Recently, Takahashi established a new approximation theory for finitely generated modules over commutative Noetherian rings, which unifies the spherical approximation theorem due to Auslan- der and Bridger and the Cohen-Macaulay approximation theorem due to Auslander and Buchweitz. In this paper we generalize these results to much more general case over non-commutative rings. As an application, we establish a relation between the injective dimension of a generalized tilting module w and the finitistic dimension with respect to w.
Let H be a finite-dimensional Hopf algebra and assume that both H and H＊ are semisimple. The main result of this paper is to show that the representation dimension is an invariant under cleft extensions of H, that is, rep.dim（A） = rep.dim（A^CaH）. Some of the applications of this equality are also given.
SUN JuXiang & LIU GongXiang Department of Mathematics,Nanjing University,Nanjing 210093,China
