the sum of the ranks of N and M/N (let M = E/H, where E is a free module of rank n and H is a submodule of rank n of E; if N = L/H, where H ⊂ L ⊂ E, notice that, if the rank of N is p, then there exists a submodule R of H, of rank n − p, admitting a complement S in L such that S ∩ H is a complement of R in H; moreover, (E ∩ KR)/R has rank ≥ n − p and is isomorphic to a quotient module of E/L (Ex. 10, a)); complete the proof using Ex. 8, b)). b) Let N be a submodule of M of rank p, and let q be the rank of M/N. Let Aa, (1 ≤ i ≤ n) be the invariant factors of M and Aβ_k (1 ≤ k ≤ p) those of N, arranged in decreasing order. Show that β_k is a multiple of a, (p + q − n) (use a) and Ex. 8, c), noticing that (λM)/(λN) is isomorphic to a quotient module of M/N).