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 Flat dimensions of injective modules over domains Bull. Korean Math. Soc. 2020 Vol. 57, No. 4, 1075-1081 https://doi.org/10.4134/BKMS.b190757Published online December 4, 2019Printed July 31, 2020 Kui Hu, Jung Wook Lim, De Chuan Zhou Kyungpook National University; Kyungpook National University; Southwest University of Science and Technology Abstract : Let $R$ be a domain. It is proved that $R$ is coherent when $IFD(R)\lst1$, and $R$ is Noetherian when $IPD(R)\lst1$. Consequently, $R$ is a $G$-Pr$\rm\ddot{u}$fer domain if and only if $IFD(R)\lst1$, if and only if ${\rm wG\mbox{-}gldim}(R)\lst1$; and $R$ is a $G$-Dedekind domain if and only if $IPD(R)\lst1$. Keywords : ${\rm wG\mbox{-}gldim}(R)$, $G$-Pr$\rm\ddot{u}$fer domain, $IFD(R)$, $IPD(R)$ MSC numbers : 13G05, 13D03 Supported by : This work was partially supported by the Department of Mathematics of Kyungpook National University and National Natural Science Foundation of China(Grant No. 11671283). The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2017R1C1B1008085). Downloads: Full-text PDF   Full-text HTML