Bulletin of the
Korean Mathematical Society BKMS

In this paper, we introduce and study the $w$-weak global dimension $\wwd(R)$ of a commutative ring $R$. As an application, it is shown that an integral domain $R$ is a Pr\"{u}fer $v$-multiplication domain if and only if $\wwd(R)\leqslant 1$. We also show that there is a large class of domains in which Hilbert's syzygy Theorem for the $w$-weak global dimension does not hold. Namely, we prove that if $R$ is an integral domain (but not a field) for which the polynomial ring $R[x]$ is $w$-coherent, then $\wwd(R[x])=\wwd(R)$.

Keywords: GV-torsionfree module, $w$-module, $w$-flat module, $w$-flat dimension, $w$-weak global dimension

MSC numbers: Primary 13D05, 13A15; Secondary 13F05