Let(X, Y) be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to(X, Y), and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the X-resolution dimension of Y(resp. Y-coresolution dimension of X)is finite, then the bounded homotopy category of Y(resp. X) is contained in that of X(resp. Y). As a consequence, we get that the right X-singularity category coincides with the left Y-singularity category if the X-resolution dimension of Y and the Y-coresolution dimension of X are finite.
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.
In this paper,for an arbitrary prime p,the square mapping graph of M2（Zp;s） is investigated and the formula of the decomposition of the square mapping graph of M2（Zp;s） is established.
