Approximations and the Little Finitistic Dimension of Artinian Rings

Let R be a ring and let simp-R be a representative set of all simple (right R-) modules. Denote by < ω the class of all modules which are finitely generated and have finite projective dimension. The little finitistic dimension of R is defined by fdim(R) = sup{proj.dim(M) M < ω}. Let ( , ) be the complete cotorsion theory cogenerated by < ω. For each S simp-R, let fS: XS → S be a special -precover of S. We prove that fdim (R) = max{proj.dim(XS) S simp-R} provided that R is right artinian. As a corollary, we extend to right artinian rings the well-known Auslander–Reiten sufficient condition for finiteness of the little finitistic dimension.