Gorenstein conditions over triangular matrix rings

Abstract

A ring is left Gorenstein regular if the classes of left modules with finite projective dimension and finite injective dimension coincide and the injective and projective finitistic left dimensions are finite. Let A and B be rings and U a (B,A)-bimodule such that UB has finite projective dimension and UA has finite flat dimension. In this paper we characterize when the ring T=(A0UB) is left Gorenstein regular and, over such rings, when a left T-module is Gorenstein projective or Gorenstein injective. As applications of these results, we characterize when T is left CM-free and give a necessary condition for existence of an infinite cardinal λ such that each Gorenstein projective module is a direct sum of λ<-generated modules.