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 $\text{fPD}(R)⩽1.$

令R为具有身份的交换环。表示为$\mathcal{F}\mathcal{P}\mathcal{[R}（\mathrm{[R}）$允许有限投影分辨率的所有R模块的集合，该分辨率由有限生成的投影模块组成。然后将R的较小的有限维定义为$\text{fPD}（\mathrm{[R}）=\mathrm{SUP}\left\{{\text{pd}}_{\mathrm{[R}}\mathrm{中号}\right|\mathrm{中号}\in \mathcal{F}\mathcal{P}\mathcal{[R}（\mathrm{[R}）\}。$Cahen等。提出了一个开放问题，如下所示：令R为Prüfer环。是$\text{fPD}（\mathrm{[R}）⩽\mathrm{1个}$？在本文中，我们表明该问题的答案是否定的。在解决问题的过程中，我们需要给出有限分数环的模块理论特征。此外，我们介绍的FT-平坦模块和一个环的全球FT-平面尺寸的概念，得到Prüfer样结构域的特征- [R与$\text{fPD}（\mathrm{[R}）⩽\mathrm{1个}。$