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 Finitely generated $G$-projective modules over PVMDs Bull. Korean Math. Soc.Published online December 4, 2019 Kui Hu, Jung Wook Lim, and Shiqi Xing Southwest University of Science and Technology, Kyungpook National University, Chengdu University of Information Technology Abstract : Let $M$ be a finitely generated $G$-projective $R$-modules 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 $G{\mbox-}gldim(R) < \infty$ and $Ext_ R^i(M,M) = 0 (i \geq1)$, then $M$ is projective. Keywords : Gorenstein projective module, projective modules, PVMD MSC numbers : 13G05, 13D03 Full-Text :