On $\phi$-$w$-Flat modules and Their Homological Dimensions
Bull. Korean Math. Soc.
Published online June 7, 2021
Xiaolei Zhang and Wei Zhao
Chengdu Aeronautic Polytechnic, ABa Teachers University
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 commutative ring $R$ is also introduced and studied. We show that,
for a $\phi$-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.
Keywords : $\phi$-$w$-flat module; $\phi$-$w$-weak global dimension; $\phi$-von Neumann ring; $\phi$-$PvMR$.
MSC numbers : 13A15; 13F05.
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