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

Let R be a commutative ring with identity. Denote by F P 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 fPD (R) = sup { pd R M | M ∈ F P R (R) }. Cahen et al. posed an open problem as follows: Let R be a Prüfer ring. Is 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 fPD (R) ⩽ 1. [ABSTRACT FROM AUTHOR]