Bulletin of the
Korean Mathematical Society BKMS

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).