Bulletin of the
Korean Mathematical Society BKMS

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).