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

TL;DR

This paper introduces $ ho$-$w$-flat modules, generalizing existing flat modules, and explores their homological dimensions, establishing key equivalences for strongly $ ho$-rings related to $ ho$-von Neumann and PvMR properties.

Contribution

It defines $ ho$-$w$-flat modules and investigates their homological dimensions, linking these concepts to $ ho$-von Neumann and PvMR rings, extending the theory of flat modules.

Findings

$ ho$-$w$-w.gl.dim$(R)=0$ iff $w$-$dim(R)=0$ iff $R$ is a $ ho$-von Neumann ring.

$ ho$-$w$-w.gl.dim$(R) extless=1$ iff nonnil ideals are $ ho$-$w$-flat, equivalent to $R$ being a $ ho$-PvMR.

Established equivalences connect homological dimensions with ring properties.

Abstract

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