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 Finitely generated $G$-projective modules over PVMDs Bull. Korean Math. Soc. 2020 Vol. 57, No. 3, 803-813 https://doi.org/10.4134/BKMS.b190531Published online December 4, 2019Printed May 31, 2020 Kui Hu, Jung Wook Lim, Shiqi Xing Kyungpook National University; Kyungpook National University; Chengdu University of Information Technology Abstract : Let $M$ be a finitely generated $G$-projective $R$-module over a PVMD $R$. We prove that $M$ is projective if and only if the canonical map $\theta: M\bigotimes_R M^* \rightarrow \Hom_R(\Hom_R(M,M),R)$ is a surjective homomorphism. Particularly, if $G{\mbox-}gldim(R) \lst \infty$ and $\Ext_ R^i(M,M) = 0$ $(i \gst 1)$, then $M$ is projective. Keywords : Gorenstein projective module, projective module, PVMD 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