A note on ω-GD domains
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Let $S$ and $T$ be $w$-linked extension domains of a domain $R$ with $S\suse T$. In this paper, we define what satisfying the $w_R$-GD property for $S \subseteq T$ means and what being $w_R$- or $w$-GD domains for $T$ means. Then some sufficient conditions are given for the $w_R$-GD property and $w_R$-GD domains. For example, if $T$ is $w_R$-integral over $S$ and $S$ is integrally closed, then the $w_R$-GD property holds. It is also given that $S$ is a $w_R$-GD domain if and only if $S\suse T$ satisfies the $w_R$-GD property for each $w_R$-linked valuation overring $T$ of $S$, if and only if $S\suse (S[u])_{w}$ satisfies the $w_R$-GD property for each element $u$ in the quotient field of $S$, if and only if $S_{\mathfrak{m}}$ is a GD domain for each maximal $w_R$-ideal $\mathfrak{m}$ of $S$. Then we focus on discussing the relationship among GD domains, $w$-GD domains, $w_R$-GD domains, Pr\"ufer domains, P$v$MDs and P$w_R$MDs, and also provide some relevant counterexamples. As an application, we give a new characterization of P$w_R$MDs. We show that $S$ is a P$w_R$MD if and only if $S$ is a $w_R$-GD domain and every $w_R$-linked overring of $S$ that satisfies the $w_R$-GD property is $w_R$-flat over $S$. Furthermore, examples are provided to show these two conditions are necessary for P$w_R$MDs.
Abstract
Let $S$ and $T$ be $w$-linked extension domains of a domain $R$ with $S\suse T$. In this paper, we define what satisfying the $w_R$-GD property for $S \subseteq T$ means and what being $w_R$- or $w$-GD domains for $T$ means. Then some sufficient conditions are given for the $w_R$-GD property and $w_R$-GD domains. For example, if $T$ is $w_R$-integral over $S$ and $S$ is integrally closed, then the $w_R$-GD property holds. It is also given that $S$ is a $w_R$-GD domain if and only if $S\suse T$ satisfies the $w_R$-GD property for each $w_R$-linked valuation overring $T$ of $S$, if and only if $S\suse (S[u])_{w}$ satisfies the $w_R$-GD property for each element $u$ in the quotient field of $S$, if and only if $S_{\mathfrak{m}}$ is a GD domain for each maximal $w_R$-ideal $\mathfrak{m}$ of $S$. Then we focus on discussing the relationship among GD domains, $w$-GD domains, $w_R$-GD domains, Pr\"ufer domains, P$v$MDs and P$w_R$MDs, and also provide some relevant counterexamples. As an application, we give a new characterization of P$w_R$MDs. We show that $S$ is a P$w_R$MD if and only if $S$ is a $w_R$-GD domain and every $w_R$-linked overring of $S$ that satisfies the $w_R$-GD property is $w_R$-flat over $S$. Furthermore, examples are provided to show these two conditions are necessary for P$w_R$MDs.
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