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 Flat dimensions of injective modules over domains Bull. Korean Math. Soc.Published online December 4, 2019 Kui Hu, Jung Wook Lim, and De Chuan Zhou Southwest University of Science and Technology, Kyungpook National University Abstract : Let $R$ be a domain. It is proved that $R$ is coherent when $IFD(R)\leq 1$, and $R$ is Noetherian when $IPD(R)\leq 1$. Consequently, $R$ is a $G$-Pr$\rm\ddot{u}$fer domain if and only if $IFD(R)\leq 1$, if and only if ${\rm wG\mbox{-}gldim}(R)\leq 1$; and $R$ is a $G$-Dedekind domain if and only if $IPD(R)\leq 1$. Keywords : ${\rm wG\mbox{-}gldim}(R)$, $G$-Pr$\rm\ddot{u}$fer domain, $IFD(R)$, $IPD(R)$ MSC numbers : 13G05, 13D03 Full-Text :