LOCALLY GCD DOMAINS AND THE RING D + XDS[X].

An integral domain D is called a locally GCD domain if DM is a GCD domain for every maximal ideal M of D. We study some ringtheoretic properties of locally GCD domains. For example, 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 at 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 Prüfer 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 ex- amples of locally GCD domains that are not GCD domains. [ABSTRACT FROM AUTHOR]