Exercise 15, § 5

Let M be a torsion module over a principal ideal domain. Prove that the ring of homotheties of M is dense in its bicommutant.

(Reduce to the case when M is $\pi$-primary for an irreducible element $\pi$ of A. Then, prove that every finite subset S of M is contained in a direct factor submodule that is the direct sum of a finitely generated module and a finite number of indecomposable divisible modules. Use induction on the cardinal of S with the help of Exercises 3 of VII, §2, p. 54 and 8 of VII, §2, p. 56.)