DUALIZING MODULES AND n-PERFECT RINGS

Abstract

In this article we extend the results about Gorenstein modules and Foxby duality to a noncommutative setting. This is done in § 3 of the paper, where we characterize the Auslander and Bass classes which arise whenever we have a dualizing module associated with a pair of rings. In this situation it is known that flat modules have finite projective dimension. Since this property of a ring is of interest in its own right, we devote § 2 of the paper to a consideration of such rings. Finally, in the paper's final section, we consider a natural generalization of the notions of Gorenstein modules which arises when we are in the situation of § 3, i.e. when we have a dualizing module.

A noncommutative generalization of Auslander′s last theorem

“…We now recall from [6] that a ring R is said to be left (right) n-perfect if every left (right) flat R-module has projective dimension less than or equal to n.…”

“…Also, if R is left n-perfect, then R[x], R [[x]], the crossed product R * ᐁ(g), and the Weyl algebra A k (R) are left k-perfect for some k (cf. [6] …”

“…Using Christensen [3], we here introduce the notion of a dualizing bimodule associated with a pair of Noetherian rings (but not necessarily commutative ones). In [6], it was shown that in this situation, every module in the Auslander class defined by the pair of rings admits a Gorenstein projective precover. Now we give examples where the dualizing bimodule has a double structure over the same noncommutative Noetherian ring and that in this case, if the ring also admits a Matlis dualizing module, (cf.…”

“…[5]). The preceding definition is given in [6] for a bimodule S V R , where S and R are left and right Noetherian rings, respectively, but through this paper, we will consider the case S = R.…”

Covers and preenvelopes by $V$-Gorenstein flat modules

“…Let R be a left and right Noetherian ring and let V be an (R, R)-bimodule such that End ( R V ) = R and End (V R ) = R . Then V is said to be a dualizing module [8] if it satisfies the following 3 conditions:…”