Approximations and the little finitistic dimension of artinian rings

[摘要] Let R be a ring and let simp-R be a representative set of all simple (right R-) modules. Denote by P-< omega 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 is an element of P-< omega}. Let (A, B) be the complete cotorsion theory cogenerated by P-< omega. For each S is an element of simp-R, let f(S): X-S --> S be a special A-precover of S. We prove that fdim(R) = max{proj.dim(X-S) / S is an element of simp-R} provided that R is right artinian. As a corollary, we extents to right artinian rings the well-known Auslander-Reiten sufficient condition for finiteness of the little finitistic dimension. (C) 2001 Elsevier Science.

[发布日期] 2001-12-01