The $w$-weak global dimension of commutative rings
Bull. Korean Math. Soc. 2015 Vol. 52, No. 4, 1327-1338
https://doi.org/10.4134/BKMS.2015.52.4.1327
Published online July 31, 2015
Fanggui Wang and Lei Qiao
Sichuan Normal University, Sichuan Normal University
Abstract : 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
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