Exercise 21, § 4

Let P be a projective module over a Dedekind domain A. Show that, if P is not finitely generated, it is free. (By virtue of Exercise 20, we may write P as an infinite direct sum $\bigoplus_{\lambda} (b, \bigoplus c_\lambda)$, where, for each A, b_\lambda and c_\lambda are ideals of A. Using Exercise 19, P = L $\oplus$ Q, where L is isomorphic to $A^{(I)}$ (I infinite) and $Q = \bigoplus_{a \in I} a$ where the a are ideals of A. Then apply Exercise 3 of Algebra, Chapter II, § 2.)