Locally GCD domains and the ring $D+XD_S[X]$

Document Type : Research Paper


1 Department of Mathematics Education‎, ‎Incheon National University‎, ‎Incheon 406-772‎, ‎Republic of Korea.

2 Facultatea de Matematica si Informatica‎, ‎University of‎ ‎Bucharest‎, ‎14 Academiei Str.‎, ‎Bucharest‎, ‎RO 010014‎, ‎Romania

3 Department of Mathematics‎, ‎Idaho State University‎, ‎Poca-tello‎, ‎ID 83209‎, ‎USA


An integral domain $D$ is called a emph{locally GCD domain} if $D_{M}$ 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 $aD\cap bD$ is locally principal for all $0\neq a,b\in D$, 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 $D+XD_{S}[X]$ construction to give some interesting examples of locally GCD domains that are not GCD domains.


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