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$(R)\le d$ if and only if for any finitely generated ideal $I$ of $R$, if $Ex{t}_R^i(R\mathrm{/}I,R)=0$ for each $i=0,\dots ,d$, then $Ex{t}_R^i(R\mathrm{/}I,R)=0$ for all $i\ge 0.$ As applications, we obtain that, for any commutative ring $R$, fPD$(R)\le FP-I{d}_{R}R$, 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.