Show that if $R$ is a Dedekind domain, then every projective $R$-module (not necessarily finitely generated) is a direct sum of ideals of $R$.
I have spent a while on this problem and I wonder if it is true that every nonzero (integral) ideal of $R$ is projective as an $R$ module. If so, can we use this fact to prove the result above?
I figured it out: First, one can show that R is Dedekind domain if and only if every (integral) ideal is invertible if and only if every (integral) ideal is projective when seen as a $R$-module. Following the proof outlined in Hilton and Stammbach's A Course in Homological Algebra for the fact that every submodule of a free $R$-module $M$ where $R$ is a PID is free, one can show every submodule of a $R$-module $M$ for $R$ a domain in which every ideal is projective when seen as an $R$-module, is isomorphic to a direct sum of the ideals of $R$, which then implies the result about projective modules over Dedekind domains we hope to prove.