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arXiv · 2107.12643

On $ϕ$-$w$-Flat modules and Their Homological Dimensions

Xiaolei Zhang·Wei Zhao
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Abstract

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

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Xiaolei Zhang, Wei Zhao. 2023-02-22. On $ϕ$-$w$-Flat modules and Their Homological Dimensions. https://arxiv.org/abs/2107.12643

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