Let R be an integral domain.
R is a Prufer domain if and only if for every maximal ideal M of R the localization R_m is a valuation domain.
=> Let R be a Prufer domain. We know every ideal is invertible. R_m (R localized at a maximal ideal m) is a domain. How do I prove something is a VD? I know I need to show that out of every element and its inverse in the field of fractions of R_m, at least one is in R_m. But I don't see how to do that. Is there an equivalent definition of VD's that is not so unwieldy?
<= I have no idea how to start this one. I don't see how we can extract information from a ring by looking at a property that all of its maximal-ideal localizations have.