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

‎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. ‎For example‎, ‎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 \small{$D+XD_{S}[X]$} construction to give some‎ ‎interesting examples of locally GCD domains that are not GCD domains‎.