On $\varphi$-$w$-Flat modules and Their Homological Dimensions

Abstract

In this paper, we introduce and study the class of $\varphi$-$w$-flat modules which are generalizations of both $\varphi$-flat modules and $w$-flat modules. The $\varphi$-$w$-weak global dimension $\varphi$-$w$-w.gl.dim$(R)$ of a strongly $\varphi$-ring $R$ is also introduced and studied. We show that, for a strongly $\varphi$-ring $R$, $\varphi$-$w$-w.gl.dim$(R)=0$ if and only if $w$-$dim(R)=0$ if and only if $R$ is a $\varphi$-von Neumann ring. It is also proved that, for a strongly $\varphi$-ring $R$, $\varphi$-$w$-w.gl.dim$(R)\le 1$ if and only if each nonnil ideal of $R$ is $\varphi$-$w$-flat, if and only if $R$ is a $\varphi$-PvMR, if and only if $R$ is a PvMR.

arXiv commentRevise the published version. Note in this version, when we study $\varphi$-$w$-weak global dimension, we always require the ring is a a strongly $\varphi$-ring. arXiv admin note: text overlap with arXiv:2107.11907