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

摘要

Let R be a commutative ring with identity. Denote by FPR(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 vertical bar M is an element of FPR(R)}. Cahen et al. posed an open problem as follows: Let R be a Prufer 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 Prufer-like characterization of the domains R with fPD(R) <= 1.