Every Prüfer ring does not have small finitistic dimension at most one

Abstract

Let R be a commutative ring with identity. Denote by $\mathcal{F}\mathcal{P}\mathcal{R}(R)$ the set of all R-modules admitting a finite projective resolution consisting of finitely generated projective modules. Then the small finitistic dimension of R is defined as $\text{fPD}(R)=\sup \left\{{\text{pd}}_{R}M\right|M\in \mathcal{F}\mathcal{P}\mathcal{R}(R)\}.$ Cahen et al. posed an open problem as follows: Let R be a Prüfer ring. Is $\text{fPD}(R)⩽1$? In this paper, we show that the answer to this problem is negative. In the process of solving the problem, we need to give module-theoretic characterizations of the ring of finite fractions. Moreover, we introduce the concepts of FT-flat modules and the global FT-flat dimension of a ring to give a Prüfer-like characterization of the domains R with $\text{fPD}(R)⩽1.$

Keywords:

• Small finististic dimension
• DQ ring
• DW-ring
• FT-flat module
• global FT-flat dimension
• Prüfer ring
• ring of finite fractions

2000 Mathematics Subject Classification::

• 13D05
• 13F05

Additional information

Funding

This work was partially supported by NSFC (No. 11671283). This research was also supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A3B03033342).