A note on $phi$-Prufer $v$-multiplication rings

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