A note on $\varphi$-Prüfer $v$-multiplication rings

Abstract

In this note, we show that a strongly $\varphi$-ring $R$ is a $\varphi$-PvMR if and only if any $\varphi$-torsion free $R$-module is $\varphi$-$w$-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.

arXiv admin note: text overlap with arXiv:2103.08278