Bulletin of the
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Bull. Korean Math. Soc. 2020; 57(4): 1075-1081

Online first article December 4, 2019      Printed July 31, 2020

https://doi.org/10.4134/BKMS.b190757

Copyright © The Korean Mathematical Society.

Flat dimensions of injective modules over domains

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).

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