Exercise 3, § 2

Let A be a principal ideal domain. a) An A-module M is injective (II, p. 389, Ex. 11) if and only if, for all x ∈ M and all a ≠ 0 in A, there exists y ∈ M such that x = ay; the A-module M is then said to be divisible. If E is an arbitrary A-module, then the sum of all the divisible submodules of E is the greatest divisible submodule Δ(E) of E. We have Δ(F) ⊂ A(E) for every submodule F of E. b) Let M be a divisible A-module, let π be an irreducible element of A and let $x_0$ be an element of M whose annihilator has the form Aπ^n (n > 0 an integer). Define a sequence (x_s) of elements of M recursively by the conditions $x_s = \pi x_{s-1}$ for each integer s ≥ 0. Show that the submodule of M generated by the sequence (x_s) is isomorphic to the π-primary component U_π of the module U = K/A, where K is the field of fractions of A. c) Show that U_π is a divisible module, and that every submodule of U_π other than U_π and {0} is cyclic and generated by the class mod A of some power of π; deduce that U_π is indecomposable (II, p. 393, Ex. 21). d) Show that every indecomposable divisible A-module is isomorphic either to K or to one of the modules U_π (use a) and b)). e) Show that every divisible A-module is isomorphic to the direct sum of a family of indecomposable divisible submodules (apply Zorn's Lemma to families of indecomposable divisible submodules whose sums are direct).