Let S and T be w-linked extension domains of a domain R with S ? T.In this paper, we define what satisfying the wR-GD property for S ? T means and what being wR- or w-GD domains for T means.Then some sufficient conditions are given for the wR-GD property and wR-GD domains.For example, if T is wR-integral over S and S is integrally closed, then the wR-GD property holds.It is also given that S is a wR-GD domain if and only if S ? T satisfies the wR-GD property for each wR-linked valuation overring T of S, if and only if S ? (S[u])w satisfies the wR-GD property for each element u in the quotient field of S, if and only if Sm is a GD domain for each maximal wR-ideal m of S.Then we focus on discussing the relationship among GD domains, w-GD domains, wR-GD domains, Pr¨ufer domains, PvMDs and PwRMDs, and also provide some relevant counterexamples.As an application, we give a new characterization of PwRMDs.We show that S is a PwRMD if and only if S is a wR-GD domain and every wR-linked overring of S that satisfies the wR-GD property is wR-flat over S.Furthermore, examples are provided to show these two conditions are necessary for PwRMDs.
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