Free groups of ideals

We study the freeness of the group $\mathrm{I}\mathrm{n}\mathrm{v}(D)$ of invertible ideals of an integral domain $D$, and the freeness of some related groups of (fractional) ideals. We study the relation between $\mathrm{I}\mathrm{n}\mathrm{v}(D)$ and $\mathrm{I}\mathrm{n}\mathrm{v}(D_P)$, in particular in the locally finite case, and we analyze in more detail the case where $D$ is Noetherian (obtaining a characterization of when $\mathrm{I}\mathrm{n}\mathrm{v}(D)$ is free for one-dimensional analytically unramified Noetherian domains) and where $D$ is Prüfer.