Bulletin of the
Korean Mathematical Society BKMS

In this note, we show that a strongly $\phi$-ring $R$ is a $\phi$-$\rm PvMR$ if and only if any $\phi$-torsion-free $R$-module is $\phi$-$w$-flat, if and only if any $\rm GV$-torsion-free divisible $R$-module is nonnil-absolutely $w$-pure, if and only if any $\rm GV$-torsion-free $h$-divisible $R$-module is nonnil-absolutely $w$-pure, if and only if any finitely generated nonnil ideal of $R$ is $w$-projective.

Keywords: $\phi$-$\rm PvMR$, $\phi$-$w$-flat module, nonnil-absolutely $w$-pure module, $w$-projective module

MSC numbers: 13A15, 13F05

Supported by: The author is very grateful to the reviewer for reading carefully the paper and pointing out errors in the proofs of Theorem 2.13 and Theorem 3.13 in the original version. The author was supported by the National Natural Science Foundation of China (No. 12061001).