Example of a fractional ideal which is locally invertible everywhere

Question

There is a theorem (e.g. proposition 9.6 in Atiyah-MacDonald) that states that a fractional ideal $I$ of a domain $R$ is invertible if and only if it is finitely generated and it is locally invertible everywhere, that is $I_\mathfrak{p}$ is invertible for every prime $\mathfrak{p} \triangleleft R$.

What would be an example of an ideal which is locally invertible at every prime ideal but that is nonetheless not invertible? This ideal would necessarily have to be non-finitely generated.

I've only been able to find examples of non-invertible ideals that are non-invertible because they're non-invertible at some prime ideal.

Answer 1

An almost Dedekind domain is an integral domain $R$ with the property that all localizations $R_P$ at maximal ideals $P$ are discrete valuation rings. If such a ring is non-noetherian, then every non finitely generated ideal gives an example of the form you are searching for. Under the link

https://www.semanticscholar.org/paper/Idealtheorie-in-einem-speziellen-unendlichen-Nakano/fb07434a9a33f1fa071baa44da9993b321d44f60

you can find an article, in which such a domain is constructed. Essentially it is the integral closure of $\mathbb{Z}$ in a well-chosen infinite algebraic extension of $\mathbb{Q}$.