Recently, Takahashi established a new approximation theory for finitely generated modules over commutative Noetherian rings, which unifies the spherical approximation theorem due to Auslander 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, w...

We construct four families of Artin-Schelter regular algebras of global dimension four. This is a complete list of Artin-Schelter regular algebras of global dimension four that are generated by two elements of degree 1 and whose Ext-algebra satisfies certain “generic” conditions. These algebras are also strongly noetherian, Auslander regular and Cohen-Macaulay. One of the main tools is Keller’s...

Let (R, m) be a Noetherian local ring of depth d and C a semidualizing R-complex. Let M be a finite R-module and t an integer between 0 and d. If GC -dimension of M/aM is finite for all ideals a generated by an R-regular sequence of length at most d − t then either GC -dimension of M is at most t or C is a dualizing complex. Analogous results for other homological dimensions are also given.

The algebra Sn of one-sided inverses of a polynomial algebra Pn in n variables is obtained from Pn by adding commuting, left (but not two-sided) inverses of the canonical generators of the algebra Pn. The algebra Sn is a noncommutative, non-Noetherian algebra of classical Krull dimension 2n and of global dimension n and is not a domain. If the ground field K has characteristic zero then the alg...

Abstract. We construct several families of Artin-Schelter regular algebras of global dimension four using double Ore extension and then prove that all these algebras are strongly noetherian, Auslander regular, Koszul and CohenMacaulay domains. Many regular algebras constructed in the paper are new and are not isomorphic to either a normal extension or an Ore extension of an Artin-Schelter regul...

Let R be a Noetherian ring . We prove that has global dimension at most two if, and only every prime ideal of is linear type. Similarly, we show three syzygetic. As consequence, derive characterization these rings using the André-Quillen homology.