THE SMALL FINITISTIC DIMENSIONS OF COMMUTATIVE RINGS

Abstract

Let R be a commutative ring with identity. The small finitistic dimension fPD(R) of R is defined to be the supremum of projective dimensions of R-modules with finite projective resolutions. In this paper, we characterize a ring R with fPD(R)≤n using finitely generated semiregular ideals, tilting modules, cotilting modules of cofinite type and vaguely associated prime ideals. As an application, we obtain that if R is a Noetherian ring, then fPD(R)=sup{grade(𝔪,R)|𝔪∈Max(R)} where grade(𝔪,R) is the grade of 𝔪 on R. We also show that a ring R satisfies fPD(R)≤1 if and only if R is a DW ring. As applications, we show that the small finitistic dimensions of strong Prüfer rings and LPVDs are at most one. Moreover, for any given n∈ℕ, we obtain a total ring of quotients R satisfying fPD(R)=n.