Locally GCD domains and the ring D+XDS[X]

An integral domain D is called a emph{locally GCD domain} if DM is a GCD domain for every maximal ideal M of D. We study some ring-theoretic properties of locally GCD domains. E.g., we show that D is a locally GCD domain if and only if aD∩bD is locally principal for all 0≠a,b∈D, and flat overrings of a locally GCD domain are locally GCD. We also show that the t-class group of a locally GCD domain is just its Picard group. We study when a locally GCD domain is Pr {u}fer or a generalized GCD domain. We also characterize locally factorial domains as domains D whose minimal prime ideals of a nonzero principal ideal are locally principal and discuss conditions that make them Krull domains. We use the D+XDS[X] construction to give some interesting examples of locally GCD domains that are not GCD domains.