The small finitistic dimensions of commutative rings, III

The small finitistic dimension fPD$(R)$ of a ring $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we show that a commutative ring $R$ has fPD$(R)\leq d$ if and only if for any finitely generated ideal $I$ of $R$, if $Ext_R^i(R/I,R)=0$ for each $i=0,\dots,d$, then $Ext_R^i(R/I,R)=0$ for all $i\geq 0.$ As applications, we obtain that, for any commutative ring $R$, fPD$(R)\leq \mbox{FP-}Id_RR$, the self-FP-injective dimension of $R$. We also give some applications of these results to (weak) $(n,d)$-rings, DW-rings and rings of Prufer type.