On $\phi$-$w$-flat modules and their homological dimensions
Bull. Korean Math. Soc. 2021 Vol. 58, No. 4, 1039-1052
https://doi.org/10.4134/BKMS.b200816
Published online June 7, 2021
Printed July 31, 2021
Xiaolei Zhang, 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 $\phi$-$\rm PvMR$, if and only if $R$ is a $\rm PvMR$.
Keywords : $\phi$-$w$-flat module, $\phi$-$w$-weak global dimension, $\phi$-von Neumann ring, $\phi$-$\rm PvMR$
MSC numbers : Primary 13A15; Secondary 13F05
Supported by : The first author was supported by the Natural Science Foundation of Chengdu Aeronautic Polytechnic (No.~062026). The second author was supported by the National Natural Science Foundation of China (No.~12061001).
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