Exercise 20, § 4

Let A be a Dedekind domain, P a projective A-module and N a submodule of P. Show that N is projective and a direct sum of modules isomorphic to ideals of A. (Attention may be confined to the case where P is free. Then proceed as in Exercise 16, using a transfinite induction argument based on that of Algebra, Chapter VII, § 3, Theorem 1.)